Content-Type: multipart/mixed; boundary="-------------0303200902948" This is a multi-part message in MIME format. ---------------0303200902948 Content-Type: text/plain; name="03-125.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-125.keywords" Hamilton-Jacobi equation, Lax-Oleinik semigroup, periodic solutions. ---------------0303200902948 Content-Type: application/postscript; name="HJ.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="HJ.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.85 Copyright 1999 Radical Eye Software %%Title: v4.dvi %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR10 CMCSC10 CMR12 CMMI12 CMMI8 CMSY10 CMR8 %%+ CMTI12 CMSY8 CMEX10 CMSY6 CMMI6 CMR6 CMBX8 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o HJ.ps v4 %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 2003.03.20:1109 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All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e28519671e8def84a1526a8b89450ef2bd624857da91e67a35269b877f161 ead3b9314f0ebb23d1a803608fb8bed47522b2a7814eba409f33973ca88ebb86 148d3d0d91d8056e573e23f17b04c8dfeb3d040cab25f1da3caad77b904b64f1 98dc18b1abbbeb700e3bcad39189c15c31bdc0009f94094d223f303408ab2800 533c624b77a8b576fcb13a492a5aac1035d7766bf5e2ba738c1de82bcb395cb4 344d6f452463521d7a6d8d979625e6900ac2b993de3d6a15cf4473176be41862 e61b3f9e109d500405e6f01936bd4cd77ca687b490df2a2e501e141fbecc7082 86f85fa428a8121af362ef3bee57af 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 90 /integraldisplay put dup 92 /intersectiondisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83fbb931da9ee6ade43de532d888f4d9aeca091a3d519db29d75b39638ced3c3 dd54562084b7d0594bcb71c2273878e6429a00e78e3df927c5e276799a142f93 dd69b7de61df8c84236ec2702253c2b6cdcdcd96bfb0db41a8b6adad587203d2 af7ecca414c8f1c7c3559638ddc55411a127f81eea0ac4674fde89ea374b709f 52db20250b72537cbb9bb806f87a117a4aa71c152e22e4f84c7726b6284460db 36a2fc20b815e41a2cc8bb6cb5b7c3f5610b213d57ba6b542d7c6b4d06beda73 3629ac2990ecaa063a88904798aa389e949963678ecf4972316462c6cd9ab4cc ce8375362d2d7f2988c1325b077b297ccdf5d4b53247232fe865de139c64c8e9 572f587f1c86cb9684a3107c8dacb0e664c5125047433dda9778f70b32a1f0af 2605c67356807c46e1e2474a0873e273175722c0c8cddabee7f1739868714199 92986e683562f67567cabff69fe7afc26242d8c4f2be76058343336d261ad97e 0169614c45ad56f5fa102f589038cb392d2cd87926cc470db946d6d706ab8041 2505b69d027dfa9a2161481f03eaf966f29f1f08988f0f123398165d36635721 6532353c955724e3497a02dcc29a1f301d3cf5d1c8c6a169fd4f0637140c1e5a 0917cf65e769db4ebaff4aeed1aed018e9e6ed3aca315904021c4b71a9517b0a 631b52fe452d1f5eef0dbfcf1a99fdc2dd2c5d2e4c3cd8b596bd0628f082cc63 ab8b9f41e44bcd9ee3b5348db3435d28805d9566a600f53ffa48fcf97140c66b 384742bc69793f978234043a1b81d90b809154e737bfb4d2d968a9460f67ac1f 96e6710178e18ff7a412b5e7b42c5f247709e120c86e3d0e0e24ec7caae00e9a fa9825acc3a6883862f939d8095531940e4be04c39135f6d34d0c9f907174927 1b74e12e5844f00c2f7b03c46fad5297ae3ce824f6f508ac0e0727b062c30c3a 0f1c3d7a83795bfaa0745d8bbb44945b07e77eae909acaaeafcc021a79e3b978 11376a14cdb532647345c2c9d0c0f311e34c061825a95e97f0b0f3021cd5146d dafbb85f95d73cdde2516de961a63dd4a95283bba9a03ca4520e486323e368c9 65dc6c04386919f7943038683cd03ec3d27346398922f081a8a7f8513f631d00 814f64877334644edf5e3a145f0bd4efc58e9eab543445bbffc8cfc506a083bc 8cf3fbd90ddeabda46b422dda7e1fc4f917ff9142ddc1e84283b03bd09f3c858 80ca84a013e7f2587382665a1b4e2e68d1449964efb83bf5ee23fb4388617b44 3ac71b07fe9a3b3ac5a6450fc3732bb8c3c8e33fd6d3845b961145aafd445dd9 4333813c3e1e3d80cfe35f4228d7402be78a5d87348acde1106017075f41854a e6cf534b324acafb68aaf02283ef0774a9dc15ceb6e9e800c85d352f2d1ce2e1 ba82613a386355688bf763a13d6456de3d1471fa0404ad67d34c701912393ec2 ac2ee4fdcaa499137c8fd73e15f34f0bb8626a25bad178cd68d10f9433317099 1cdaa75d62a8f70ef1d5f6119b717a27149529b8595550386c653c50113be7c5 5d77206bb51d5e23c1571879ae18cad0bdbbd2443b0050f24cd7d8d895e712d2 954ddd8558c10c32f43613e1c487f59ea0ad6482bec4221e8b37a7406f4658c4 a2c431157c2b1e319f1e5f17e7ed846781914cfbfe9428b29d5065ce42fff8 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 65 /A put dup 77 /M put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03acaa5e71066673665ebc99c1671730a922347e0df8497c3ca74d55d3 32625a2d47d41d7ab8ee776eb4fab79116f2e8018fb519002418aa46a669068f 27c27cae5f5aa453d8de98319e56ce64386181c9c8069aa47ff27668714eba0d d39d8c8ffa6df2a93689f079a64e39c0046fd520e1c7cf4eb748308325f4ba6d 2913e24747ee098ef833ad776c54d723ea367499f0d8d79961d81d9569b0bd25 09f28cc67d48938ea8fd075274d458fcab2f0a5a60eb313742ece2f5143c24bd 08cd1618d9ff74bbfc87b2f6408b1a3f944788e02e1f869b84798a9b47cdecf3 6346b6aefe355d01ab7547a8910c02e6668f7d1346bc5378f463473278972116 db1fe8ea7ec54651dcd42848091be5e6acdc9386d12f4aa3d65b68e24677490b 66be55778e55bfc9cdd58ad41d0cf85ade353cd30db1744d032051b40828b741 1047e7f9dd1a1c4d69967e9988983adc4614d87d3e0983327a2fff6f8bf0eb60 56a25d1a0e8fba691f32ce43b1948bbb80a717e0176d8c1a91b19dcb0eb399fa 628ee3366aaaf0047164799bc19a85727c4b59f7e1ef7743aa6080a613c121e6 81277146a3216383c681bbddd80bea53d37379eb5cb2712bd06f49f4bece717c aec669e434901822eeee8d6becbf4b06be53fe46941d0f783eaccf729d609267 3f74fa2fc14f176f3e0adb534c2134293e7371c2b4511e61183d7866d0879ca7 6c2757d71460e0bdd75f727a9d017d50063ee6fc782c8e6e47a4205b8be24161 c8b21c563458186ace01157e890e62a08636f5befb20100b6336b167e5077779 aae41db1415cb9767d60e0af91a28077ff5bb8928787947216f6b3a08a258180 01e248518d3ecdcd8dd0b9d208d0f59ff996f3f4254f14ae42380d2fe7296634 a89478a7ffb267757cb42db6ad4baf46976d775f877f2ccd2cba0f8eef6eca66 db072bd533acaa37a0b291e43db775931547e1aff9d5322bc3c4b96253441809 5ec34827773ddd46bb49463f7c7af7d519fb34966d2d4c00fba59ef463d564b8 ab5d405bd14fcfe4b79651fcdff7d56739b92fb45bc7f4ef9154112a36e98e9f e3bebbb86f8808474b1655eeb073f39f4c40e17689938eb16c7e6f04c69ba399 1525e0734b5f8721ddc8de10cb3017c72a268275b28b99606e6484e112dff55a 95c1fe2fc0c78320447da6d589af070f82426190ad038a4b2b46e81ebfe31238 557227eae45863d812b03c259706d002cabeb0ae64d49c8a76f4b9e4bf10e25c 11226920af5c1420abd646f2f3bd9138bf305a29a95e29b3b8234d213bf67954 580a70a34f8305bab6ff8ddadb5a720bdafebcea582458885c03e77a5963d1b5 5995ead368048e3eaab8285f2e9f81194b40ac2db7321822af5c41955c65a91f 6cd8268e54087be3e8d44b2c97af2df7ffd1578464887367546ec5e54fff716c b0766098430f8579e5fb9d5c71511562026fb469fe36d75b71c9626ab1a78e97 44961f55fa49ed6fb55ba76d845b51cb76470d30ad7d1ab67b292670580e9cb5 488b9f8eb79dcc4fa0d18abb7ac4ba8274dd08e894f8c70ff185f95be9b5da6c 6c35fd0f5d52c78f117e74b3ce2f9b321ecb73ee7f5737318996f66244ec73db d9776a215f45b3aeaf26a7ef4fe2afdad204072d60398a59a8cefda213b6dd81 958bd171717a066cd048483c06158abd5159805edaa98fab211f2b82023f415e 4b5b288d92f4bbd7f4475dbbbe6a7b8c80792485e1b506db42f61a53070581c6 c2a2166e0c6384fb1b722d62c8fac389f70949029a66e7bc7984d507f625b7b4 dbfcbe077692c614aac7667e07b52633f65d7b414205d22b2a0c2b37161ffff4 fecabddade35a51906abb7961e788c2a5084bd6c63aa8af2754269bde9a4edda 4579ade158dc4f07a88d 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 59 /semicolon put dup 69 /E put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueXX 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36e555668d9578b338402f8c259f6813b0b71e6cbd95e19c20ced8495 2559e47577e612a4d94de3c1b4c6c41a6a3fa9e2313457242de74c7e4c0d722b 3e224d18baa726369542682e9459b951b2ac638494880c42be4e2175a3c067e5 945bfc1a05d3de8c3e8ab97f80ab6000e1f78fc258b85d3c84b038d7d0f5be85 0d43fe03e26cdfbb41c36d14cb552e8447ca678efea1d275e85503123a2263b1 