On concavity of solution of dirichlet problem for the equation (−∆)1/2φ = 1 in a convex planar region

For a sufficiently regular open bounded set D ⊂ R let us consider the equation (−∆)φ(x) = 1, x ∈ D with the Dirichlet exterior condition φ(x) = 0, x ∈ D. φ is the expected value of the first exit time from D of the Cauchy process in R. We prove that if D ⊂ R is a convex bounded domain then φ is concave on D. To show it we study the Hessian matrix of the harmonic extension of φ. The key idea of the proof is based on a deep result of Hans Lewy concerning determinants of Hessian matrices of harmonic functions.