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
