A convexity problem for a semi-linear PDE

Complete Solution

Posted online: 2019-09-22 09:23:05Z by layan El Hajj 147

Cite as: S-190922.1

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Convexity and uniqueness for a semilinear PDE

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".

Solution Description

We first prove that the largest subsolution $u$ of the problem exists and is star-shaped with respect to all points in $D$. This in turn implies that the free boundary $\partial \{u >0\}$ is a locally Lipschitz graph. Classical results of free boundary are used to show that the free boundary is $C^{1,\alpha}$. From here we prove uniqueness by a scaling argument , a comparison principle and using the smoothness of the free boundary. We then use quasi-concave envelope $u^*$ of $u$ which is a subsolution of the problem but larger than our solution. This contradicts the fact that our solution is the largest one and we conclude that $u^*=u$.

  1. Article A convexity Problem for a semi linear PDE

    Applicable ana, 2019fulltext

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  • Created at: 2019-09-22 09:23:05Z