Positivity and regularity of a solution of an elliptic PDE

Question

Let $\varepsilon > 0$ be fixed, and let $\Omega \subset \mathbb{R}^d$ be a bounded domain with smooth boundary. Consider the following (Neumann) elliptic problem \begin{cases} \varepsilon \Delta u - u + f = 0 \quad \text{in} \ \Omega\\ \partial_n u = 0 \quad \text{on} \ \partial \Omega \end{cases} where $f \in C^{0, 1}(\overline{\Omega})$ is non-negative on $\overline{\Omega}$ and not identically $0$, and $\partial_n$ denotes the normal derivative (taken with respect to the outward normal vector). I managed to prove, using a Lax-Milgram argument, that the problem has a unique solution $u \in \{u \in H^1(\Omega) \colon -\Delta u \in L^2(\Omega)\}$. How may I show that the solution $u$ is positive? Furthermore, how may we deduce (using some sort of elliptic regularity) that the solution is $C^2(\overline{\Omega})$? On a side note, may we say that $u \in \{u \in H^1(\Omega) \colon -\Delta u \in L^2(\Omega)\}$ implies $u \in H^2(\Omega)$?

Answer 1

The weak solution $u\in H^1(\Omega)$ of your PDE is the unique minimizer of

$$I(u) = \int_\Omega \frac{1}{2}\varepsilon |\nabla u|^2 + \frac{1}{2} u^2 - uf \, dx.$$

If $f$ is nonnegative, then $I(u_+)\leq I(u)$ for all $u\in H^1(\Omega)$ where $u_+=\max\{u,0\}$. This shows that $u\geq 0$ almost everywhere.

I don't have a copy of Gilbarg-Trudinger with me. Look in the Schauder estimates section for estimates that control the $C^{2,\alpha}$ norm of $u$ by the $C^{0,\alpha}$ norm of $f$. This will prove $u\in C^{2,\alpha}$.