Content-Type: multipart/mixed; boundary="-------------0902020444540" This is a multi-part message in MIME format. ---------------0902020444540 Content-Type: text/plain; name="09-16.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="09-16.comments" 5 pages ---------------0902020444540 Content-Type: text/plain; name="09-16.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="09-16.keywords" Duality, Non-smooth functional ---------------0902020444540 Content-Type: application/x-tex; name="Hikmet.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Hikmet.tex" % Generated by GrindEQ Word-to-LaTeX 2008 % ========== UNREGISTERED! ========== Please register! ========== % LaTeX/AMS-LaTeX \documentclass{article} %%% remove comment delimiter ('%') and specify encoding parameter if required, %%% see TeX documentation for additional info (cp1252-Western,cp1251-Cyrillic) %\usepackage[cp1252]{inputenc} %%% remove comment delimiter ('%') and select language if required %\usepackage[english,spanish]{babel} \usepackage{amssymb} \usepackage{amsmath} \usepackage[dvips]{graphicx} \numberwithin{equation}{section} %%% remove comment delimiter ('%') and specify parameters if required %\usepackage[dvips]{graphics} \begin{document} %%% remove comment delimiter ('%') and select language if required %\selectlanguage{spanish} \begin{center} \textbf{H.S. AKHUNDOV}\end{center} \begin{center} \textbf{Minmization of the non-smooth functional using a adjoint function }\end{center} \noindent \textbf{} \noindent \textit{ In the paper} \textit{we obtain} a \textit{necessary extremum condition in the non-smooth problem of} \textit{ variational calculus } \[J(u)=\frac{\alpha }{2} \int _{\Omega }\left|grad\, u(x)\right|^{2} dx+\beta \int _{\Omega }\left|grad\, u(x)\right|dx-\int _{\Omega }f(x)u(x)dx \to \mathop{\inf }\limits_{u\in H^{1} _{0} (\Omega )}\] \textbf{\textit{ Key words}: }\textit{ duality, non-smooth functional.\underbar{}} \noindent \textit{} \section{Introduction } This problem has been considered in [1, ñ.95-100] and is called the Mosolov problem . In the paper [1] a duality problem is received by two ways. In the present paper we consider another duality problem, obtain a necessary and sufficient exstremum condition. \section{ Problem statement } Let $f\in L_{2} (\Omega )$ be a given function , $\Omega \in C^{1} \, \, $be an open and bounded set,$\alpha $and$\beta $be positive constants, $u_{x} (x)=(u_{x_{1} } (x),...,u_{x_{n} } (x))=grad\, \, u(x).