On concavity of solutions of the Dirichlet problem for the equation in convex planar regions
Tadeusz Kulczycki
Wroclaw University of Technology, Poland
![On concavity of solutions of the Dirichlet problem for the equation $(-\Delta)^{1/2} \varphi = 1$ in convex planar regions cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-jems-volume-19-issue-5.png&w=3840&q=90)
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