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

Abstract

For a sufficiently regular open bounded set $D\subset R^2$ let us consider the equation $(-\Delta {)}^{1/2}\phi (x)=1$, $x\in D$ with the Dirichlet exterior condition $\phi (x)=0$, $x\in D^c$. $\phi$ 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 $\phi$ is concave on $D$. To show it 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 determinants of Hessian matrices of harmonic functions.

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