On concavity of solution of Dirichlet problem for the equation (-Δ)^1/2 φ= 1 in a convex planar region

2014/05/15 by Tadeusz Kulczycki, Kulczycki, Tadeusz
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1405.3846

Abstract

For a sufficiently regular open bounded set D ⊂ R² let us consider the equation (-Δ)^1/2 φ(x) = 1, x ∈ D with the Dirichlet exterior condition φ(x) = 0, x ∈ D^c. φ is the expected value of the first exit time from D of the Cauchy process in R². We prove that if D ⊂ R² 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.

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