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

Abstract

For a sufficiently regular open bounded set $D \subset R^2$ let us consider the equation $(-\Delta)^{1/2} \varphi(x) = 1$, $x \in D$ with the Dirichlet exterior condition $\varphi(x) = 0$, $x \in D^c$. $\varphi$ is the expected value of the first exit time from $D$ of the Cauchy process in $R^2$. We prove that if $D \subset R^2$ is a convex bounded domain then $\varphi$ is concave on $D$. To show it 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 determinants of Hessian matrices of harmonic functions.