Question: for any bounded Lipschitz domain $\Omega\subset\mathbb{R}^d$, does there always exist a nonnegative function $\phi\in C^2(\Omega)$ such that
- $\phi$ vanishes on $\partial\Omega$
- the normal inward derivative $\partial_n\phi(x)>0$ for $\mathcal H^{d-1}$-almost every boundary point $x\in\partial\Omega$
- $-\Delta\phi (x)\geq c_{\Omega}>0$ is uniformly bounded away from zero in $\Omega$, for some constant $c_{\Omega}$ possibly depending on the domain.
If really needed I would be content with just $\phi\in W^{2,\infty}$ instead of $C^2$. I know how to do this in smooth domains (see below), but can this be done in the universal class of Lipschitz domains? If not, can anyone give a sufficient weaker condition on $\Omega$ (for example locally smooth with finitely many corners, or whatever...)
Some context first: for purely technical purposes I need such a function to serve as a "universal" bound from below, in order to control some other (possibly nasty) supersolutions. In particular I really need my supersolutions to grow at least linearly close to the boundary, and it is moreover absolutely crucial for my specific application that the (minus) Laplacian be bounded from below as in my condition 3.
Some clues: for smooth domains one can simply pick $\phi$ by solving the elliptic problem $$ \begin{cases} -\Delta\phi =1 & \mbox{in }\Omega\\ \phi =0 & \mbox{on }\partial\Omega \end{cases}. $$ In smooth domains the usual elliptic regularity guarantees that $\phi\in C^\infty(\bar\Omega)$, and Hopf's boundary lemma gives moreover $\partial_n\phi(x)>0$ everywhere on the boundary so we are done.
Some issues: For nonsmooth domains this natural but naive approach completely fails (for example $\Delta\phi\in C^\infty(\bar\Omega)$ does not even guarantee $\phi\in W^{1,p}$, at least for some $p$). And the Hopf boundary lemma is unclear to me. A reasonable and apparently (more or less) standard approach is to try using instead the first Dirichlet-Laplacian eigenvalue $-\Delta\phi_1=\lambda_1\phi_1$ as a "universal lower bound", see for example Guido Sweer's paper "Hopf’s lemma and two dimensional domains with corners" available here. Unfortunately this is not good enough for me, since unevitably the Laplacian $-\Delta\phi_1=\lambda_1\phi_1$ vanishes on the boundary (by definition $\phi_1|_{\partial\Omega}=0$) and thus violates my requirement 3. But perhaps there is room for maneuver: maybe such a function $\phi$ can be constructed in some other way, not just by solving a nice elliptic equation. For example, note carfully that I am not requiring any upper bound on $-\Delta\phi$, this might help.
Any help is appreciated!
