Starshape of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings

TL;DR

This paper investigates the geometric shape of superlevel sets of solutions to fractional Laplacian equations in starshaped rings, proving they inherit starshapedness under certain conditions on the domain and nonlinearity.

Contribution

It establishes that solutions' superlevel sets are starshaped when the domain is starshaped, extending geometric properties to fractional Laplacian problems with various nonlinearities.

Findings

Superlevel sets are starshaped if the domain is starshaped.

Results hold for fractional Laplacian with  in (0,2).

Applicable to different nonlinearities in the equation.

Abstract

In the present work we study solutions of the problem $-(-\Delta)^{\alpha/2}u = f(x,u)$ in $D_0\setminus \overline{D}_1$, with exterior conditions $u = 0$ in $R^N \setminus D_0$ and $u = 1$ in $\overline{D}_1$, where $D_1, D_0 \subset R^N$ are open sets such that $\overline{D}_1 \subset D_0$, $\alpha \in (0,2)$, and $f$ is a nonlinearity. Under different assumptions on $f$ we prove that, if $D_0$ and $D_1$ are starshaped with respect to the same point $\bar{x} \in \overline{D}_1$, then the same occurs for every superlevel set of $u$.