On concavity of solutions of the Dirichlet problem for the equation \((-\Delta)^{1/2} \varphi = 1\) in convex planar regions (Q2628336)

Summary: 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.