On concavity of solutions of the Dirichlet problem for the equation $(-\Delta)^{1/2} \varphi = 1$ in convex planar regions
Tadeusz Kulczycki · Journal of the European Mathematical Society · 2017
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.