4077587683de69a8f5a88d666172668411d0d0fed3e037ee468627207505dbb3 59cfdc5d7f526fca5416d741073284f5dd2471e09f5916f4bff1a90eea4d1283 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8090228e5ee5fc39e7f395114c854d8f01f7d195d5a6b4855a963891ce878903 019586d6b438b484c21c04c4ee4378064e0d7de964ccc178b66a731d051bb1ba c6603309a14d8d75ef350d1a5b674f52f56e728ee23629dbaf56dfde842a4709 b6103bc197f7ab37ad85cbfb42bca6f4a23f98f420392695a2d517660d553541 68951b24ea5498dd21c3b557e72766a30c7fe9a6e98d9d743eb1460f77d85172 248e993019c8775b723b17fc66b11bedf15271199e6c972c3f4420ed8d3152e0 7977633548bb91443157528807001d5ebbce69d80ce5ee472ab09a505c6422ec ea3005dc3d03189783f5f93054d2de93184946adbdbfd6050461ff3cf55897a8 dd1f046bb6ee78a05061118b688183cac8b307eb65dd6f61f038d0d2a2d4d738 c2361afa58f24795c4df2268e8c7edfeaf1ed1167fc50fc9a290a48d97f87b9a 3f3d18e2cb37fbf984734e722d7867af3ac2a5af5e40614d44ee2a4cee2d4b53 6592007791045a89f373243b5a4628c37e1f39798082c73986ae81e428f07d64 f99c26cc483e2573eeb60f483d29091dbe02a4ff010c2f287c5840c2be06f6f0 49ced810f6950739265992f525f5fdf7370d6a3b790602d99a55d6470cc9c864 50ac2c33d66bbf02307cec6d9d6a09eec21d8a39919d5e5e88af3ac1696463ec dbc511b8037c4c5aee83903dd993a21ea74a4230f9214cc3ad73339cf5b717ff 71b4dbc34b0d4fbfc4cb721255eadf42abddea89bbe0b4256ba4a093b540cdef bc992f07cbe547b67a264ad243b1c7fda0623cb6bf1ab5a220c850b7dfba9f1c c9e88468a6995cd2a77a6aa28349bd6878f341d74fa99ea85036b6053ace1532 a914b0811d1d12b4c30b7f635c8e47013fdd230d290c4a1b8a11cf68b7bb0800 fa33379c4810293856b7af016bef1b6ca50f61df0d6f0816981fd49d460102eb dee8e2f01451be8e0346b78cf488c39edaf1d0125264ef28039a517dbe30426a e6ec05cd4b0c16da892e5d75c259869c456b8cf4e3f371e6ee7295264d86cc93 524e3bb1b4d86fed6356281f1aa95131bbb724d42b26c4562582b9ae6073affb 8e31eba45198a75f967dd5c0c359f85034acb3233fdce1600cd92954c3957a57 837cf2460c9729c6052d8e417642843a862a1d27cc42bcc9ddb829f295afab6f 3699e752b1df86d11dd4f1ec5a345cd03652ef0d6b7f1ca31b2ad468e7d4057c 5850d60ae582872c9af9b2e499433fbd75773931ea58057c2d55fcb0438a79ee c2b0a46456aa75a8ad00b4587128300c8e5954b330e1938513f2cacecf132847 d10c3db543c4a8aaf94a1846f9957aa49d4da29dfba07b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 f55dce17acd996c1f5e023c9e5a18cbeecc3097f23763acb86cdd7cd13381ae7 4e48495ec7fa520539d87f8a8dcb3c826275469b6800876a457e7d1e5be867c7 b1ccad69742a8c9b0ad943482bf2a4ad0aed40baeb69a0233bad36b4ca2d2da7 322956c70375d152653500b2f22d2ab6990cadde2da14b4917f7515e64bc3d96 bf775258fc7dae4e42a4c9b6da8eddec4a800c8aadc8d75e48cae52137e05c03 677f5d6a82fa46d9f2fc7f56d62e5c605a1b7898b8d1401c2cac1a0122a2c8a7 aae09607f2c5f29293a09b9959399283be89051452898238b777db9830ff4318 a298b221c4a820c700ca964fc99e6b1d9eb0bafc39be9aa9dffa2fa326b2a466 1e67261bb3dbc85844e76f9313ab533ee16bcb478166de5502b68e6ec1ea6110 f53206cf1cfea419571b0243244b32ea38b828a01e62b03bcf476a32c2341cec b8a0fa2a3f3f339315d5f3aa7704408f6617d026ffee4925aa48537a4b2e556d 225be09392546ad561cd46cf766f0ba14c6854c5af1a528c3842b0c5aeefefcf 19f19cbce5d967153c4f88e51c776cfbe10d39328466f7a3b83d85601ef4d1ce 0cc136610fa6a74d9676e4e99291d565cad27b9eb2337b0eba4148efd0eb2866 ebf0a4209ba135ad1376cf9cc2d4110f36121055caa3f2e125a42a237f8aa6c1 6fdcb0b5c8da4238991cd0193e4dd3aea198abf32a449ec1085b906fb406de98 b8d181ee02a346773e3b491158b881f3e94fa5110b2e8123d3e726c1f9ff14b0 85bc8aec42dc9be91d0627aa2df2b373dda3f1171f528406937aabbeb5d1357b d18946fa955dc011a572c7fecde23921a68607d008fca3677208775d28107cb6 c468c3333e9b3733b67170deba1fe165dfe61106ba1b8df26183b3bb2d877276 86b737900072dd58a2e3f27a7e8ba39e055ddd2639d5296f0c32b28b71fc76cd 647b7a6068c07e70f1a772c72320b2b1cc1e51b0f82b9e8881d069eefeb7b499 c97c8e896f55e1d952e10714d29fe2c4b43c6f449a1e8d608573f8147c716069 63c34e6a68344811394fa0a8c23444ff5833d98a9bbf25d1a1ebaccf5e84bca1 6ea2f0ab94cf3abdefe9fad8769845aa9c2ef6409d5ef54fafdd027b3ebd918d 829a6cf804ae96411433a188f74822eaae2c81dc9d1a2edb08e7a41c074dcc85 cded5b11144609c40bdeee9f53dde09170241e5886c4961ba7514e941b167e6d 48a2a96bcc2c098f027db43808de8871e416075cf49053a3adf857c2f5392188 ba55f13d93f135bd62a956e11c50760626cdc93427eae8c44b9610b910ffbdf9 e79429a07671b937bdfe7bd47288e1e2c2944d119a3c0df2c135ca3ec606dba7 96712c170dc693f7375c79a0fde295c9f86adeeb66f22814f136fc1b80ed7c4f 35ac3416c2ed22adff25e1d80d31b03ed16bdaf71d01e3bbfcec0f970d044952 d905bd177a8cc79f3cc814c1743dc0133268b43656fd6db1682c03314b3b2011 79ba4523bb2d22a90edca23a7d8267a7c2d3feac3de18fe74f5c9a74baf5d60a 4e2989d8184582589a158a13f25b745c41517ed201175ce78467a9582786abd7 2d8bf695d6d1a8ee6ee40a2ff958d17d2993ab102ce94a3e0e27fb5f7fb110ee 8375f0cf0634257f3dfc9542f06f9edbdc09 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 14 /openbullet put dup 15 /bullet put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 65 /A put dup 68 /D put dup 77 /M put dup 84 /T put dup 106 /bar put dup 107 /bardbl put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 b64c4301d00e489955b2ce2d9040bb11626f0633ded65ae053753f4af570b6da 256678ac3c38dbf0c3558b68ffd0eed303e91d8e6f146bfc1423c8e355d8d909 4db424413d8433175726ae6fb7d5792226a4d8896d0f0fd5c10d1ce91bc356e9 a1a60c5b02c1ba6e83fac1b4e1adf675cdff13275bd85356c60d41f37dbff65e 7d5bb2f0be0b8a5ce57cb202dc9954dcb75f66ae7b5cc218baf1d4a3c16c247b 544f56afd6ff691c69772f0677847d783b97849523358847c420324c934ec864 55f5d47ec0c20d898ac46e23581b1c5f2dd9fb0a314cb776aead333df3440ae0 4d65893f25c3c6baa08b2064a6fd1aea02b20aad83f8530bf5cc90aa92148090 8e364aae0b157e9f6e86050f3fe00df09b4d6f722c1c7809269c7ccc994f6950 3dae8508266c6b8b05cf2ef900ea2ec4e24c190b508eb20d4698cd66f9f7164c bddffa1a1dc860cd1ccf7e752e0c0625fb7288ffa36c3d0b3f2ab58cd1a2ad97 f825eb318a463184af549244a63c26fb93c3b6543a4f1ef247768e259c39d241 45a28d0f6af1baf482f7d891a91db7a623051015c9ddb29ae71e86353b7b6651 9132908423fb1f01d72e227c59cae0db0e276c316c147e2d66ca1bff04c11362 f84c856c0efca5680be8ddaca4aa114142ed821c4e29af8a91c88df67f10de43 0e4aa8cdfb91d2f191fdbe6eeb41482e38a3a34b614cbd40c511993714890853 290adc202301ec2dd1760054103831f12bc5729561dc0f683acfe1dee1495bcb 1253bf13bb20666dfbe42e4b6378185a4e7f3f55df787500184ecfe4879991ea ab6be0701a2c124267add7171d5a22b05aba1eb5e71963540b88566fb37e2ac6 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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 13 /gamma put dup 30 /phi put dup 32 /psi put dup 73 /I put dup 82 /R put dup 107 /k put dup 110 /n put dup 112 /p put dup 115 /s put dup 116 /t put dup 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 13 /gamma put dup 15 /epsilon1 put dup 21 /lambda put dup 26 /rho put dup 27 /sigma put dup 30 /phi put dup 32 /psi put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 66 /B put dup 67 /C put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 78 /N put dup 82 /R put dup 83 /S put dup 84 /T put dup 88 /X put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 102 /f put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7bcb09b7064e91fa0ded228e42 09aae407a7aaca60b1076299ac4abd23ef02f108765f0e3d91f92f3afbfded37 2fcf6e4b1416901517da8f2fb3c9fe7a87bcbe6fd36cd5b5823fdb74229036a6 3c3346a1093e6b1036902c1bf42fc317c80abf04020a47b344c36de42f05c490 a0ff44ab6d5249e9f552a8707bb7661e242644814001c8430ebd5e5f0b944ceb 666ee64359d663e355b2f17093a964139d17287f6ca6a024767eba4fe4873855 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954bf68d22166b3f1936dc2a336f1e22079d8dacfafdabfb2fab837b8972aac0 4a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 18 /grave put dup 19 /acute put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 19 /acute put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 86 /V put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y put dup 126 /tilde put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueXX 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3037a112066957 7889d67fff57549064b985d67ede2fb03272083723541051ebe936eff01a4ab3 cd845d565dfdf25d9ddd2792db17f52f894bea484480f06c832140fdd6c3af0e fef0584c6b78b5e2bbb9a752f10581fd1c48cb498a822afb18cddebba34695b9 b418010368fdfb7bf758d50f3af03360b81c776325c33c9c81130b1e1be840cf 68c71ffd091dccc0df7b507fe8953dc3836af0da08ff9ff3e9cf7288d9eb8baf ffac546a0fd987ae8e0bcf7b759f4408e13524f478de5896d816cb0ad2904271 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%%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 18 /grave put dup 19 /acute put dup 39 /quoteright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 66 /B put dup 67 /C put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4d81cf7fbc8de5 2d2e7bb57142033993f9c08c315abade8dbc4a732e84e142d3bee51557910e12 cd8aa37c459a5e6b7f5269f59078aba3be4641a11ac48d0b625c8325b38ec08e 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 56b0035ddfbf206c3074beeb0739dcbca272a6e629fb7aea2c5ba7bae50c7b4c 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/CMBX8 rf /Fb 207[30 48[{}1 49.8132 /CMR6 rf /Fc 148[33 107[{}1 49.8132 /CMMI6 rf /Fd 207[18 48[{}1 49.8132 /CMSY6 rf /Fe 163[92 1[46 70[61 61 50 50 16[{}6 83.022 /CMEX10 rf /Ff 149[20 28[84 11[56 14[47 71 19 14[71 32[55{}8 66.4176 /CMSY8 rf /Fg 133[40 47 45 65 45 52 32 40 41 45 50 50 55 80 25 2[30 50 45 30 45 50 45 45 50 12[70 7[61 1[51 38 72 75 1[66 9[30 9[50 2[30 35 30 2[40 40 28[60 11[{}39 99.6264 /CMTI12 rf /Fh 205[35 35 35 4[55 43[{}4 66.4176 /CMR8 rf /Fi 141[83 6[50 28 21[54 6[120 8[77 2[80 9[0 0 3[66 100 7[100 100 6[100 5[77 77 4[77 77 4[50 50 11[77 1[77{}22 99.6264 /CMSY10 rf /Fj 134[35 40 1[34 1[25 33 2[36 1[43 2[37 24[53 8[31 40[46 1[42 16[36 13[{}13 66.4176 /CMMI8 rf /Fk 134[48 55 1[47 56 35 46 1[43 49 1[58 85 29 5[48 1[51 42 41 51 8[81 3[57 60 74 3[78 1[66 83 1[43 81 4[70 74 1[51 1[76 49 1[27 27 24[61 63 1[58 2[55 50 4[57 5[39 1[51 1[62 11[{}41 99.6264 /CMMI12 rf /Fl 129[49 2[49 43 51 51 70 51 54 38 38 38 51 54 49 54 81 27 51 30 27 54 49 30 43 54 43 54 49 1[27 1[27 1[27 60 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Fl(Let)f(us)f(start)g(with)h(some)g(w)m(ell)g(understo)s(o)s(d)f (prop)s(erties)h(of)f(viscosit)m(y)i(solutions)f(without)106 1892 y(pro)s(of,)41 b(see)g([14)o(],)h([6)o(])e(or)f([9].)64 b(W)-8 b(e)40 b(denote)g Fi(T)26 b Fl(\()p Fk(s;)17 b(t)p Fl(\))39 b(:)h Fk(C)7 b Fl(\()p Fo(T)p Fk(;)17 b(I)-25 b(R)q Fl(\))38 b Fi(\000)-16 b(!)39 b Fk(C)7 b Fl(\()p Fo(T)p Fk(;)17 b(I)-25 b(R)q Fl(\))39 b(the)h(map-)106 2013 y(ping)d(whic)m(h)g(asso)s(ciates)g(to)f(eac)m(h)h(function)g Fk(u)1849 2028 y Fh(0)1921 2013 y Fi(2)d Fk(C)7 b Fl(\()p Fo(T)p Fk(;)17 b(I)-25 b(R)q Fl(\))35 b(the)i(function)f Fk(u)p Fl(\()p Fk(t;)17 b(:)p Fl(\),)36 b(where)106 2133 y Fk(u)p Fl(\()p Fk(t;)17 b(x)p Fl(\))28 b Fi(2)g Fk(C)7 b Fl(\([)p Fk(s;)17 b Fl(+)p Fi(1)p Fl([)p Fi(\002)p Fo(T)p Fk(;)g(I)-25 b(R)q Fl(\))24 b(is)i(the)g(viscosit)m(y)i (solution)e(of)e(\(1.1\))h(suc)m(h)i(that)e Fk(u)p Fl(\()p Fk(s;)17 b(:)p Fl(\))27 b(=)h Fk(u)3466 2148 y Fh(0)3505 2133 y Fl(.)106 2253 y(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f(Mark)m(o)m(v)h (prop)s(ert)m(y)1268 2466 y Fi(T)26 b Fl(\()p Fk(t;)17 b(t)1500 2425 y Ff(0)1523 2466 y Fl(\))22 b Fi(\016)g(T)k Fl(\()p Fk(s;)17 b(t)p Fl(\))27 b(=)h Fi(T)d Fl(\()p Fk(s;)17 b(t)2309 2425 y Ff(0)2332 2466 y Fl(\))106 2679 y(for)38 b Fk(s)g Fi(\024)g Fk(t)g Fi(\024)g Fk(t)683 2643 y Ff(0)707 2679 y Fl(,)i(hence)g(the)f(mappings)g Fi(T)25 b Fl(\(0)p Fk(;)17 b(n)p Fl(\))38 b(=)f Fi(T)26 b Fl(\(0)p Fk(;)17 b Fl(1\))2425 2643 y Fj(n)2471 2679 y Fl(,)40 b Fk(n)e Fi(2)g Fk(I)-25 b(N)49 b Fl(form)38 b(a)g(discrete)106 2799 y(semi-group.)45 b(W)-8 b(e)32 b(will)i(note)f Fi(T)58 b Fl(for)32 b Fi(T)25 b Fl(\(0)p Fk(;)17 b Fl(1\))32 b(for)g(simplicit)m(y)-8 b(.)253 2920 y(The)33 b(mappings)h Fi(T)25 b Fl(\()p Fk(s;)17 b(t)p Fl(\))33 b(are)f(con)m(tractions,)1056 3133 y Fi(kT)26 b Fl(\()p Fk(s;)17 b(t)p Fl(\))p Fk(u)k Fi(\000)i(T)i Fl(\()p Fk(s;)17 b(t)p Fl(\))p Fk(v)t Fi(k)1945 3162 y Ff(1)2047 3133 y Fi(\024)28 b(k)p Fk(u)22 b Fi(\000)g Fk(v)t Fi(k)2480 3162 y Ff(1)2555 3133 y Fk(:)253 3345 y Fl(The)33 b(mappings)g Fi(T)25 b Fl(\()p Fk(s;)17 b(t)p Fl(\))32 b(are)g(compact.)44 b(More)32 b(precisely)-8 b(,)35 b(giv)m(en)e(a)f(b)s(ounded)h(set)f Fk(B)h Fi(\032)106 3466 y Fk(C)7 b Fl(\()p Fo(T)p Fk(;)17 b(I)-25 b(R)q Fl(\),)48 b(there)e(exists)h(a)e(p)s(ositiv)m(e)h(nondecreasing)h (function)f Fk(K)7 b Fl(\()p Fk(\017)p Fl(\))49 b(:]0)p Fk(;)17 b Fi(1)p Fl([)p Fi(\000)-16 b(!)p Fl(]0)p Fk(;)17 b Fi(1)p Fl([)106 3586 y(suc)m(h)34 b(that)f Fi(T)25 b Fl(\()p Fk(s;)17 b(t)p Fl(\))p Fk(u)32 b Fl(is)h Fk(K)7 b Fl(\()p Fk(\017)p Fl(\)-Lipsc)m(hitz)35 b(for)d(all)g Fk(u)27 b Fi(2)i Fk(B)37 b Fl(and)c(all)f Fk(t)c Fi(\025)g Fk(s)22 b Fl(+)g Fk(\017)p Fl(.)253 3707 y(The)41 b(mappings)h Fi(T)25 b Fl(\()p Fk(s;)17 b(t)p Fl(\))41 b(are)f(order-preserving)i (and)f(satisfy)h Fi(T)25 b Fl(\()p Fk(s;)17 b(t)p Fl(\)\()p Fk(u)27 b Fl(+)h Fk(c)p Fl(\))41 b(=)g Fk(c)28 b Fl(+)106 3827 y Fi(T)e Fl(\()p Fk(s;)17 b(t)p Fl(\)\()p Fk(u)p Fl(\))40 b(for)g(all)h(real)g Fk(c)p Fl(.)69 b(Let)41 b(us)g(end)h(this)f(list)h(b)m(y)f(recalling)h(the)f(Hopf-Lax-Oleinik) 106 3947 y(form)m(ula:)878 4208 y Fi(T)25 b Fl(\()p Fk(s;)17 b(t)p Fl(\))p Fk(u)1214 4223 y Fh(0)1281 4208 y Fl(=)27 b(inf)1423 4257 y Fj(\015)1502 4086 y Fe(\022)1564 4208 y Fk(u)1620 4223 y Fh(0)1659 4208 y Fl(\()p Fk(\015)5 b Fl(\()p Fk(s)p Fl(\)\))21 b(+)2032 4091 y Fe(Z)2115 4117 y Fj(t)2078 4279 y(s)2162 4208 y Fk(L)p Fl(\()p Fk(\033)n(;)c(\015)5 b(;)31 b Fl(_)-41 b Fk(\015)5 b Fl(\))32 b Fk(d\033)2699 4086 y Fe(\023)106 4208 y Fl(\(2.1\))106 4421 y(where)38 b(the)f(in\014m)m(um)h(is)g(tak)m(en)f(on)g(the)g(set)g (of)f(piecewise)j Fk(C)2409 4384 y Fh(1)2485 4421 y Fl(curv)m(es)g (with)e(v)-5 b(alues)38 b(in)f Fo(T)p Fl(,)106 4541 y(suc)m(h)46 b(that)e Fk(\015)5 b Fl(\()p Fk(t)p Fl(\))48 b(=)f Fk(x)p Fl(.)78 b(In)45 b(the)g(ab)s(o)m(v)m(e,)i(the)e(Lagrangian)e Fk(L)p Fl(\()p Fk(t;)17 b(x;)g(v)t Fl(\))45 b(is)f(de\014ned)i(as)e (the)106 4661 y(Legendre)34 b(transform)f(of)f Fk(H)8 b Fl(:)1096 4906 y Fk(L)p Fl(\()p Fk(t;)17 b(x;)g(v)t Fl(\))28 b(=)f(max)1615 4965 y Fj(p)p Ff(2)p Fj(I)-18 b(R)1780 4785 y Fe(\022)1841 4906 y Fk(pv)26 b Fi(\000)c Fk(H)8 b Fl(\()p Fk(t;)17 b(x;)g(p)p Fl(\))2454 4785 y Fe(\023)2515 4906 y Fk(:)1795 5291 y Fl(3)p eop %%Page: 4 4 4 3 bop 106 76 a Fl(It)28 b(is)f(a)g(classical)i(result)f(from)f(the)g (calculus)i(of)d(v)-5 b(ariations)28 b(that,)g(under)g(our)f (assumptions,)106 196 y(the)33 b(in\014m)m(um)i(in)d(\(2.1\))g(is)h (reac)m(hed)h(b)m(y)g Fk(C)1669 160 y Fh(2)1741 196 y Fl(curv)m(es.)253 316 y(The)k(main)f(p)s(oin)m(t)g(of)f(this)h (paragraph)g(is)g(the)g(follo)m(wing)g(simple)i(prop)s(osition)e(b)s (elo)m(w.)106 437 y(Note)26 b(that)f(it)g(also)h(implies)g(the)g (existence)i(of)d(a)g(\014xed)h(p)s(oin)m(t)f(of)g(the)h(semi-group)g (generated)106 557 y(b)m(y)34 b Fi(T)26 b Fl(,)32 b Fg(i.e.)43 b Fl(the)33 b(existence)i(of)d(a)h(1-p)s(erio)s(dic)f(viscosit)m(y)j (solution)e(of)f(\(1.1\).)106 769 y Fo(Prop)s(osition)38 