Let $\Omega$ be a bounded domain with smooth boundary in $R^n$. Fix $u_0 \in H^1(\Omega) - \{ 0\}$.
Define
$$ J(u):= \int_{\Omega} \langle A(x) \nabla u(x), \nabla u(x)\rangle dx,$$ $$u \in K:=\{ v \in H^1(\Omega); v-u_0 \in H^{1}_{0}(\Omega)\}$$
where $A(x) = a_{ij}(x), i,j=1,...n , x \in \Omega$ is a matrix with smooth coeficients and there are constants $0<\lambda_1 <\lambda_2 < +\infty$ such that
$$\lambda_1 |\xi|^2 \leq \langle A(x) \xi,\xi\rangle \leq \lambda_2 |\xi|^2 $$
Does the funcional $J$ admits a minimum? Probably the answer is yes, because it is a natural question. I tried to prove this but I am not seeing how to do this. Someone could help me to prove this?
1) Note that $K$ is a closed convex set.
2) The hypothesis on $A$ ($A$ is an strictly elliptic operator) implies that $J:K\to\mathbb{R}$ is a strictly convex, weak lower semicontinuous and coercive operator.
3) Combine 1) and 2) to find a unique minimum for $J$ in $K$.