$ \noindent Let' s consider the following problem : to minimize the functional \begin{equation} \label{GrindEQ__2_1_} J(u)=\frac{\alpha }{2} \left[\int _{\Omega }\left|grad\, u(x)\right|^{2} dx+\beta \int _{\Omega }\left|grad\, u(x)\right|dx-\int _{\Omega }f(x)u(x)dx \right] \end{equation} \noindent on $ H^{1} _{0} (\Omega )$ \section{ Problem solution} Introduce \textbf{ } a function $\Phi :H_{0}^{1} (\Omega )\times H_{0}^{1} (\Omega )\to \overline{R}$ , such that \[\Phi (u,\vartheta )=\frac{\alpha }{2} \int _{\Omega }\left|grad(\, \, u(x)+\vartheta (x))\right|^{2} dx+\beta \int _{\Omega }\left|grad\, \, (u(x)+\vartheta (x))\right|dx-\int _{\Omega }f(x)u(x)dx. \] Clearly \[\Phi (u,0)=J(u)=\frac{\alpha }{2} \int _{\Omega }\left|grad\, u(x)\right|^{2} dx+\beta \int _{\Omega }\left|grad\, u(x)\right|dx-\int _{\Omega }f(x)u(x)dx. \] For any $p\in Y=H_{0}^{1} (\Omega )$ consider the minimization problem \[\mathop{\inf }\limits_{u\in V} \Phi (u,p),\, \, \, \, \, V=H_{0}^{1} (\Omega ) (P_{p} ).\] Clearly, for $p=0$ the problem $P_{0} $ is none other than the problem$P$. \noindent The set of the problems ${\rm P} _{p} $ is said to be perturbation of the problem$P$. Now, by the given problem $P$ and its perturbation $P_{p} $we can give definition of the dual problem. To do this we consider the function$\Phi ^{*} \in \Gamma \left(V^{*} \times Y^{*} \right)$, conjugated to $\Phi $ in duality between $H_{0}^{1} (\Omega )\times H_{0}^{1} (\Omega )$ and $(H_{0}^{1} (\Omega ))^{*} \times (H_{0}^{1} (\Omega ))^{*} $. The problem \[\mathop{\sup }\limits_{\vartheta ^{*} \in Y^{*} } \left\{-\Phi ^{*} (0,\vartheta ^{*} )\right\}\] is called , to be dual to $P$ with respect to the given function$\Phi $. In the problem $P^{*} $ the upper bound will be denoted by $\sup P^{*} $, and any element $\vartheta ^{*} $ on$\left(H_{0}^{1} (\Omega )\right)^{*} $, for which \begin{equation} \label{GrindEQ__2_2_} -\Phi ^{*} (0,\vartheta ^{*} )=\sup P^{*} \end{equation} will be called a solution of the problem $P^{*} $. \noindent Clearly \[\begin{array}{l} {\Phi ^{*} (0,\vartheta ^{*} )=\mathop{\mathop{\sup }\limits_{u\in H_{0}^{'} (\Omega )} }\limits_{\vartheta \in H_{0}^{'} (Q)} \left\{\int _{\Omega }\vartheta (x)y_{0} (x)dx+\int _{\Omega }<\vartheta _{x} (x),y(x)>dx-\frac{\alpha }{2} \int _{\Omega }\left|grad\, (u(x)+\vartheta (x))\right|^{2} dx -\, \right. } \\ {-\beta \int _{\Omega }\left|grad\, (u(x)+\vartheta (x))\right|dx +\left. \int _{\Omega }f(x)u(x)dx \right\},} \end{array}\] where, $y_{0} (\cdot )\in L_{2} (\Omega ),\, \, y_{i} (\cdot )\in L_{2} (\Omega )$, $u_{x} (x)=(u_{x_{1} } (x),...,u_{x_{n} } (x))=grad\, \, u(x).