On concavity of solutions of the Dirichlet problem for the equation $(-\Delta {)}^{1/2}\phi =1$ in convex planar regions

Abstract

For a sufficiently regular open bounded set $D\subset {\mathbb{R}}^2$ let us consider the equation $(-\Delta {)}^{1/2}\phi (x)=1$ for $x\in D$ with the Dirichlet exterior condition $\phi (x)=0$ for $x\in D^c$. Its solution $\phi (x)$ is the expected value of the first exit time from $D$ of the Cauchy process in ${\mathbb{R}}^2$. We prove that if $D\subset {\mathbb{R}}^2$ is a convex bounded domain then $\phi$ is concave on $D$. To do so we study the Hessian matrix of the harmonic extension of $\phi$. The key idea of the proof is based on a deep result of Hans Lewy concerning the determinants of Hessian matrices of harmonic functions.