On concavity of solutions of the Dirichlet problem for the equation in convex planar regions

  • Tadeusz Kulczycki

    Wroclaw University of Technology, Poland

Abstract

For a sufficiently regular open bounded set let us consider the equation for with the Dirichlet exterior condition for . Its solution is the expected value of the first exit time from of the Cauchy process in . We prove that if is a convex bounded domain then is concave on . To do so we study the Hessian matrix of the harmonic extension of . The key idea of the proof is based on a deep result of Hans Lewy concerning the determinants of Hessian matrices of harmonic functions.

Cite this article

Tadeusz Kulczycki, On concavity of solutions of the Dirichlet problem for the equation in convex planar regions. J. Eur. Math. Soc. 19 (2017), no. 5, pp. 1361–1420

DOI 10.4171/JEMS/695