b(2.1)49 b Fg(L)-5 b(et)38 b Fk(u)p Fl(\()p Fk(t;)17 b(x)p Fl(\))34 b Fi(2)g Fk(C)7 b Fl(\([)p Fk(s;)17 b Fi(1)p Fl([)p Fi(\002)p Fo(T)p Fk(;)g(I)-25 b(R)q Fl(\))37 b Fg(b)-5 b(e)37 b(a)h(visc)-5 b(osity)38 b(solution)g(of)g(\(1.1\);) 106 889 y(r)-5 b(e)g(c)g(al)5 b(l)35 b(that)g(it)g(is)g(b)-5 b(ounde)g(d)34 b(on)g Fl([)p Fk(s;)17 b Fi(1)p Fl([)p Fi(\002)p Fo(T)p Fg(.)45 b(Then)34 b(the)h Fl(1)p Fg(-p)-5 b(erio)g(dic)33 b(function)1254 1095 y Fk(\036)p Fl(\()p Fk(t;)17 b(x)p Fl(\))28 b(=)f(lim)17 b(inf)1669 1149 y Fj(n)p Ff(!)p Fh(+)p Ff(1)1940 1095 y Fk(u)p Fl(\()p Fk(t)22 b Fl(+)g Fk(n;)17 b(x)p Fl(\))106 1332 y Fg(is)35 b(a)g(visc)-5 b(osity)34 b(solution)h(of)f(\(1.1\).)106 1544 y Fm(Pr)n(oof.)44 b Fl(W)-8 b(e)33 b(ha)m(v)m(e)h(to)e(pro)m(v)m (e)i(that)f Fi(T)25 b Fl(\()p Fk(s;)17 b(t)p Fl(\))p Fk(\036)p Fl(\()p Fk(s;)g(:)p Fl(\))27 b(=)h Fk(\036)p Fl(\()p Fk(t;)17 b(:)p Fl(\))32 b(for)g(all)g Fk(s)c Fi(\024)g Fk(t)p Fl(.)253 1665 y(It)40 b(follo)m(ws)h(from)f(the)h (Barles-P)m(erthame)h(Lemma)f([1])f(that)g Fk(\036)g Fl(is)h(a)f(viscosit)m(y)i(sup)s(er-)106 1785 y(solution)e(of)e (\(1.1\),)i Fg(i.e.)62 b Fl(that)38 b Fi(T)26 b Fl(\()p Fk(s;)17 b(t)p Fl(\))p Fk(\036)p Fl(\()p Fk(s;)g(:)p Fl(\))37 b Fi(\024)i Fk(\036)p Fl(\()p Fk(t;)17 b(:)p Fl(\).)62 b(This)40 b(fact,)g(whic)m(h)g(is)g(a)e(general)106 1906 y(feature)30 b(of)f(viscosit)m(y)j(solutions)e(of)f(\014rst)h(or)f (second)i(order)f(equations,)h(can)f(b)s(e)g(easily)g(seen)106 2026 y(on)36 b(\(2.1\).)52 b(In)35 b(order)h(to)f(do)g(so,)i(w)m(e)f (\014x)h(\()p Fk(t;)17 b(x)p Fl(\))35 b(and)h(consider)h(an)e (increasing)i(sequence)h Fk(n)3489 2041 y Fj(k)106 2146 y Fl(of)f(in)m(tegers)g(suc)m(h)h(that)f Fk(u)p Fl(\()p Fk(t)24 b Fl(+)h Fk(n)1338 2161 y Fj(k)1381 2146 y Fk(;)17 b(x)p Fl(\))34 b Fi(\000)-16 b(!)34 b Fk(\036)p Fl(\()p Fk(t;)17 b(x)p Fl(\).)56 b(There)38 b(exists)g(a)e(sequence)j(of)e (curv)m(es)106 2267 y Fk(\015)157 2282 y Fj(k)227 2267 y Fl(:)28 b([)p Fk(s;)17 b(t)p Fl(])28 b Fi(\000)-16 b(!)27 b Fo(T)33 b Fl(suc)m(h)h(that)448 2517 y Fk(u)p Fl(\()p Fk(t)22 b Fl(+)g Fk(n)755 2532 y Fj(k)798 2517 y Fk(;)17 b(x)p Fl(\))28 b(=)f Fk(u)p Fl(\()p Fk(s)22 b Fl(+)g Fk(n)1384 2532 y Fj(k)1427 2517 y Fk(;)17 b(\015)1522 2532 y Fj(k)1564 2517 y Fl(\(0\)\))k(+)1846 2400 y Fe(Z)1929 2426 y Fj(t)1893 2588 y(s)1976 2517 y Fk(L)p Fl(\()p Fk(\033)n(;)c(\015)2228 2532 y Fj(k)2271 2517 y Fl(\()p Fk(\033)26 b Fl(+)c Fk(t)p Fl(\))p Fk(;)31 b Fl(_)-41 b Fk(\015)2656 2532 y Fj(k)2698 2517 y Fl(\()p Fk(\033)26 b Fl(+)c Fk(t)p Fl(\)\))32 b Fk(d\033)n(:)106 2761 y Fl(The)h(sequence)h Fk(\015)760 2776 y Fj(k)834 2761 y Fl(is)f(compact)f(for)f(the)h Fk(C)1713 2725 y Fh(1)1784 2761 y Fl(top)s(ology)-8 b(,)31 b(and)h(w)m(e)h(will)f(assume)i(b)m(y)e (p)s(ossibly)106 2881 y(taking)39 b(a)f(subsequence)k(in)d Fk(n)1235 2896 y Fj(k)1317 2881 y Fl(that)f(it)g(is)h(con)m(v)m(ergen)m (t,)k(and)38 b(note)h Fk(\015)k Fl(the)c(limit.)62 b(T)-8 b(aking)106 3002 y(the)33 b(lim)17 b(inf)40 b(in)33 b(the)g(equalit)m (y)h(ab)s(o)m(v)m(e)g(giv)m(es)322 3246 y Fk(\036)p Fl(\()p Fk(t;)17 b(x)p Fl(\))28 b Fi(\025)g Fk(\036)p Fl(\()p Fk(s;)17 b(\015)5 b Fl(\(0\)\))21 b(+)1247 3129 y Fe(Z)1330 3155 y Fj(t)1293 3318 y(s)1376 3246 y Fk(L)p Fl(\()p Fk(\033)n(;)c(\015)5 b Fl(\()p Fk(\033)27 b Fl(+)22 b Fk(t)p Fl(\))p Fk(;)31 b Fl(_)-41 b Fk(\015)5 b Fl(\()p Fk(\033)26 b Fl(+)c Fk(t)p Fl(\)\))32 b Fk(d\033)g Fi(\025)c(T)e Fl(\()p Fk(s;)17 b(t)p Fl(\))p Fk(\036)p Fl(\()p Fk(s;)g(:)p Fl(\)\()p Fk(x)p Fl(\))p Fk(:)106 3486 y Fl(W)-8 b(e)36 b(ha)m(v)m(e)h(used)f(that)f(the)h(functions)h Fk(u)p Fl(\()p Fk(s)23 b Fl(+)h Fk(n)1860 3501 y Fj(k)1903 3486 y Fk(;)17 b(:)p Fl(\))35 b(ha)m(v)m(e)h(a)g(common)f(Lipsc)m(hitz)j (constan)m(t)106 3606 y(to)33 b(conclude)h(that)e(lim)17 b(inf)24 b Fk(u)p Fl(\()p Fk(s)d Fl(+)h Fk(n)1442 3621 y Fj(k)1485 3606 y Fk(;)17 b(\015)1580 3621 y Fj(k)1622 3606 y Fl(\(0\)\))27 b(=)h(lim)17 b(inf)24 b Fk(u)p Fl(\()p Fk(s)d Fl(+)h Fk(n)2520 3621 y Fj(k)2563 3606 y Fk(;)17 b(\015)5 b Fl(\(0\)\))27 b Fi(\025)h Fk(\036)p Fl(\()p Fk(s;)17 b(\015)5 b Fl(\(0\)\).)253 3727 y(The)24 b(rev)m(erse)i (inequalit)m(y)g(is)e(sp)s(eci\014c)h(to)e(Hamilton-Jacobi)h (equations,)i(and)e(explicitely)106 3847 y(relies)34 b(on)f(\(2.1\).)43 b(Note)32 b(that)h(for)f(all)g(curv)m(e)i Fk(\015)f Fl(:)28 b([)p Fk(s;)17 b(t)p Fl(])28 b Fi(\000)-16 b(!)27 b Fo(T)p Fl(,)32 b(w)m(e)i(ha)m(v)m(e)546 4097 y Fk(u)p Fl(\()p Fk(t)22 b Fl(+)g Fk(n;)17 b(x)p Fl(\))28 b Fi(\024)g Fk(u)p Fl(\()p Fk(s)21 b Fl(+)h Fk(n;)17 b(\015)5 b Fl(\(0\)\))22 b(+)1823 3980 y Fe(Z)1906 4007 y Fj(t)1869 4169 y(s)1952 4097 y Fk(L)p Fl(\()p Fk(\033)n(;)17 b(\015)5 b Fl(\()p Fk(\033)27 b Fl(+)22 b Fk(t)p Fl(\))p Fk(;)31 b Fl(_)-41 b Fk(\015)5 b Fl(\()p Fk(\033)26 b Fl(+)c Fk(t)p Fl(\)\))33 b Fk(d\033)n(:)106 4332 y Fl(T)-8 b(aking)34 b(the)f(lim)17 b(inf)7 b(,)32 b(w)m(e)i(obtain)733 4576 y Fk(\036)p Fl(\()p Fk(t;)17 b(x)p Fl(\))28 b Fi(\024)g Fk(\036)p Fl(\()p Fk(s;)17 b(\015)5 b Fl(\(0\)\))21 b(+)1658 4459 y Fe(Z)1741 4485 y Fj(t)1704 4648 y(s)1787 4576 y Fk(L)p Fl(\()p Fk(\033)n(;)c(\015)5 b Fl(\()p Fk(\033)27 b Fl(+)22 b Fk(t)p Fl(\))p Fk(;)31 b Fl(_)-41 b Fk(\015)5 b Fl(\()p Fk(\033)26 b Fl(+)c Fk(t)p Fl(\)\))32 b Fk(d\033)106 4816 y Fl(for)g(eac)m(h)i(curv)m(e)g Fk(\015)5 b Fl(,)33 b(hence)h Fk(\036)p Fl(\()p Fk(t;)17 b(:)p Fl(\))27 b Fi(\024)h(T)e Fl(\()p Fk(s;)17 b(t)p Fl(\))p Fk(\036)p Fl(\()p Fk(s;)g(:)p Fl(\),)31 b(whic)m(h)k(is)e(the)g(desired)h (inequalit)m(y)-8 b(.)p 3475 4791 57 4 v 3475 4841 4 50 v 3528 4841 V 3475 4844 57 4 v 1795 5291 a(4)p eop %%Page: 5 5 5 4 bop 253 76 a Fl(The)36 b(basic)f(ob)5 b(jects)36 b(to)f(understand)h(in)f(order)g(to)f(study)i(the)g(asymptotic)g(b)s (eha)m(viour)106 196 y(of)30 b(solutions)g(of)f(\(1.1\))g(are)h(the)g Fk(!)t Fl(-limit)f(solutions.)43 b(Recall)30 b(that)g(a)f(solution)h Fk(u)p Fl(\()p Fk(t;)17 b(x)p Fl(\))28 b(:)f Fk(I)-25 b(R)17 b Fi(\002)106 316 y Fo(T)28 b Fi(\000)-16 b(!)27 b Fk(I)-25 b(R)24 b Fl(is)g(called)g(an)f Fk(!)t Fl(-limit)g(solution)h (if)f(there)h(exists)h(a)e(solution)h Fk(v)31 b Fl(:)d([)p Fk(s;)17 b Fi(1)p Fl([)p Fi(\002)p Fo(T)28 b Fi(\000)-16 b(!)27 b Fk(I)-25 b(R)106 437 y Fl(and)33 b(an)g(increasing)g(sequence) j Fk(n)1350 452 y Fj(k)1425 437 y Fl(of)c(in)m(tegers)i(suc)m(h)g(that) 1246 657 y Fk(u)p Fl(\()p Fk(t;)17 b(x)p Fl(\))28 b(=)71 b(lim)1643 717 y Fj(k)r Ff(\000)-12 b(!1)1883 657 y Fk(v)t Fl(\()p Fk(t)22 b Fl(+)g Fk(n)2185 672 y Fj(k)2228 657 y Fk(;)17 b(x)p Fl(\))p Fk(:)106 908 y Fl(In)44 b(other)f(w)m(ords,)k Fk(!)t Fl(-limit)42 b(solutions)i(are)f(solutions)h(whose)g(initial)g (v)-5 b(alue)43 b Fk(u)p Fl(\(0)p Fk(;)17 b(:)p Fl(\))42 b(is)i(an)106 1028 y Fk(!)t Fl(-limit)32 b(of)g(the)h(semi-group)h Fi(T)25 b Fl(.)106 1361 y Fp(3.)161 b(Calibrated)53 b(curv)l(es)f(and)i (Uniqueness)d(set)106 1580 y Fl(In)44 b(this)g(section,)j(w)m(e)d(giv)m (e)g(-)e(without)i(man)m(y)g(pro)s(ofs,)h(for)e(they)h(are)f(already)h (stated)f(in)106 1700 y([6])36 b(or)g([18)o(])g(-)g(some)g(salien)m(t)h (features)g(of)f Fk(!)t Fl(-limit)f(solutions)i(that)e(do)h(not)g(dep)s (end)h(of)e(the)106 1821 y(dimension)g(of)d(the)h(am)m(bien)m(t)h (space.)253 1991 y(Let)g Fk(u)d Fl(:)g([)p Fk(s;)17 b Fi(1)p Fl([)p Fi(\002)p Fo(T)31 b Fi(\000)-17 b(!)31 b Fk(I)-25 b(R)35 b Fl(b)s(e)g(a)f(viscosit)m(y)j(solution)e(of)f (\(1.1\).)48 b(A)35 b(curv)m(e)h Fk(\015)g Fl(:)31 b([)p Fk(s;)17 b Fi(1)p