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Let $U$ be an open subset of $\mathbb{R}^n$
Consider the minimization problem for
\begin{equation} I[w]:=\int_UL(Dw(x),w(x),x)dx \end{equation} where $w:U\rightarrow \mathbb{R}^m$, $L:\mathbb{M}^{m\times n}\times\mathbb{R}^m\times \bar{U}\rightarrow\mathbb{R}$ are functionals on $U$ and $\mathbb{M}^{m\times n}\times\mathbb{R}^m\times \bar{U}$ respectively.
Evans says in his book 'Partial Differential Equations' that if $L(P,z,x)$ is convex with respect to $P$ and satisfies that \begin{equation} L(P,z,x)\ge\alpha|P|^q-\beta \end{equation} for some $\alpha>0, \beta\ge0$ and $1<q<\infty$, and the set $A=\{w\in W^{1,q}(U;\mathbb{R}^m): w=g$ on $\partial U$ in the trace sence$\}$ for given $g:\partial U\rightarrow \mathbb{R}^m$ is nonempty, then there is $u\in A$ such that \begin{equation} I[u]=min\{I[w]: w\in A\} \end{equation}

I wonder whether this holds for the version that $L$ is a functional on $\mathbb{R}^m\times\bar{U}$, i.e., minimize \begin{equation} I[w]:=\int_UL(w(x),x)dx \end{equation} with the similar condition above. Any reference or ideas will be appreciated, thanks.

CSH
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1 Answers1

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Let us treat it as some kind of free local minimum problem: Here is a classical theorem:

If $F:X\to \mathbb{R}$ is a functional over a Banach space, then $u_0$ is a local minimum of $F$ provided that

(1) for all $h\in X$ and some $c>0$, $$ \delta F(u_0,h)=0,\quad \delta^2 F(u_0,h)\geq c\|h\|^2 $$

(2) $u\to \delta^2F(u;h)$ is continuous at $u_0$

I think "Zeidler's Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization " is a very good reference for minimization problems. Evans's approach is indeed one of the major methods: direct method upone compactness principle.

Assume that $L$ is smooth enough. Then we have $$ \delta I(w;h)=\int \nabla L(w,x)\cdot h(x) dx=\sum \int \partial_i L(w,x) h_i(x) dx, $$ and $$ \delta^2 I(w;h)=\int h(x)^T \nabla^2 L(w,x)h(x) dx=\sum \int \partial_{ij} L(w,x) h_i(x)h_j(x) dx. $$ If we apply the Theorem above, then $L$ is required to be $C^2$ and convex and $$ \nabla L(w,x)=0 $$ admits some solution in some function spaces (say $C(U;\mathbb{R}^n)$ or $L^p(U;\mathbb{R}^n)$ )

Ice sea
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