On concavity of solutions of the Dirichlet problem for the equation (Δ)1/2φ=1(-\Delta)^{1/2} \varphi = 1 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
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Abstract

For a sufficiently regular open bounded set DR2D \subset \mathbb R^2 let us consider the equation (Δ)1/2φ(x)=1(-\Delta)^{1/2} \varphi(x) = 1 for xDx \in D with the Dirichlet exterior condition φ(x)=0\varphi(x) = 0 for xDcx \in D^c. Its solution φ(x)\varphi(x) is the expected value of the first exit time from DD of the Cauchy process in R2\mathbb R^2. We prove that if DR2D \subset \mathbb R^2 is a convex bounded domain then φ\varphi is concave on DD. To do so we study the Hessian matrix of the harmonic extension of φ\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.

Cite this article

Tadeusz Kulczycki, On concavity of solutions of the Dirichlet problem for the equation (Δ)1/2φ=1(-\Delta)^{1/2} \varphi = 1 in convex planar regions. J. Eur. Math. Soc. 19 (2017), no. 5, pp. 1361–1420

DOI 10.4171/JEMS/695