Fl([)p Fi(\033)106 2111 y Fl([)p Fk(t;)g(t)247 2075 y Ff(0)271 2111 y Fl(])28 b Fi(\000)-16 b(!)27 b Fo(T)32 b Fl(is)i(said)f(calibrated)g(b)m(y)g Fk(u)f Fl(if)937 2386 y Fk(u)p Fl(\()p Fk(t)1066 2345 y Ff(0)1089 2386 y Fk(;)17 b(\015)5 b Fl(\()p Fk(t)1262 2345 y Ff(0)1285 2386 y Fl(\)\))28 b(=)f Fk(u)p Fl(\()p Fk(t;)17 b(\015)5 b Fl(\()p Fk(t)p Fl(\)\))22 b(+)1990 2269 y Fe(Z)2073 2295 y Fj(t)2098 2272 y Fd(0)2036 2457 y Fj(t)2142 2386 y Fk(L)p Fl(\()p Fk(\033)n(;)17 b(\015)5 b(;)31 b Fl(_)-41 b Fk(\015)5 b Fl(\))32 b Fk(d\033)n(:)106 2640 y Fl(F)-8 b(rom)33 b(F)-8 b(athi)33 b([6],)h(if)f Fk(\015)h Fl(:)29 b([)p Fk(t;)17 b(t)1156 2604 y Ff(0)1179 2640 y Fl(])30 b Fi(\000)-17 b(!)29 b Fo(T)k Fl(is)h(calibrated)g(b)m(y)g Fk(u)p Fl(,)f(then)h Fk(u)2622 2655 y Fj(x)2699 2640 y Fl(exists)h(at)e(eac)m(h)h(p)s(oin)m(t)106 2760 y(\()p Fk(s;)17 b(\015)5 b Fl(\()p Fk(s)p Fl(\)\))p Fk(;)17 b(s)27 b Fi(2)h Fl([)p Fk(t;)17 b(t)802 2724 y Ff(0)826 2760 y Fl([)32 b(and)h(satis\014es)453 2980 y Fk(u)509 2995 y Fj(x)552 2980 y Fl(\()p Fk(s;)17 b(\015)5 b Fl(\()p Fk(s)p Fl(\)\))27 b(=)h Fk(L)1093 2995 y Fj(v)1134 2980 y Fl(\()p Fk(s;)17 b(\015)5 b Fl(\()p Fk(s)p Fl(\))p Fk(;)31 b Fl(_)-41 b Fk(\015)t Fl(\()p Fk(s)p Fl(\)\))28 b Fi(\()-17 b(\))41 b Fl(_)-41 b Fk(\015)5 b Fl(\()p Fk(s)p Fl(\))27 b(=)h Fk(H)2327 2995 y Fj(p)2366 2980 y Fl(\()p Fk(s;)17 b(\015)5 b Fl(\()p Fk(s)p Fl(\))p Fk(;)17 b(u)2772 2995 y Fj(x)2815 2980 y Fl(\()p Fk(s;)g(\015)5 b Fl(\()p Fk(s)p Fl(\)\))p Fk(:)253 3200 y Fl(Suc)m(h)33 b(a)f(prop)s(ert)m(y)g(w)m(as)h(already)g(understo)s(o)s(d)f(in)g([17]) g(for)f(the)i(equation)g Fi(jr)p Fk(u)p Fi(j)26 b Fl(=)i Fk(f)11 b Fl(\()p Fk(x)p Fl(\))106 3321 y(on)30 b(the)f(sphere,)j Fk(f)40 b Fl(nonnegativ)m(e)31 b(with)e(nonempt)m(y)i(zero)f(set.)43 b(It)30 b(w)m(as)g(not,)g(ho)m(w)m(ev)m(er,)i(made)106 3441 y(that)h(systematic.)253 3611 y(W)-8 b(e)38 b(no)m(w)h(c)m(ho)s (ose)g(once)f(and)g(for)g(all)g(a)g(1-p)s(erio)s(dic)f(solution)i Fk(\036)p Fl(\()p Fk(t;)17 b(x)p Fl(\))38 b(of)f(\(1.1\).)59 b(First,)106 3732 y(let)40 b(us)f(note)g(that)f(a)h(classical)h (compactness)h(argumen)m(t)f(giv)m(es)g(the)f(existence)i(of)e(curv)m (es)106 3852 y Fk(\015)33 b Fl(:)28 b Fk(I)-25 b(R)28 b Fi(\000)-16 b(!)28 b Fo(T)k Fl(whic)m(h)i(are)f(calibrated)g(b)m(y)g Fk(\036)g Fl(on)f(all)h(compact)g(in)m(terv)-5 b(al.)253 3973 y(It)25 b(is)g(a)g(consequence)j(of)d([6)o(],)i(or)e(Mather's)h (shortening)g(lemma,)h(see)f([12])f(that)f(t)m(w)m(o)i(suc)m(h)106 4093 y(curv)m(es)40 b(cannnot)e(in)m(tersect.)60 b(More)38 b(precisely)h(if)f Fk(\015)2041 4108 y Fh(1)2117 4093 y Fl(and)g Fk(\015)2363 4108 y Fh(2)2438 4093 y Fl(:)e Fk(I)-25 b(R)37 b Fi(\000)-17 b(!)36 b Fo(T)h Fl(are)h(calibrated)106 4213 y(b)m(y)c Fk(\036)p Fl(,)e(and)h(if)f(there)i(exists)g(a)e Fk(t)h Fl(suc)m(h)h(that)e Fk(\015)1788 4228 y Fh(1)1827 4213 y Fl(\()p Fk(t)p Fl(\))c(=)g Fk(\015)2121 4228 y Fh(2)2160 4213 y Fl(\()p Fk(t)p Fl(\),)33 b(then)g Fk(\015)2604 4228 y Fh(1)2670 4213 y Fl(=)28 b Fk(\015)2825 4228 y Fh(2)2864 4213 y Fl(.)253 4334 y(Let)1567 4454 y Fi(D)i(\032)e Fk(I)-25 b(R)23 b Fi(\002)g Fo(T)106 4628 y Fl(b)s(e)32 b(the)g(union)f(of)g(the)h(graphs)g(of)e(these)j(orbits,)f(and)g Fi(D)2192 4643 y Fh(0)2258 4628 y Fi(\032)d Fo(T)i Fl(b)s(e)g(the)h (set)g(of)f(p)s(oin)m(ts)h Fk(\015)5 b Fl(\(0\),)106 4749 y(where)39 b Fk(\015)h Fl(:)35 b Fk(I)-25 b(R)36 b Fi(\000)-16 b(!)35 b Fo(T)i Fl(is)g(calibrated.)57 b(This)38 b(is)g(a)f(nonempt)m(y)h(compact)g(set.)57 b(F)-8 b(or)36 b(eac)m(h)i Fk(t)p Fl(,)106 4869 y(w)m(e)c(de\014ne)g (the)f(mapping)h Fk(S)1168 4833 y Fj(t)1225 4869 y Fl(:)28 b Fi(D)1357 4884 y Fh(0)1424 4869 y Fi(\000)-16 b(!)28 b Fo(T)k Fl(whic)m(h)i(asso)s(ciates)g(to)f(eac)m(h)g Fk(x)c Fi(2)f(D)3045 4884 y Fh(0)3084 4869 y Fl(,)33 b(the)g(v)-5 b(alue)1795 5291 y(5)p eop %%Page: 6 6 6 5 bop 106 76 a Fk(\015)5 b Fl(\()p Fk(t)p Fl(\),)34 b(where)h Fk(\015)f Fl(:)29 b Fk(I)-25 b(R)29 b Fi(\000)-16 b(!)29 b Fo(T)k Fl(is)h(the)f(unique)i(calibrated)f(curv)m(e)h (satisfying)f Fk(\015)5 b Fl(\(0\))29 b(=)g Fk(x)p Fl(.)46 b(It)33 b(is)106 196 y(a)g(bi-Lipsc)m(hitz)h(homeomorphism)g(on)m(to)f (its)g(image.)253 316 y(Clearly)-8 b(,)46 b Fk(S)689 280 y Fh(1)771 316 y Fl(is)d(a)f(homeomorphism)i(of)e Fi(D)1910 331 y Fh(0)1949 316 y Fl(.)74 b(Let)42 b(us)i(note)e Fi(M)2716 331 y Fh(0)2798 316 y Fl(its)h Fk(!)t Fl(-limit.)72 b(This)106 437 y(is)41 b(the)g(closure)h(in)e Fo(T)h Fl(the)f(set)i(of)d(p)s(oin)m(ts)i Fk(x)h Fi(2)f(D)1998 452 y Fh(0)2078 437 y Fl(whic)m(h)h(are)e(the)h(limit)g(of)f(a)g (sequence)106 557 y Fk(S)172 521 y Fj(n)215 533 y Fc(k)257 557 y Fl(\()p Fk(y)t Fl(\))f(with)h Fk(y)j Fi(2)e(D)928 572 y Fh(0)1007 557 y Fl(and)f Fk(n)1262 572 y Fj(k)1344 557 y Fl(an)g(increasing)h(sequence)h(of)e(in)m(tegers.)66 b(The)41 b(set)f Fi(M)3387 572 y Fh(0)3466 557 y Fl(is)106 677 y(non-empt)m(y)29 b(and)f(compact.)43 b(W)-8 b(e)28 b(call)g Fi(M)f Fl(the)h(union,)i(in)d Fk(I)-25 b(R)13 b Fi(\002)f Fo(T)p Fl(,)30 b(of)d(the)h(graphs)g(of)g(curv)m(es)106 798 y Fk(S)172 762 y Fj(t)202 798 y Fl(\()p Fk(x)p Fl(\))p Fk(;)17 b(x)28 b Fi(2)g(M)674 813 y Fh(0)713 798 y Fl(.)253 918 y(The)45 b(follo)m(wing)g(remark,)j(noticed)e(in)e([18],)j(is)e (the)g(k)m(ey)h(p)s(oin)m(t)f(to)f(the)h(con)m(v)m(ergence)106 1039 y(pro)s(of:)106 1242 y Fo(Lemma)40 b(3.1)49 b Fg(L)-5 b(et)43 b Fk(u)p Fl(\()p Fk(t;)17 b(x)p Fl(\))42 b(:)h([)p Fk(s;)17 b Fi(1)p Fl([)p Fi(\002)p Fo(T)42 b Fi(\000)-16 b(!)42 b Fk(I)-25 b(R)44 b Fg(b)-5 b(e)42 b(a)h(visc)-5 b(osity)42 b(solution)h(of)g(\(1.1\),)h(let)106 1362 y Fk(x)29 b Fi(2)f(D)361 1377 y Fh(0)400 1362 y Fg(.)45 b(Then)34 b(the)h(function)f Fk(t)28 b Fi(7!)f Fk(u)p Fl(\()p Fk(t;)17 b(S)1703 1326 y Fj(t)1732 1362 y Fl(\()p Fk(x)p Fl(\)\))23 b Fi(\000)f Fk(\036)p Fl(\()p Fk(t;)17 b(S)2264 1326 y Fj(t)2293 1362 y Fl(\()p Fk(x)p Fl(\)\))35 b Fg(is)g(non-incr)-5 b(e)g(asing.)106 1566 y Fl(The)34 b(imp)s(ortan)m(t)f(consequence)i(b)s(elo)m(w)f(is)f(also)f(pro)m(v)m (ed)i(in)f([18]:)106 1769 y Fo(Corollary)38 b(3.1)49 b Fg(L)-5 b(et)40 b Fk(u)p Fl(\()p Fk(t;)17 b(x)p Fl(\))39 b Fg(b)-5 b(e)38 b(an)h Fk(!)t Fg(-limit)f(visc)-5 b(osity)39 b(solution)g(of)f(\(1.1\),)i(and)e(let)h Fk(x)d Fi(2)106 1890 y(M)226 1905 y Fh(0)266 1890 y Fg(,)e(then)h(the)g(function)f Fk(t)28 b Fi(7!)g Fk(u)p Fl(\()p Fk(t;)17 b(S)1522 1853 y Fj(t)1550 1890 y Fl(\()p Fk(x)p Fl(\)\))23 b Fi(\000)f Fk(\036)p Fl(\()p Fk(t;)17 b(S)2082 1853 y Fj(t)2111 1890 y Fl(\()p Fk(x)p Fl(\)\))36 b Fg(is)e(c)-5 b(onstant.)253 2093 y Fl(It)32 b(follo)m(ws)i(that)e(the)h(curv)m(e)h Fk(S)1384 2057 y Fj(t)1413 2093 y Fl(\()p Fk(x)p Fl(\))f(is)g (calibrated)g(b)m(y)h Fk(u)p Fl(;)e(hence)i Fk(u)e Fl(and)g Fk(\036)h Fl(are)f(di\013eren-)106 2213 y(tiable)h(on)g Fi(M)f Fl(b)m(y)i(the)f(ab)s(o)m(v)m(e-stated)g(regularit)m(y)h (results)g(of)e(F)-8 b(athi.)106 2417 y Fo(Corollary)38 b(3.2)49 b Fg(L)-5 b(et)47 b Fk(u)p Fl(\()p Fk(t;)17 b(x)p Fl(\))46 b Fg(b)-5 b(e)47 b(an)f Fk(!)t Fg(-limit)f(visc)-5 b(osity)46 b(solution)h(of)f(\(1.1\).)79 b(Then)46 b(the)106 2537 y(functions)35 b Fk(u)p Fl(\()p Fk(t;)17 b(x)p Fl(\))35 b Fg(and)f Fk(\036)p Fl(\()p Fk(t;)17 b(x)p Fl(\))35 b Fg(and)f(di\013er)-5 b(entiable)33 b(on)i Fi(M)p Fg(,)f(and)h(we)f (have)1054 2757 y Fk(@)1105 2772 y Fj(t)1135 2757 y Fl(\()p Fk(u)22 b Fi(\000)g Fk(\036)p Fl(\)\()p Fk(t;)17 b(x)p Fl(\))28 b(=)g Fk(@)1839 2772 y Fj(x)1883 2757 y Fl(\()p Fk(u)22 b