Extension of solution PDE

Question

Let us consider a non-negative function $u \in C^{0,\alpha}(B_1)$ such that $\Delta u=1$ in the set $\{u>0\}\cap B_1$. Is it true that $$ \Delta u = \chi_{\{u>0\}} \quad\mbox{in }B_1? $$ In the previous equation we use the notation $\chi_{\{u>0\}}$ for the characteristi function of the set $\{u>0\}$, defined by $$ \chi_{\{u>0\}}(x)= \begin{cases} 1 & \mbox{if }x \in \{u>0\}\\ 0 & \mbox{if }x \in \{u=0\}. \end{cases} $$

Answer 1

a couple of comments that I hope will shed some light on the question. First off, I think it should be clarified in which sense you want $$\Delta u=\chi_{\{u>0\}}$$ to make sense. Clearly you won't be able to make sense of it neither point-wise nor a.e., as second derivatives of $u$ don't have to exist in any sense, a priori.

From the "variational" point of view, you could try to prove that

$$\int_{B_1} \nabla u\cdot \nabla\psi\:dx=-\int_{\{u>0\}}\psi\:dx$$

for any $\psi\in W^{1,2}_0(B_1)$ (the Sobolev space). In order to do that you would have to require further that $u\in W^{1,2}_{\text{loc}}(B_1)$, check Exercise 2.6 in Petrosyan's book, "Regularity of free boundaries in obstacle-type problems", I believe the proof works mutatis mutandis. This is the natural way of thinking about this problem if you arrive at it from a minimization procedure. I'm not sure if you can push it further to drop the assumption of $u$ having a weak derivative.

Finally, as you used the tag "viscosity solutions": that would be an entirely different approach. If $\psi$ is a paraboloid touching $u$ by above at $x_0$, two things can happen: either $u(x_0)>0$ or $u(x_0)=0$. In the first case, we can use the equation. In the second case, we know that $\psi\geq0$ and has a minimum at $x_0$, hence $\Delta \psi(x_0)\geq0$ as desired in this case as well. I would be careful here with the notion of viscosity solution, though, as continuity is usually required and the characteristic function has a jump. You can check out the paper "On viscosity solutions of fully nonlinear equations with measurable ingredients", by Caffarelli et al. I think they relax the continuity assumptions on the right hand side there, but I haven't read the paper so I don't know what further things are needed.

Okay, I hope this was comment is of some help.