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Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is aLipschitz graph. $S \subset \partial \Omega$ is measurabel and $H^{n-1}(S)>0.$ The Dirichlet data on $S$ are given by non-negative function $u^0 \in ^{1}_{Loc}(\Omega)$ with $\nabla u^0 \in L^{2}(\Omega)$. The given force function $Q$ is non-negative and measurable.

Consider the convex set \begin{equation} K:=\{ v \in L^{1}_{Loc}(\Omega): \nabla v \in L^{2}(\Omega) \quad \mbox{and} \quad v=u^0 \quad \mbox{on} S\}. \end{equation} We are looking for an absolute minimum of the functional \begin{equation} J(v):= \int_{\Omega}(|\nabla v|^{2} + \chi(\{v>0\})Q^2) \end{equation} in the class $K$.

I wish someone would remake this theorem explaining the details. I'll be very grateful.

Existence theorem. If $J(u_0)< \infty$ then exist an absolute minimum $u \in K$ of the functional $J$.

Since $J$ is non-negative there is a minimal sequence $u_k, k \in \mathbb{N}$. Then $\nabla u_k$ are bounded in $L^{2}(\Omega)$ and since $H^{n-1}(S)$ is positive $u_k -u^0$ are bounded in $L^2(B_r \cap \Omega)$ for large $R$. Therefore there is an $u \in K$, such that for a subsequence \begin{equation} \nabla u_k \rightarrow \nabla u \quad \mbox{weakly in} \quad L^2(\Omega), \ u_k \rightarrow u \quad \mbox{almost everywhere in} \ \Omega. \end{equation} Moreover there is a function $\gamma \in L^\infty(\Omega)$ with $0 \le \gamma \le 1$, such that \begin{equation} \chi(\{u_k>0\}) \rightarrow \gamma \quad \mbox{weakly star in} \quad L^\infty(\Omega). \end{equation} Then for $R>0$ \begin{eqnarray} \int_{B_R \cap \Omega}(|\nabla u|^2 + \gamma \min({Q,R})^2 &\le& \liminf_{k} \int_{B_R \cap \Omega}(|\nabla u_k|^2 + \lim_{k}\int_{B_R \cap \Omega} \chi(\{u_k>0\}) \min(Q,R)^2 \\ &\le & \lim_{k} J(u_k). \\ \end{eqnarray} Letting $R\rightarrow \infty$, and since $\gamma = 1$ almost everywhere in $\{u>0\}$ we conclude \begin{equation} J(u) \le \int_{\Omega}(|\nabla u|^2 + \gamma Q^2) \le \lim_{k} J(u_k). \end{equation}

The details can be found in the article Alt, H. M. and Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325, (1981), 105–144. in the page 3

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I don't have the time to write a full answer now, but just to point you in a direction which I hope may help: essentially the existence theorem is a statement of Dirichlet's principle. First look through the "The Method" section on the linked Wiki page to see if you understand the general picture. – Willie Wong Sep 3 '12 at 13:36
Your question would be easier to answer if you can pin point where you do not understand. In particular, do you have question with (a) the "big picture" (see my previous comment) or (b) How to implement the "big picture" or (c) A particular inequality in the implementation of the big picture? – Willie Wong Sep 3 '12 at 13:37
First thank you for your attention. 1.Since $H^{n-1}(S)$ is positive $u_k -u^0$ are bounded in $L^2(B_r \cap \Omega)$ for large $R$. 2.there is a function $\gamma \in L^\infty(\Omega)$ with $0 \le \gamma \le 1$, such that \begin{equation} \chi(\{u_k>0\}) \rightarrow \gamma \quad \mbox{weakly star in} \quad L^\infty(\Omega). \end{equation} – user29999 Sep 3 '12 at 14:42
Are those two the statements you are having problem with? If so, for the sake of other readers (and with the benefit of bumping your question to the front page), please edit the question to include that information. Thanks! – Willie Wong Sep 4 '12 at 8:56

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