Fi(\000)g Fk(\036)p Fl(\)\()p Fk(t;)17 b(x)p Fl(\))28 b(=)f(0)106 2977 y Fg(for)35 b(al)5 b(l)35 b Fl(\()p Fk(t;)17 b(x)p Fl(\))28 b Fi(2)g(M)p Fg(.)253 3181 y Fl(The)41 b(next)g(step)g(is)f(to)g(de\014ne)h(a)f(uniqueness)j (set,)g(i.e.)66 b(a)40 b(set)h(suc)m(h)g(that)f(t)m(w)m(o)h(global)106 3301 y(solutions)f(of)d(\(1.1\))h(coinciding)i(on)e(this)h(set)g (coincide)g(ev)m(erywhere.)64 b(W)-8 b(e)39 b(form)m(ulate)f(the)106 3421 y(results)c(in)f(the)g(general,)g(non-autonomous,)g(setting.)106 3650 y Fo(Prop)s(osition)38 b(3.2)49 b Fg(L)-5 b(et)41 b Fk(u)p Fl(\()p Fk(t;)17 b(x)p Fl(\))37 b(:)h Fk(I)-25 b(R)27 b Fi(\002)g Fo(T)38 b Fi(\000)-16 b(!)38 b Fk(I)-25 b(R)41 b Fg(b)-5 b(e)40 b(a)g(glob)-5 b(al)39 b(and)h(b)-5 b(ounde)g(d)40 b(visc)-5 b(osity)106 3770 y(solution)35 b(of)g(\(1.1\))f(such)g(that)i Fk(u)27 b Fl(=)g Fk(\036)35 b Fg(on)f Fi(M)p Fg(,)h(then)g Fk(u)27 b Fl(=)g Fk(\036)p Fg(.)106 3998 y Fl(This)34 b(prop)s(osition)f(easily)h(follo)m(ws)f (from)f(the)h(follo)m(wing)g(lemma:)106 4227 y Fo(Lemma)40 b(3.2)49 b Fg(L)-5 b(et)29 b Fk(u)p Fl(\()p Fk(t;)17 b(x)p Fl(\))27 b(:)h Fk(I)-25 b(R)10 b Fi(\002)f Fo(T)29 b Fi(\000)-17 b(!)28 b Fk(I)-25 b(R)30 b Fg(b)-5 b(e)28 b(a)h(glob)-5 b(al)28 b(and)h(b)-5 b(ounde)g(d)28 b(visc)-5 b(osity)29 b(solution)106 4347 y(of)39 b(\(1.1\))e(and)h Fk(\015)5 b Fl(\()p Fk(t)p Fl(\))35 b(:])25 b Fi(\000)g(1)p Fk(;)17 b(s)p Fl(])34 b Fi(\000)-16 b(!)34 b Fo(T)k Fg(b)-5 b(e)39 b(a)f(curve)g(c)-5 b(alibr)g(ate)g(d)38 b(by)h Fk(u)p Fg(.)55 b(Then)38 b(ther)-5 b(e)38 b(exists)106 4467 y(an)d(incr)-5 b(e)g(asing)33 b(se)-5 b(quenc)g(e)34 b Fk(n)1159 4482 y Fj(k)1237 4467 y Fg(of)h(inte)-5 b(gers)34 b(such)h(that)g Fk(\015)5 b Fl(\()p Fi(\000)p Fk(n)2364 4482 y Fj(k)2407 4467 y Fl(\))28 b Fi(\000)-16 b(!)27 b Fk(x)h Fi(2)g(M)2958 4482 y Fh(0)2998 4467 y Fg(.)106 4696 y Fm(Pr)n(oof.)53 b Fl(In)35 b(view)i(of)e(Lemma)h(3.1,)g(the)g (function)f Fk(t)e Fi(7\000)-16 b(!)32 b Fk(u)p Fl(\()p Fk(t;)17 b(\015)5 b Fl(\()p Fk(t)p Fl(\)\))24 b Fi(\000)g Fk(\036)p Fl(\()p Fk(t;)17 b(\015)5 b Fl(\()p Fk(t)p Fl(\)\))35 b(is)h(non-)106 4816 y(increasing,)j(and)e(b)s(ounded.)57 b(Hence)38 b(this)f(function)g(has)g(a)f(limit)i(as)e Fk(t)f Fi(\000)-16 b(!)34 b(\0001)p Fl(.)56 b(Let)37 b(us)106 4936 y(c)m(ho)s(ose)i(an)e(increasing)i(sequence)h Fk(n)1489 4951 y Fj(k)1569 4936 y Fl(of)d(in)m(tegers)i(suc)m(h)f(that) g(the)f(curv)m(es)j Fk(\015)5 b Fl(\()p Fk(t)26 b Fi(\000)f Fk(n)3283 4951 y Fj(k)3326 4936 y Fl(\))38 b(are)1795 5291 y(6)p eop %%Page: 7 7 7 6 bop 106 76 a Fl(con)m(v)m(erging)40 b(uniformly)f(on)f(compact)g (sets)h(to)f(a)g(limit)g Fk(\015)2280 91 y Ff(1)2391 76 y Fl(:)f Fk(I)-25 b(R)38 b Fi(\000)-17 b(!)37 b Fo(T)p Fl(.)59 b(The)39 b(follo)m(wing)106 196 y(calculations)34 b(sho)m(w)g(that)e(this)h(curv)m(e)i(is)e(calibrated)g(b)m(y)g Fk(\036)p Fl(:)408 407 y Fk(\036)p Fl(\()p Fk(t)539 366 y Ff(0)562 407 y Fk(;)17 b(\015)657 422 y Ff(1)731 407 y Fl(\()p Fk(t)804 366 y Ff(0)828 407 y Fl(\)\))22 b Fi(\000)h Fk(\036)p Fl(\()p Fk(t;)17 b(\015)1252 422 y Ff(1)1325 407 y Fl(\()p Fk(t)p Fl(\)\))28 b(=)g(lim)1758 311 y Fe(\020)1807 407 y Fk(\036)p Fl(\()p Fk(t)1938 366 y Ff(0)1962 407 y Fk(;)17 b(\015)5 b Fl(\()p Fk(t)2135 366 y Ff(0)2180 407 y Fi(\000)22 b Fk(n)2337 422 y Fj(k)2380 407 y Fl(\)\))g Fi(\000)h Fk(\036)p Fl(\()p Fk(t;)17 b(\015)5 b Fl(\()p Fk(t)22 b Fi(\000)h Fk(n)3062 422 y Fj(k)3104 407 y Fl(\)\))3180 311 y Fe(\021)733 656 y Fl(=)k(lim)988 560 y Fe(\020)1038 656 y Fk(\036)p Fl(\()p Fk(t)1169 615 y Ff(0)1214 656 y Fi(\000)c Fk(n)1372 671 y Fj(k)1415 656 y Fk(;)17 b(\015)5 b Fl(\()p Fk(t)1588 615 y Ff(0)1633 656 y Fi(\000)23 b Fk(n)1791 671 y Fj(k)1833 656 y Fl(\)\))f Fi(\000)h Fk(\036)p Fl(\()p Fk(t)f Fi(\000)h Fk(n)2342 671 y Fj(k)2385 656 y Fk(;)17 b(\015)5 b Fl(\()p Fk(t)22 b Fi(\000)g Fk(n)2737 671 y Fj(k)2780 656 y Fl(\)\))2856 560 y Fe(\021)735 864 y Fl(=)27 b(lim)990 768 y Fe(\020)1040 864 y Fk(u)p Fl(\()p Fk(t)1169 823 y Ff(0)1214 864 y Fi(\000)c Fk(n)1372 879 y Fj(k)1415 864 y Fk(;)17 b(\015)5 b Fl(\()p Fk(t)1588 823 y Ff(0)1633 864 y Fi(\000)23 b Fk(n)1791 879 y Fj(k)1833 864 y Fl(\)\))f Fi(\000)h Fk(u)p Fl(\()p Fk(t)f Fi(\000)h Fk(n)2340 879 y Fj(k)2382 864 y Fk(;)17 b(\015)5 b Fl(\()p Fk(t)22 b Fi(\000)h Fk(n)2735 879 y Fj(k)2778 864 y Fl(\)\))2854 768 y Fe(\021)396 1115 y Fl(=)28 b(lim)652 997 y Fe(Z)735 1024 y Fj(t)760 1000 y Fd(0)698 1186 y Fj(t)804 1115 y Fk(L)p Fl(\()p Fk(\033)n(;)17 b(\015)5 b Fl(\()p Fk(\033)26 b Fi(\000)d Fk(n)1338 1130 y Fj(k)1381 1115 y Fl(\))p Fk(;)31 b Fl(_)-41 b Fk(\015)5 b Fl(\()p Fk(\033)26 b Fi(\000)c Fk(n)1795 1130 y Fj(k)1838 1115 y Fl(\)\))p Fk(d\033)31 b Fl(=)2155 997 y Fe(Z)2238 1024 y Fj(t)2263 1000 y Fd(0)2201 1186 y Fj(t)2307 1115 y Fk(L)p Fl(\()p Fk(\033)n(;)17 b(\015)2559 1130 y Ff(1)2634 1115 y Fl(\()p Fk(\033)t Fl(\))p Fk(;)31 b Fl(_)-41 b Fk(\015)2864 1130 y Ff(1)2937 1115 y Fl(\()p Fk(\033)t Fl(\)\))p Fk(d\033)n(:)106 1319 y Fl(As)38 b(a)f(consequence,)k(the)d(curv)m(e)h Fk(t)c Fi(\000)-16 b(!)35 b Fl(\()p Fk(t;)17 b(\015)1796 1334 y Ff(1)1871 1319 y Fl(\()p Fk(t)p Fl(\)\))37 b(is)g(asymptotic)i(to)e Fi(M)p Fl(,)h(so)f(that)g(there)106 1440 y(exists)29 b(an)d(increasing)i(sequences)h Fk(m)1470 1455 y Fj(k)1539 1440 y Fl(of)d(in)m(tegers)i(suc)m(h)g(that)e(that)g Fk(\015)2674 1455 y Ff(1)2749 1440 y Fl(\()p Fk(m)2872 1455 y Fj(k)2914 1440 y Fl(\))i Fi(\000)-16 b(!)27 b Fk(x)h Fi(2)g(M)3465 1455 y Fh(0)3505 1440 y Fl(.)106 1560 y(P)m(ossibly)35 b(taking)e(a)f(subsequence)k(of)d Fk(n)1591 1575 y Fj(k)1633 1560 y Fl(,)g(w)m(e)h(obtain)e(that)h Fk(\015)5 b Fl(\()p Fk(m)2531 1575 y Fj(k)2596 1560 y Fi(\000)22 b Fk(n)2753 1575 y Fj(k)2796 1560 y Fl(\))28 b Fi(\000)-16 b(!)27 b Fk(x)h Fi(2)g(M)p Fk(:)p 3475 1535 57 4 v 3475 1585 4 50 v 3528 1585 V 3475 1588 57 4 v 253 1801 a Fl(In)39 b(the)g(autonomous)f(case,)j(where)f(the)f (Hamiltonian)g Fk(H)46 b Fl(do)s(es)39 b(not)f(dep)s(end)i(on)e(the)106 1921 y(v)-5 b(ariable)40 b Fk(t)p Fl(,)h(the)e(ab)s(o)m(v)m(e)h (remarks)g(imply)h(that)d(an)m(y)i Fk(!)t Fl(-limit)f(viscosit)m(y)i (solution)f Fk(u)e Fl(is)i(in-)106 2042 y(dep)s(enden)m(t)d(of)d Fk(t)g Fl(on)g Fi(M)d Fl(=)f Fk(I)-25 b(R)24 b Fi(\002)g(M)1493 2057 y Fh(0)1532 2042 y Fl(.)49 b(One)34 b(can)h(then)g(conclude)h (that)e(the)g(solution)h Fk(u)f Fl(is)106 2162 y(indep)s(enden)m(t)h (of)d Fk(t)p Fl(.)253 2330 y(The)k(time-p)s(erio)s(dic)f(case)h(ho)m(w) m(ev)m(er)h(is)f(more)f(complicated,)i(and)e(w)m(e)h(are)f(not)g(able)g (to)106 2450 y(giv)m(e)29 b(a)e(description)i(of)f Fk(!)t Fl(-limit)e(orbits)i(without)h(using)f(some)g(sp)s(eci\014c)i(features) e(of)f(the)h(lo)m(w)106 2571 y(dimension.)46 b(This)33 b(will)h(b)s(e)e(done)h(in)g(the)g(next)h(section.)253 2691 y(Before)42 b(w)m(e)i(con)m(tin)m(ue,)i(let)d(us)g(giv)m(e)g(an)f (imp)s(ortan)m(t)h(remark.)73 b(All)43 b(the)f(ob)5 b(jects)44 b(con-)106 2811 y(structed)35 b(in)f(this)g(section,)h(the)f(sets)h Fi(D)h Fl(and)d Fi(M)g Fl(and)h(the)g(mappings)g Fk(S)2830 2775 y Fj(t)2859 2811 y Fl(,)g(dep)s(end)h(on)e(the)106 2932 y(p)s(erio)s(dic)39 b(solution)f Fk(\036)g Fl(that)g(w)m(as)h(c)m (hosen)g(in)f(the)h(b)s(eginning.)60 b(Let)38 b(us)h(note)f Fi(D)3087 2947 y Fj(\036)3171 2932 y Fl(and)g Fi(M)3486 2947 y Fj(\036)106 3052 y Fl(and)27 b Fk(S)356 3016 y Fj(t)350 3077 y(\036)422 3052 y Fl(in)g(order)f(to)g(emphasize)i(this)f (dep)s(endence.)44 b(If)26 b Fk( )k Fl(is)d(another)f(1-p)s(erio)s(dic) g(viscosit)m(y)106 