Variational formulation of a pde

Question

Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, and: $$ X_f=\{V\in C^\infty(\mathbb{R}^n\times(0,\infty))\cap C^0(\mathbb{R}^n\times[0,\infty)): E_a(V)<\infty, V(x,0)=f(x),\forall x\in\mathbb{R}^n\}, $$ where: $$ E_a(V):=\int_{\mathbb{R}^n\times(0,\infty)}y^a|\nabla_{x,y}V(x,y)|^2\,dx\,dy. $$ I want to prove that if $U\in X_f$ is a classical solution of: $$ \text{div}(y^a\nabla_{x,y}V)=0,\quad\text{on }\mathbb{R}^n\times(0,\infty),$$ $$ V(x,0)=f(x),\quad\forall x\in\mathbb{R}^n,$$ then we have that: $$ E_a(U)=\min_{V\in X_f}E_a(V).$$ How i can prove this fact? I have no idea, any help is appreciated.