Convexity and uniqueness for a semilinear PDE

Problem's Description

Let $D$ be a given convex and bounded domain in ${\mathbb R}^n$. Then there is always a solution to the following semilinear PDE $$ \Delta u = \chi_{\{0< u < 1\}}, \quad \hbox{in } {\mathbb R}^n \setminus D, \qquad u= 2 \quad \hbox{on } D , $$ and $u$ continuous in ${\mathbb R}^n$, with compact support.

1) Is the solution unique?

2) Is the set $\Omega:= \{u>0 \}$ convex?

The same question can be asked for more general operators, such as p-laplacian, and Fully-nonlinear PDEs.

In case $D$ is a ball this was proven in [1, Appendix]. The proof uses explicit computations, and seems not useful for general convex cases. Even a non-computational proof for the case of ball would be interesting to see.

There are several results of this type in papers that appear in the literature, under the title "convexity and uniqueness for Bernoulli type problems".

1. ArticleIs an originPerturbed divisible sandpiles and quadrature surfaces

   • Hayk Aleksanyan,
   • Henrik Shahgholian

   pp. 1-29, year of publication: 2017arXiv