3172 y(solution,)33 b(then)g(w)m(e)g(see)h(from)d (Corollary)i(3.1)f(that)g(the)g(orbits)h(of)e Fi(M)2727 3187 y Fj( )2812 3172 y Fl(are)h(calibrated)h(b)m(y)106 3293 y Fk(\036)p Fl(.)44 b(It)32 b(follo)m(ws)i(that)e(the)h(set)1584 3413 y Fi(A)27 b Fl(=)1795 3330 y Fe(\\)1820 3515 y Fj(\036)1904 3413 y Fi(D)1981 3428 y Fj(\036)2027 3413 y Fk(;)106 3664 y Fl(is)42 b(not)f(empt)m(y)-8 b(,)45 b(where)e(the)f(in)m (tersection)h(is)f(tak)m(en)g(on)g(the)f(set)h(of)f(1-p)s(erio)s(dic)g (viscosit)m(y)106 3784 y(solutions.)57 b(This)38 b(set)f(is)g(usually)h (called)g(the)f(Aubry)g(set.)57 b(The)37 b(mappings)h Fk(S)3064 3748 y Fj(t)3058 3811 y(\036)p Ff(jA)3176 3820 y Fb(0)3251 3784 y Fl(do)f(not)106 3904 y(dep)s(end)d(on)f Fk(\036)p Fl(,)f(and)h(for)f(all)g Fk(\036)p Fl(,)h(w)m(e)g(ha)m(v)m(e) h Fi(M)1778 3919 y Fj(\036)1851 3904 y Fi(\032)29 b(A)p Fl(.)106 4236 y Fp(4.)161 b(Rotation)53 b(n)l(um)l(b)t(er)g(and)g(con)l (v)l(ergence)106 4455 y Fl(In)42 b(this)g(section,)j(w)m(e)d(shall)g (tak)m(e)g(adv)-5 b(an)m(tage)42 b(of)f(the)g(lo)m(w)h(dimension.)72 b(More)41 b(precisely)-8 b(,)106 4575 y(w)m(e)39 b(shall)e(mak)m(e)i (use)f(of)f(P)m(oincar)m(\023)-46 b(e)38 b(theory)g(of)e (homeomorphisms)k(of)c(the)i(circle,)i(see)e([11])106 4696 y(for)g(example.)62 b(W)-8 b(e)38 b(ha)m(v)m(e)h(constructed)h(in) e(the)h(previous)h(section)f(a)f(closed)h(subset)h Fi(A)d Fl(of)106 4816 y Fk(I)-25 b(R)25 b Fi(\002)f Fo(T)p Fl(,)35 b(whic)m(h)h(is)f(the)g(disjoin)m(t)h(union)f(of)f(graphs)h(of)f (calibrated)h(curv)m(es.)52 b(These)37 b(curv)m(es)106 4937 y(ha)m(v)m(e)42 b(a)e(w)m(ell)i(de\014ned)g(rotation)e(n)m(um)m(b) s(er)i Fk(\032)f Fi(2)g Fk(I)-25 b(R)q Fl(.)67 b(Recall)41 b(that)f(this)h(rotation)f(n)m(um)m(b)s(er)1795 5291 y(7)p eop %%Page: 8 8 8 7 bop 106 76 a Fl(can)35 b(b)s(e)g(de\014ned)h(b)m(y)g(lifting)e(the) h(calibrated)g(curv)m(es)i(to)d(the)h(univ)m(ersal)i(co)m(v)m(er)f Fk(I)-25 b(R)35 b Fl(of)f Fo(T)p Fl(.)50 b(If)110 196 y(~)-53 b Fk(\015)34 b Fl(:)29 b Fk(I)-25 b(R)29 b Fi(\000)-16 b(!)28 b Fk(I)-25 b(R)34 b Fl(is)f(one)g(of)g(these)h(lifted)g(curv)m (es,)h(w)m(e)f(de\014ne)h Fk(\032)28 b Fl(=)h(lim)21 b(~)-53 b Fk(\015)5 b Fl(\()p Fk(t)p Fl(\))p Fk(=t)p Fl(.)45 b(This)34 b(n)m(um)m(b)s(er)106 316 y(do)s(es)f(not)g(dep)s (end)g(on)g(the)g(curv)m(e)h(c)m(hosen,)g(hence)g(it)e(dep)s(ends)i (only)f(on)f(the)h(Hamiltonian)106 437 y Fk(H)8 b Fl(.)106 604 y Fm(Pr)n(oof)52 b(of)g(the)e(theorem.)81 b Fl(Let)45 b Fk(u)p Fl(\()p Fk(t;)17 b(x)p Fl(\))45 b(b)s(e)g(an)g Fk(!)t Fl(-limit)f(viscosit)m(y)k(solution.)81 b(Let)106 724 y Fk(\036)p Fl(\()p Fk(t;)17 b(x)p Fl(\))37 b(=)f(lim)17 b(inf)24 b Fk(u)p Fl(\()p Fk(t)i Fl(+)f Fk(n;)17 b(x)p Fl(\))38 b(b)s(e)g(as)g(in)g(Prop)s(osition)g(2.1.)59 b(W)-8 b(e)38 b(will)h(pro)m(v)m(e)g(that)e Fk(u)f Fl(=)h Fk(\036)106 844 y Fl(in)g(the)g(t)m(w)m(o)g(cases)h Fk(\032)c Fl(=)g(0)i(and)g Fk(\032)h Fl(irrational.)54 b(The)37 b(general)g(case)g(of)f(a)g(rational)g(rotation)106 965 y(n)m(um)m(b)s(er)44 b Fk(\032)g Fl(=)f Fk(a=b)p Fl(,)i Fk(a)f Fi(6)p Fl(=)g(0)e(can)g(b)s(e)g(reduced)i(to)d(the)i(case)g Fk(\032)h Fl(=)f(0)f(b)m(y)h(considering)g(the)106 1085 y(Hamiltonian)1201 1180 y(~)1176 1206 y Fk(H)8 b Fl(\()p Fk(t;)17 b(x;)g(p)p Fl(\))27 b(=)h Fk(aH)8 b Fl(\()p Fk(at;)17 b(bx)23 b Fi(\000)f Fk(at;)2365 1138 y(p)p 2365 1182 49 4 v 2369 1274 a(b)2424 1206 y Fl(\))106 1406 y(and)45 b(noticing)g(that)f(the)h(function)f Fk(u)p Fl(\()p Fk(t;)17 b(x)p Fl(\))45 b(is)f(a)g(solution)h(of)f(the)h (equation)g(\(1.1\))f(with)106 1526 y(Hamiltonian)33 b Fk(H)39 b Fl(if)31 b(and)h(only)h(if)e(the)i(function)38 b(~)-55 b Fk(u)o Fl(\()p Fk(t;)17 b(x)p Fl(\))28 b(=)g Fk(u)p Fl(\()p Fk(at;)17 b(bx)k Fi(\000)g Fk(at)p Fl(\))32 b(is)g(a)g(solution)g(of)106 1647 y(the)e(equation)g(\(1.1\))f(with)h (Hamiltonian)1698 1622 y(~)1673 1647 y Fk(H)7 b Fl(,)30 b(and)g(that)f(the)g(rotation)g(n)m(um)m(b)s(er)i(asso)s(ciated)106 1767 y(to)i(this)g(second)h(equation)f(is)g Fk(\032)28 b Fl(=)g(0.)253 1888 y(In)41 b(view)i(of)d(Lemma)i(3.2,)h(it)e(is)h (enough)f(to)g(pro)m(v)m(e)i(equalit)m(y)f(on)f Fi(M)2888 1903 y Fj(\036)2934 1888 y Fl(.)69 b(Let)42 b(us)f(note)106 2008 y Fk(d)28 b Fl(=)f Fk(u)22 b Fi(\000)h Fk(\036)p Fl(.)43 b(This)34 b(function)f(is)g(di\013eren)m(tiable)h(on)f Fi(M)2138 2023 y Fj(\036)2216 2008 y Fl(and)f(satis\014es)1511 2217 y Fk(@)1562 2232 y Fj(t)1592 2217 y Fk(d)27 b Fl(=)h Fk(@)1825 2232 y Fj(x)1869 2217 y Fk(d)g Fl(=)f(0)p Fk(:)106 2426 y Fl(Recall)j(that)f(there)g(exists)i(a)d(constan)m(t)i Fk(K)36 b Fl(suc)m(h)31 b(that)d(the)i(function)f Fk(d)p Fl(\()p Fk(t;)17 b(:)p Fl(\))29 b(is)g Fk(K)7 b Fl(-Lipsc)m(hitz)106 2546 y(for)32 b(eac)m(h)i Fk(t)p Fl(.)106 2666 y Fo(Case)k(1.)43 b Fk(\032)28 b Fl(=)g(0.)43 b(In)33 b(this)f(case,)i Fi(M)1470 2681 y Fh(0)1541 2666 y Fl(is)e(a)g(union)h(of)e(\014xed)j(p) s(oin)m(ts)e(of)g Fk(S)2805 2630 y Fh(1)2844 2666 y Fl(,)g(or)g(equiv) -5 b(alen)m(tly)106 2787 y(the)47 b(orbits)g(in)f Fi(M)g Fl(are)g(1-p)s(erio)s(dic.)84 b(The)47 b(function)g Fk(d)f Fl(is)g(constan)m(t)h(on)g(eac)m(h)g(of)e(these)106 2907 y(p)s(erio)s(dic)24 b(orbits,)h(so)f(that)e Fk(d)p Fl(\()p Fk(n;)17 b(x)p Fl(\))28 b(=)f Fk(d)p Fl(\(0)p Fk(;)17 b(x)p Fl(\))23 b(for)f(eac)m(h)i Fk(x)29 b Fi(2)f(M)2439 2922 y Fh(0)2478 2907 y Fl(.)40 b(It)23 b(follo)m(ws)h(that)f Fk(d)p Fl(\(0)p Fk(;)17 b(x)p Fl(\))27 b(=)106 3028 y(0)33 b(since)h(lim)17 b(inf)23 b Fk(d)p Fl(\()p Fk(n;)17 b(x)p Fl(\))28 b(=)g(0.)43 b(as)33 b(a)f(consequence,)k Fk(d)27 b Fl(=)h(0)k(on)g Fi(M)p Fl(.)106 3195 y Fo(Case)54 b(2.)85 b Fk(\032)46 b Fl(is)h(irrational.)84 b(The)48 b(pro)s(of)d(that)h Fk(d)p Fi(j)2064 3210 y Ff(M)2202 3195 y Fl(=)51 b(0)46 b(is)h(similar)g(to)f(the)h(pro)s(of)e(of)106 3315 y(Prop)s(osition)f (6.5.)75 b(in)43 b([4].)75 b(Let)44 b(us)g(lift)f(the)g(set)h Fi(M)f Fl(to)g(the)g(univ)m(ersal)i(co)m(v)m(er)g Fk(I)-25 b(R)30 b Fi(\002)g Fk(I)-25 b(R)q Fl(,)106 3435 y(as)45 b(w)m(ell)g(as)g(the)g(function)f Fk(d)p Fl(.)79 b(The)45 b(set)g Fi(M)f Fl(is)h(a)f(disjoin)m(t)h(union)f(of)g(graphs)h(of)f (curv)m(es)110 3556 y(~)-53 b Fk(\015)5 b Fl(\()p Fk(t)p Fl(\))30 b(:)g Fk(I)-25 b(R)30 b Fi(\000)-16 b(!)29 b Fk(I)-25 b(R)q Fl(,)33 b(whic)m(h)i(are)f(the)g(liftings)g(of)f (calibrated)h(curv)m(es)i(in)e Fo(T)p Fl(.)46 b(The)35 b(function)f Fk(d)106 3676 y Fl(is)g(constan)m(t)g(on)f(eac)m(h)i(of)d (these)j(graphs.)46 b(Let)33 b(us)h(set)g Fk(d)2178 3691 y Fh(0)2246 3676 y Fl(=)28 b Fk(d)p Fl(\(0)p Fk(;)17 b(:)p Fl(\))28 b(:)h Fk(I)-25 b(R)30 b Fi(\000)-16 b(!)28 b Fk(I)-25 b(R)q Fl(.)45 b(W)-8 b(e)34 b(w)m(an)m(t)106 3797 y(to)39 b(pro)m(v)m(e)i(that)e Fk(d)771 3812 y Fh(0)p Ff(jM)910 3821 y Fb(0)987 3797 y Fl(is)h(constan)m(t.)64 b(In)39 b(order)g(to)g(do)g(so,)i(let)f(us)g(consider)g(a)f(connected) 106 3917 y(comp)s(onen)m(t)f(])p Fk(x;)17 b(y)t Fl([)35 b(of)h(the)h(complemen)m(t)h(of)e Fi(M)1919 3932 y Fh(0)1994 3917 y Fl(in)h Fk(I)-25 b(R)q Fl(,)37 b(and)f(let)k(~)-52 b Fk(\015)2658 3932 y Fj(x)2737 3917 y Fl(and)40 b(~)-52 b Fk(\015)2982 3932 y Fj(y)3057 3917 y Fl(:)34 b Fk(I)-25 b(R)34 b Fi(\000)-16 b(!)34 b Fk(I)-25 b(R)106 4037 y Fl(b)s(e)34 b(the)g(liftings)g(of)f(the)h(calibrated)g(curv)m(es)h(suc) m(h)g(that)i(~)-53 b Fk(\015)2253 4052 y Fj(x)2297 4037 y Fl(\(0\))28 b(=)h Fk(x)34 b Fl(and)j(~)-52 b Fk(\015)2886 4052 y Fj(y)2927 4037 y Fl(\(0\))28 b(=)h Fk(y)t Fl(.)45 b(Since)106 4158 y(inf)24 