On page 8 of the book Regularity of Free Boundaries in Obstacle-type Problems, it considered a functional $$J(u):=\int_D (|\nabla u|^2+2fu)\,dx \qquad D=B(0,1)$$ where $f\in L^\infty(D)$ is fixed. It is minimized over a set $$S:=\{u\in W^{1,2}(D): u-g\in W^{1,2}_0(D), u\ge \psi\text{ in }D\}$$ Here, $g,\psi\in L^\infty(D)$.
The author then stated, "Since $J$ is continuous and strictly convex on a convex subset $S$ of the Hilbert space $W^{1,2}(D)$, it has a unique minimizer on $S$."
I am able to show that $J$ is continuous and strictly convex, and that $S$ is a convex subset. In addition, if a minimizer exists, it must be unique due to strict convexity. However, how can I show the existence of a minimizer?
My attempt: If $J$ satisfies
- $\displaystyle\lim_{||u||_{W^{1,2}(D)}\to\infty}|J(u)|=+\infty$
- $J(u)$ is lower bounded on $S$
then the existence of minimizer follows from the following argument: take a sequence $\{u_n\}$ such that $J(u_n)\to \inf_{u\in S} J(u)$. Then, this sequence is bounded (in the $W^{1,2}(D)$ norm), and thus (by the weak compactness of the $W^{1,2}(D)$ unit ball) there exists a weakly converging subsequence $u_{n_k}\to^w u_0$. Then, by Mazur's Lemma, certain convex combinations of $u_{n_k}$'s (which also lie in $S$) converges strongly to $u_0$, and thus $u_0$ is a minimizer.
Unfortunately, I am unable to prove both (1) and (2).