- #1
Euge
Gold Member
MHB
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Let ##U## be a bounded open subset of ##\mathbb{R}^n##. Given a continuous function ##\phi : \overline{U} \to \mathbb{R}##, show that any real-valued function ##u## of class ##C^2(\overline{U})## such that ##\Delta u = \phi## in ##U## and ##u|_{\partial U} = 0## is a minimizer of the energy functional $$\mathscr{E}(u) = \int_U d^n x\, \left(\frac{1}{2}|\nabla u(x)|^2 + \phi(x) u(x)\right)$$ over the class of functions ##\Sigma = \{v\in C^2(\overline{U}) : v|_{\partial U} = 0\}##.