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

Abstract

For a sufficiently regular open bounded set $D \subset \mathbb R^2$ let us consider the equation $(-\Delta)^{1/2} \varphi(x) = 1$ for $x \in D$ with the Dirichlet exterior condition $\varphi(x) = 0$ for $x \in D^c$. Its solution $\varphi(x)$ 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 $\varphi$ is concave on $D$. To do so we study the Hessian matrix of the harmonic extension of $\varphi$. The key idea of the proof is based on a deep result of Hans Lewy concerning the determinants of Hessian matrices of harmonic functions.