$ Defining $z(x)=u(x)+\vartheta (x)$, we have \[\begin{array}{l} {\Phi ^{*} (0,\vartheta ^{*} )=\mathop{\mathop{\sup }\limits_{u\in H_{0}^{'} (\Omega )} }\limits_{z\in H_{0}^{'} (\Omega )} \left\{\int _{\Omega }(z(x)-u(x))y_{0} (x)dx+\int _{\Omega }dx- \right. } \\ {-\frac{\alpha }{2} \int _{\Omega }\left|z_{x} (x)\right|^{2} dx -\, \beta \int _{\Omega }\left|z_{x} (x)\right|dx+ \left. \int _{\Omega }f(x)u(x)dx \right\}=} \end{array}\] \[\begin{array}{l} {=\mathop{\sup }\limits_{z\in H_{0}^{'} (\Omega )} \left\{\int _{\Omega }z(x)y_{0} (x)dx+\int _{\Omega }dx-\left. \frac{\alpha }{2} \int _{\Omega }\left|z_{x} (x)\right|^{2} dx-\beta \int _{\Omega }\left|z_{x} (x)\right|dx \right\} \right. +} \\ {+\mathop{\sup }\limits_{u\in H_{0}^{'} (\Omega )} \left\{-\int _{\Omega }u(x)y_{0} (x)dx-\int _{\Omega }dx+\int _{\Omega }f(x)u(x)dx .\right. } \end{array}\] Puting $M=\left\{(\right. u(x),u_{x} (x)):u(\cdot )\in H_{0}^{1} (\Omega \left. )\right\}$, we get \[\begin{array}{l} {S(y_{0} ,y)=\mathop{\sup }\limits_{z\in H_{0}^{'} (\Omega )} \left\{\begin{array}{l} {\int _{\Omega } z(x)y_{0} (x)dx+\int _{\Omega }dx-\frac{\alpha }{2} \int _{\Omega }\left|z_{x} (x)\right|^{2} dx-\left. \beta \int _{\Omega }\left|z_{x} (x)\right|dx \right\} \, =} \\ {} \end{array}\right. \, \, } \\ {=\mathop{\sup }\limits_{(\omega _{0} ,\omega )\in L_{2}^{n+1} (\Omega )} \left\{\int _{\Omega }\omega _{0} (x)y_{0} (x)dx+\int _{\Omega }<\omega (x),y(x)>dx- \right. } \end{array}\] \[\begin{array}{l} {-\left. \frac{\alpha }{2} \int _{\Omega }\left|\omega (x)\right|^{2} dx-\beta \int _{\Omega }\left|\omega (x)\right| dx+\delta _{M} (\omega _{0} (x),\omega (\cdot )) \right\}=} \\ {=\mathop{\sup }\limits_{(\omega _{0} ,\omega )\in L_{2}^{n+1} (\Omega )} \left\{\int _{\Omega }\omega _{0} (x)(y_{0} (x)-h_{0} (x))dx+ \right. } \\ {+\left. \int _{\Omega }<\, \omega (x),y(x)-h(x)>dx-\frac{\alpha }{2} \int _{\Omega }\left|\omega (x)\right|^{2} dx-\beta \int _{\Omega }\left|\omega (x)\right| dx \right\}+} \\ {+\mathop{\sup }\limits_{u\in H_{0}^{'} (\Omega )} \left\{\int _{\Omega }u(x)h_{0} (x)dx+\int _{\Omega }dx \right\}.} \end{array}\] From $\mathop{\sup }\limits_{u\in H_{0}^{'} (\Omega )} \left\{\int _{\Omega }u(x)h_{0} (x)dx+\int _{\Omega }dx \right\}<+\infty $ follows , that \noindent $h_{0} (x)=\sum _{i=1}^{n}h_{i_{x_{i} } } (x)=div\, \, h(x) $. So ,$(h_{0} ,h)\in L_{2}^{n+1} (\Omega ),$ we have ,that î $h_{0} (\cdot )=div\, \, h\in L_{2} (\Omega )$ and $-\left(\sum _{i=1}^{n}h_{i} \nu _{i} \right)(s)=0$ for $S\in \partial \Omega $ (ñì.[3, ñ.53]). Therfore \[\begin{array}{l} {S(y_{0} ,y)=\mathop{\sup }\limits_{(\omega _{0} ,\omega )\in L_{2}^{n+1} (\Omega )} \left\{\int _{\Omega } \omega _{0} (x)(y_{0} (x)-div\, h(x))dx+\int _{\Omega }<\omega (x),y(x)-h(x)>dx- \right. } \\ {-\left. \frac{\alpha }{2} \int _{\Omega }\left|\omega (x)\right|^{2} dx-\, \beta \int _{\Omega }\left|\omega (x)\right|dx \right\}.