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Duality, Non-smooth functional
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\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,
�.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

\[\begin{array}{l} {\Phi ^{*} (0,\vartheta ^{*} )=\mathop{\mathop{\sup }\limits_{u\in H_{0}^{'} (\Omega )} }\limits_{z\in H_{0}^{'} (\Omega )} \left\{\int _{\Omega }(z(x)-u(x))y_{0} (x)dx+\int _{\Omega }<z_{x} (x)-u_{x} (x),y(x)>dx-  \right. } \\ {-\frac{\alpha }{2} \int _{\Omega }\left|z_{x} (x)\right|^{2} dx -\, \beta \int _{\Omega }\left|z_{x} (x)\right|dx+ \left. \int _{\Omega }f(x)u(x)dx \right\}=} \end{array}\]

\[\begin{array}{l} {=\mathop{\sup }\limits_{z\in H_{0}^{'} (\Omega )} \left\{\int _{\Omega }z(x)y_{0} (x)dx+\int _{\Omega }<z_{x} (x),y(x)>dx-\left. \frac{\alpha }{2} \int _{\Omega }\left|z_{x} (x)\right|^{2} dx-\beta \int _{\Omega }\left|z_{x} (x)\right|dx  \right\}  \right. +} \\ {+\mathop{\sup }\limits_{u\in H_{0}^{'} (\Omega )} \left\{-\int _{\Omega }u(x)y_{0} (x)dx-\int _{\Omega }<u_{x} (x),y(x)>dx+\int _{\Omega }f(x)u(x)dx   .\right. } \end{array}\]
   Puting $M=\left\{(\right. u(x),u_{x} (x)):u(\cdot )\in H_{0}^{1} (\Omega \left. )\right\}$, we get

\[\begin{array}{l} {S(y_{0} ,y)=\mathop{\sup }\limits_{z\in H_{0}^{'} (\Omega )} \left\{\begin{array}{l} {\int _{\Omega } z(x)y_{0} (x)dx+\int _{\Omega }<z_{x} (x),y(x)>dx-\frac{\alpha }{2} \int _{\Omega }\left|z_{x} (x)\right|^{2} dx-\left. \beta \int _{\Omega }\left|z_{x} (x)\right|dx \right\} \,  =} \\ {} \end{array}\right. \, \, } \\ {=\mathop{\sup }\limits_{(\omega _{0} ,\omega )\in L_{2}^{n+1} (\Omega )} \left\{\int _{\Omega }\omega _{0} (x)y_{0} (x)dx+\int _{\Omega }<\omega (x),y(x)>dx-  \right. } \end{array}\]

\[\begin{array}{l} {-\left. \frac{\alpha }{2} \int _{\Omega }\left|\omega (x)\right|^{2} dx-\beta \int _{\Omega }\left|\omega (x)\right| dx+\delta _{M} (\omega _{0} (x),\omega (\cdot )) \right\}=} \\ {=\mathop{\sup }\limits_{(\omega _{0} ,\omega )\in L_{2}^{n+1} (\Omega )} \left\{\int _{\Omega }\omega _{0} (x)(y_{0} (x)-h_{0} (x))dx+ \right. } \\ {+\left. \int _{\Omega }<\,  \omega (x),y(x)-h(x)>dx-\frac{\alpha }{2} \int _{\Omega }\left|\omega (x)\right|^{2} dx-\beta \int _{\Omega }\left|\omega (x)\right| dx \right\}+} \\ {+\mathop{\sup }\limits_{u\in H_{0}^{'} (\Omega )} \left\{\int _{\Omega }u(x)h_{0} (x)dx+\int _{\Omega }<u_{x} (x),h(x)>dx  \right\}.} \end{array}\]

 From  $\mathop{\sup }\limits_{u\in H_{0}^{'} (\Omega )} \left\{\int _{\Omega }u(x)h_{0} (x)dx+\int _{\Omega }<u_{x} (x),h(x)>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 �  $h_{0} (\cdot )=div\, \, h\in L_{2} (\Omega )$ and   $-\left(\sum _{i=1}^{n}h_{i} \nu _{i}  \right)(s)=0$ for  $S\in \partial \Omega $ (��.[3, �.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

\[\begin{array}{l} {\mathop{\sup }\limits_{u\in H_{0}^{'} (\Omega )} \left\{-\int _{\Omega }u(x)y_{0}
(x)dx-\int _{\Omega }<u_{x} (x),y(x)>dx+\int _{\Omega }f(x)u(x)dx   \right\}=} \\ {=\left\{\begin{array}{l}
{0:\, \, div\, y(x)-y_{0} (x)+f(x)=0\, \, \, \, \, \, \, \, \, almost\,\,\,\, everywhere\, \, \, x\in \Omega } \\
 {+\infty :div\, y(x)-y_{0} (x)+f(x)\ne 0\, \, \, \, \, \, \, almost\,\,\,\, everywhere\, \, \, \, \, x\notin \Omega }
  \end{array}\right. =} \\ {=\, \left\{\begin{array}{l} {0:\, \, div\, y(x)-div\, h(x)+f(x)=0\,
   \, \, \, \, \, \, \, \, \, almost \,\,\,\, everywhere\, \, \, x\in \Omega } \\ {+\infty :div\, y(x)-div\, h(x)+f(x)\ne
   0\, \, \, \, \, \, almost \,\,\,\, everywhere \, \, \, \, x\notin \Omega ,} \end{array}\right. } \end{array}\]
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,�.30-33] follows, that $h$ subdifferentiable at the point zero . From remark 3.2.3 [1, �.60]  and from corollary  3.2.4[1, �.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{}


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