b Fi(j)t Fl(~)-53 b Fk(\015)320 4173 y Fj(x)363 4158 y Fl(\()p Fk(t)p Fl(\))22 b Fi(\000)27 b Fl(~)-53 b Fk(\015)647 4173 y Fj(y)688 4158 y Fl(\()p Fk(t)p Fl(\))p Fi(j)28 b Fl(=)f(0,)33 b(and)f(since)543 4367 y Fi(j)p Fk(d)622 4382 y Fh(0)661 4367 y Fl(\()p Fk(y)t Fl(\))21 b Fi(\000)h Fk(d)960 4382 y Fh(0)1000 4367 y Fl(\()p Fk(x)p Fl(\))p Fi(j)27 b Fl(=)h Fi(j)p Fk(d)p Fl(\()p Fk(t;)20 b Fl(~)-52 b Fk(\015)1537 4382 y Fj(y)1577 4367 y Fl(\()p Fk(t)p Fl(\)\))22 b Fi(\000)h Fk(d)p Fl(\()p Fk(t;)d Fl(~)-52 b Fk(\015)2067 4382 y Fj(x)2110 4367 y Fl(\()p Fk(t)p Fl(\)\))p Fi(j)27 b(\024)h Fk(K)7 b Fi(j)t Fl(~)-53 b Fk(\015)2588 4382 y Fj(x)2632 4367 y Fl(\()p Fk(t)p Fl(\))22 b Fi(\000)k Fl(~)-52 b Fk(\015)2916 4382 y Fj(y)2957 4367 y Fl(\()p Fk(t)p Fl(\))p Fi(j)106 4575 y Fl(for)46 b(eac)m(h)h Fk(t)p Fl(,)j(w)m(e)d(ha)m(v)m(e) g Fk(d)1060 4590 y Fh(0)1100 4575 y Fl(\()p Fk(y)t Fl(\))i(=)i Fk(d)1455 4590 y Fh(0)1494 4575 y Fl(\()p Fk(x)p Fl(\).)84 b(Hence)48 b(there)f(exists)g(a)f(con)m(tin)m(uous)i(function)106 4696 y Fk(f)39 b Fl(:)28 b Fk(I)-25 b(R)28 b Fi(\000)-16 b(!)27 b Fk(I)-25 b(R)31 b Fl(whic)m(h)h(is)e(equal)h(to)f Fk(d)1474 4711 y Fh(0)1543 4696 y Fl(on)g Fi(M)1796 4711 y Fh(0)1835 4696 y Fl(,)h(and)f(whic)m(h)h(is)g(constan)m(t)g(on)f(all) g(connected)106 4816 y(comp)s(onen)m(ts)41 b(of)e(the)h(complemen)m(t)h (of)e Fi(M)1732 4831 y Fh(0)1771 4816 y Fl(.)64 b(This)41 b(function)f(is)f(di\013eren)m(tiable)i(and)f(sat-)106 4937 y(is\014es)e Fk(f)403 4900 y Ff(0)460 4937 y Fl(=)c(0)j(on)f(eac)m (h)h(p)s(oin)m(t)g(of)e(the)i(complemen)m(t)i(of)d Fi(M)2347 4952 y Fh(0)2386 4937 y Fl(,)h(but)g(also)f(on)g(eac)m(h)i(p)s(oin)m(t) e(of)1795 5291 y(8)p eop %%Page: 9 9 9 8 bop 106 76 a Fi(M)226 91 y Fh(0)295 76 y Fl(since)32 b Fk(d)583 39 y Ff(0)583 100 y Fh(0)649 76 y Fl(=)c(0)i(there.)43 b(As)30 b(a)g(consequence,)j(it)d(is)h(constan)m(t,)g(hence)g Fk(d)2841 91 y Fh(0)p Ff(jM)2980 100 y Fb(0)3048 76 y Fl(is)g(constan)m(t,)106 196 y(hence)i Fk(d)427 211 y Ff(j)p Fa(T)539 196 y Fl(is)f(constan)m(t,)h(this)f(constan)m(t)g(has)g (to)g(b)s(e)f(zero)h(since)h(lim)17 b(inf)24 b Fk(d)p Fl(\()p Fk(t)c Fl(+)g Fk(n;)d(x)p Fl(\))28 b(=)f(0.)p 3475 171 57 4 v 3475 221 4 50 v 3528 221 V 3475 224 57 4 v 106 536 a Fo(References)204 740 y Fl([1])49 b Fm(G.)41 b(Barles,)h(B.)f(Per)-7 b(thame)p Fl(,)38 b(Discon)m(tin)m(uous)g (solutions)f(of)f(deterministic)j(op-)356 860 y(timal)32 b(stopping)i(time)f(problems,)h(M2AN,)f Fo(21)f Fl(\(1987\),)g (557{579.)204 1063 y([2])49 b Fm(G.)39 b(Barles,)h(P.E.)g(Souganidis)p Fl(,)35 b(On)g(the)g(large)g(time)h(b)s(eha)m(viour)f(of)g(solutions) 356 1184 y(of)29 b(Hamilton-Jacobi)h(equations,)i(SIAM)e(J.)g(Math.)g (Anal.)g(31,)g Fo(4)g Fl(\(2000\),)f(925{939.)204 1387 y([3])49 b Fm(G.)35 b(Barles,)g(P.E.)f(Souganidis)p Fl(,)e(Some)f(coun) m(terexamples)i(on)d(the)h(asymptotic)356 1508 y(b)s(eha)m(viour)j(of)f (the)g(solutions)i(of)d(Hamilton-Jacobi)i(equations,)g(C.)g(R.)f(Acad.) h(Sci.,)356 1628 y(P)m(aris,)f(S)m(\023)-46 b(erie)34 b(I,)f Fo(330)p Fl(,)f(\(2000\),)g(963{968.)204 1831 y([4])49 b Fm(P.)d(Bernard)p Fl(,)e(Connecting)e(orbits)g(of)f(time)g (dep)s(enden)m(t)j(Lagrangian)c(systems.)356 1952 y(Ann.)33 b(Inst.)g(F)-8 b(ourier)33 b Fo(52)f Fl(\(2002\),)g(1533{1568.)204 2155 y([5])49 b Fm(W.)42 b(E)p Fl(,)36 b(Aubry-Mather)h(theory)g(and)g (p)s(erio)s(dic)g(solutions)g(of)f(the)h(forced)g(Burgers)356 2276 y(equation.)c(Comm.)h(Pure)f(Appl.)g(Math.)g Fo(52)g Fl(\(1999\),)e(811{828.)204 2479 y([6])49 b Fm(A.)g(F)-12 b(a)-7 b(thi)p Fl(,)47 b(Th)m(\023)-46 b(eor)m(\022)g(eme)45 b(KAM)f(faible)f(et)h(th)m(\023)-46 b(eorie)44 b(de)g(Mather)g(sur)g (les)g(syst)m(\022)-46 b(emes)356 2599 y(lagrangiens,)33 b(C.)g(R.)f(Acad.)h(Sci.)h(P)m(aris,)f(S)m(\023)-46 b(erie)34 b(I,)e Fo(324)h Fl(\(1997\),)f(1043-1046.)204 2803 y([7])49 b Fm(A.)36 b(F)-12 b(a)-7 b(thi)p Fl(,)33 b(Sur)f(la)g(con)m(v)m (ergence)j(du)d(semi-group)s(e)g(de)g(Lax-Oleinik.)h(C.)f(R.)g(Acad.) 356 2923 y(Sci.,)h(P)m(aris,)h(S)m(\023)-46 b(erie)33 b(I,)g Fo(327)p Fl(,\(1998\),)f(267-270.)204 3127 y([8])49 b Fm(A.)36 b(F)-12 b(a)-7 b(thi)38 b(and)f(E.)f(Maderna)p Fl(,)31 b(W)-8 b(eak)32 b(KAM)g(theorem)h(on)e(non)h(compact)g(man-)356 3247 y(ifolds,)h(to)f(app)s(ear,)h(\(2002\).)204 3450 y([9])49 b Fm(A.)43 b(F)-12 b(a)-7 b(thi)45 b(and)f(J.N.)g(Ma)-7 b(ther)p Fl(,)39 b(F)-8 b(ailure)38 b(of)g(con)m(v)m(ergence)i(of)e (the)h(Lax-Oleinik)356 3571 y(semi-group)56 b(in)g(the)h(time-dep)s (enden)m(t)h(case,)k(Bull.)57 b(So)s(c.)e(Math.)i(F)-8 b(rance,)62 b Fo(128)356 3691 y Fl(\(2000\),)31 b(473-483.)155 3895 y([10])49 b Fm(G.)37 b(Contreras;)g(J.)h(Delgado;)g(R.)f(Iturria)n (ga)c Fl(Lagrangian)f(\015o)m(ws:)44 b(the)33 b(dy-)356 4015 y(namics)41 b(of)g(globally)f(minimizing)i(orbits.)f(I)s(I.)g (Bol.)g(So)s(c.)f(Brasil.)h(Mat.)g(\(N.S.\))g(28)356 4135 y(\(1997\),)31 b(no.)i(2,)f(155{196.)155 4339 y([11])49 b Fm(A.)43 b(Ka)-7 b(tok)45 b(and)f(B.)f(Hasselbla)-7 b(tt)p Fl(,)39 b Fg(Intr)-5 b(o)g(duction)39 b(to)i(the)f(mo)-5 b(dern)39 b(the)-5 b(ory)40 b(of)356 4459 y(dynamic)-5 b(al)33 b(systems.)e Fl(Encyclop)s(edia)j(of)d(Mathematics)i(and)f(its) g(Applications,)h(54.)356 4579 y(Cam)m(bridge)h(Univ)m(ersit)m(y)h (Press,)f(Cam)m(bridge,)g(\(1995\).)155 4783 y([12])49 b Fm(J.)35 b(N.)g(Ma)-7 b(ther)p Fl(,)30 b(Action)h(minimizing)g(in)m (v)-5 b(arian)m(t)31 b(measures)h(for)e(p)s(ositiv)m(e)h(de\014nite)356 4903 y(Lagrangian)h(systems,)j(Math.)e(Z.)f Fo(207)p Fl(,)h(169{207)d(\(1991\))1795 5291 y(9)p eop %%Page: 10 10 10 9 bop 155 76 a Fl([13])49 b Fm(R.)42 b(Ma)673 68 y(~)670 76 y(ne)p Fl(,)c(Lagrangian)e(\015o)m(ws:)54 b(the)38 b(dynamics)h(of)e(globally)g(minimizing)i(orbits.)356 196 y(I.)33 b(Bol.)f(So)s(c.)h(Brasil.)g(Mat.)g(\(N.S.\))f Fo(28)h Fl(\(1997\),)f(no.)g(2,)h(141{153.)155 399 y([14])49 b Fm(P.-L.)60 b(Lions)p Fl(,)h Fg(Gener)-5 b(alize)g(d)54 b(solutions)h(of)g(Hamilton-Jac)-5 b(obi)54 b(Equations,)60 b Fl(Re-)356 520 y(searc)m(h)34 b(Notes)f(in)g(Mathematics,)h(Pitman.) 155 723 y([15])49 b Fm(P.-L.)31 b(Lions,)i(G.)f(P)-9 b(ap)i(anicola)n(ou,)34 b(S.R.S)d(V)-12 b(aradhan)p Fl(,)58 b(Homogenization)28 b(of)356 843 y(Hamilton-Jacobi)k(equations;)i (unpublished)h(preprin)m(t.)155 1047 y([16])49 b Fm(G.)38 b(Namah,)i(J.-M.)e(R)m(oquejoffre,)69 b Fl(Remarks)34 b(on)g(the)g(long)f(time)i(b)s(eha)m(viour)356 1167 y(of)46 b(the)h(solutions)h(of)f(Hamilton-Jacobi)g(Equations,)k(Comm)m(un.)e(P) m(artial)e(Di\013er.)356 1288 y(Equations)34 b Fo(24)e Fl(\(1999\),)g(883-893)e(.)155 1491 y([17])49 b Fm(M.)32 b(Pierre,)h(E.)f(R)m(ouy,)59 b Fl(A)28 b(tridimensional)h(in)m(v)m (erse)i(shaping)e(problem,)h(Comm.)356 1611 y(P)m(artial)j(Di\013.)f (Eq.,)h Fo(21)g Fl(\(1996\),)e(1279-1305.)155 1815 y([18])49 b Fm(J.-M.)37 b(R)m(oquejoffre,)65 b Fl(Con)m(v)m(ergence)35 b(to)e(steady)g(states)h(or)e(p)s(erio)s(dic)h(solutions)356 1935 y(in)i(a)g(class)i(of)d(Hamilton-Jacobi)i(equations,)h(J.)e(Math.) h(Pures)h(Appl.,)f Fo(80)g Fl(\(2001\),)356 2056 y(85-104.)155 2259 y([19])49 b Fm(J.-M.)106 b(R)m(oquejoffre,)115 b Fl(Comp)s(ortemen)m(t)99 b(asymptotique)h(des)f(solutions)356 2379 y(d')m(\023)-46 b(equations)53 b(de)g(Hamilton-Jacobi)f(mono)s (dimensionnelles,)61 b(C.)52 b(R.)g(Acad.)h(Sci.)356 2500 y(P)m(aris,)33 b Fo(326)p Fl(,)g(S)m(\023)-46 b(erie)34 b(I,)e(185-189)f(\(1998\).)1770 5291 y(10)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0303200902948--