} \end{array}\] Then from $S(y_{0} ,y)<+\infty $ follows that $y_{0} (x)=div\, \, h(x)$ and there exists such $l(\cdot )\in L_{2}^{n} (\Omega )$ that \[S(y_{0} ,y)=\mathop{\sup }\limits_{\omega \in L_{2}^{n} (\Omega )} \left\{\int _{\Omega }<\omega (x),y(x)-h(x)-l(x)>dx-\left. \frac{\alpha }{2} \int _{\Omega }\left|\omega (x)\right|^{2} dx \right\} \right. +\] \[\begin{array}{l} {+\mathop{\sup }\limits_{\omega \in L_{2}^{n} (\Omega )} \left\{\int _{\Omega }< \, \omega (x),l(x)>dx-\beta \int _{\Omega }\left|\omega (x)\right| dx\right\}=} \\ {=\frac{\alpha }{2} \int _{\Omega }\left|\frac{y(x)-h(x)-l(x)}{\alpha } \right|^{2} dx+\beta \int _{\Omega }\delta _{B} \left(\frac{l(x)}{\beta } \right) dx=} \\ {=\frac{1}{2\alpha } \int _{\Omega }\left|y(x)-h(x)-l(x)\right|^{2} dx +\beta \int _{\Omega }\delta _{B} \left(\frac{l(x)}{\beta } \right)dx ,} \end{array}\] $B$-unit ball . \noindent Simulary we have, that \[\begin{array}{l} {\mathop{\sup }\limits_{u\in H_{0}^{'} (\Omega )} \left\{-\int _{\Omega }u(x)y_{0} (x)dx-\int _{\Omega }dx+\int _{\Omega }f(x)u(x)dx \right\}=} \\ {=\left\{\begin{array}{l} {0:\, \, div\, y(x)-y_{0} (x)+f(x)=0\, \, \, \, \, \, \, \, \, almost\,\,\,\, everywhere\, \, \, x\in \Omega } \\ {+\infty :div\, y(x)-y_{0} (x)+f(x)\ne 0\, \, \, \, \, \, \, almost\,\,\,\, everywhere\, \, \, \, \, x\notin \Omega } \end{array}\right. =} \\ {=\, \left\{\begin{array}{l} {0:\, \, div\, y(x)-div\, h(x)+f(x)=0\, \, \, \, \, \, \, \, \, \, almost \,\,\,\, everywhere\, \, \, x\in \Omega } \\ {+\infty :div\, y(x)-div\, h(x)+f(x)\ne 0\, \, \, \, \, \, almost \,\,\,\, everywhere \, \, \, \, x\notin \Omega ,} \end{array}\right. } \end{array}\] where $\left(\sum _{i=1}^{n}y_{i} \nu _{i} \right)(s)=0$. \noindent We prove , that the problem $\, \, \, \, \, \mathop{\inf J(u)}\limits_{u\in H_{0}^{'} (\Omega )} $ is stable. \noindent Puting $h(z)=\inf \left\{\Phi (u,z)):u\in H_{0}^{1} (\Omega )\right\}$. \noindent Clearly \[\begin{array}{l} {h(z)\le \Phi (0,z)=\frac{\alpha }{2} \int _{\Omega }\left|z_{x} (x)\right|^{2} dx+\beta \int _{\Omega }\left|z_{x} (x)\right|dx\le } \\ {\le \frac{\alpha }{2} \int _{\Omega }\left|z_{x} (x)\right|^{2} dx+\beta (mes\Omega )^{\frac{1}{2} } \left(\int _{\Omega }\left|z_{x} (x)\right|^{2} dx \right) ^{\frac{1}{2} } \le \frac{\alpha }{2} +\beta \sqrt{mes\Omega } ,} \end{array}\] by $z\in \left\{w\in H_{0}^{'} (\Omega ):\left\| w\right\| \le 1\right\}$. \noindent From the theorem 2.1[2,ñ.30-33] follows, that $h$ subdifferentiable at the point zero . From remark 3.2.3 [1, ñ.60] and from corollary 3.2.4[1, ñ.62] follows that all solutions $\bar{u}(\cdot )$ of problem $\, \mathop{\inf J(u)}\limits_{u\in H_{0}^{'} (\Omega )} $ and all solutions $\bar{\vartheta }^{*} =(y_{0} ,y)$of the problems$\mathop{\sup }\limits_{\vartheta ^{*} \in H_{0}^{'} (\Omega )^{*} } \left\{-\Phi ^{*} (0,\vartheta ^{*} )\right\}$ are releted with extremal expressions \[\Phi (\bar{u},0)+\Phi ^{*} (0,\bar{\vartheta }^{*} )=0.\] From inequality Fenkhel follows , that \noindent \noindent Hense is follows ,that \[\begin{array}{l} {\frac{\alpha }{2} \int _{\Omega }\left|\bar{u}_{x} (x)\right|^{2} dx+\frac{1}{2\alpha } \int _{\Omega }\left|y(x)-h(x)-l(x)\right|^{2} dx+\beta \int _{\Omega }\left|\bar{u}_{x} (x)\right|dx+ \, \, \, \beta \int _{\Omega }\delta _{B} \left(\frac{l(x)}{\beta } \right) dx\ge } \\ {\ge \int _{\Omega }<\bar{u}_{x} (x),y(x)-h(x)>dx .} \end{array}\] As $div\, y(x)-y_{0} (x)+f(x)=0$ almost everywhere in $ \Omega $ and $y_{0} (x)=div\, h(x)$, we obtain that \noindent $\int _{\Omega }<\bar{u}_{x} (x),h(x)>dx \, \, \, -\int _{\Omega }<\bar{u}_{x} (x),y(x)>dx+\int _{\Omega }f(x)\bar{u}(x)dx=0 $,i.e., \[\int _{\Omega }<\bar{u}_{x} (x),y(x)-h(x)>dx \, =\int _{\Omega }f(x)\bar{u}(x)dx .\] Then from equality $\Phi (\bar{u},0)+\Phi ^{*} (0,\vartheta ^{*} )=0$ follows, that \[\frac{\alpha }{2} \int _{\Omega }\left|\bar{u}_{x} (x)\right|^{2} dx+\frac{1}{2\alpha } \, \, \int _{\Omega }\left|y(x)-h(x)-l(x)\right|^{2} dx=\int _{\Omega }<\bar{u}_{x} (x),y(x)-h(x)-l(x)>dx , \] \[\beta \int _{\Omega }\left|\bar{u}_{x} (x)\right|dx+ \, \, \, \beta \int _{\Omega }\delta _{B} \left(\frac{l(x)}{\beta } \right) dx=\int _{\Omega }<\bar{u}_{x} (x),l(x)>dx .\] \section{Result } If $\bar{u}(\cdot )\in H_{0}^{1} (\Omega )$ is a solutions of problem \eqref{GrindEQ__2_1_} then there exists a solution $\bar{\vartheta }^{*} =(y_{0} ,y)\in L_{2}^{n+1} (\Omega )$ of problems (2.2) and such $l(\cdot )\in L_{\infty }^{n} (\Omega )$ that $\left|l(x)\right|\le \beta $ almost everywhere in $\Omega $, $div\, \, y(x)-div\, h(x)+f(x)=0\, \, $ almost everywhere $x,$ where $y_{0} (x)=div\, h(x)$ and \[1) \frac{\alpha }{2} \left|\bar{u}_{x} (x)\right|^{2} +\frac{1}{2\alpha } \left|y(x)-h(x)-l(x)\right|^{2} +<\bar{u}_{x} (x),h(x)+l(x)-y(x)>=0\] almost everywhere in $ \Omega $ \noindent 2) $\beta \left|\bar{u}_{x} (x)\right|-<\bar{u}_{x} (x),l(x)>=0$ almost everywhere in $ \Omega $ \noindent 3)\textbf{ $\left(\sum\limits _{i=1}^{n}h_{i} \nu _{i} \right)(s)=0,\, \, \, \left(\sum \limits_{i=1}^{n}y_{i} \nu _{i} \right)(s)=0$ }for\textbf{ } $s\in \partial \Omega $. \bigskip \bigskip\bigskip \begin{center} R E F E R E N C E S \end{center} \noindent \textbf{ } \noindent 1. I.Ekeland , R.Temam . Convex analysis and variational problems.M:Mir,1979. \noindent 2. J.-P.Oben . Nonlinear analysis and its economical applications . M:Mir,1988 \noindent \noindent 3. M.A.Sadyqov . The ekstremal problems for non-smooth systems .Baku, 1996, 148 p. \noindent \textbf{} \end{document} % == UNREGISTERED! == GrindEQ Word-to-LaTeX 2008 == ---------------0902020444540--