Geodesic
On concavity of solutions of the Dirichlet problem for the equation $(-\Delta)^{1/2} \varphi = 1$ in convex planar regions
Journal of the European Mathematical Society, Tome 19 (2017) no. 5, pp. 1361-1420.

Voir la notice de l'article provenant de la source EMS Press

For a sufficiently regular open bounded set D⊂R2 let us consider the equation (−Δ)1/2φ(x)=1 for x∈D with the Dirichlet exterior condition φ(x)=0 for x∈Dc. Its solution φ(x) 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 do so 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 the determinants of Hessian matrices of harmonic functions.
DOI : 10.4171/jems/695
Classification : 35-XX, 31-XX
Keywords: Fractional Laplacian, concavity, Hessian matrix, harmonic function, Cauchy process, first exit time 
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     author = {Tadeusz Kulczycki},
     title = {On concavity of solutions of the {Dirichlet} problem for the equation $(-\Delta)^{1/2} \varphi = 1$ in convex planar regions},
     journal = {Journal of the European Mathematical Society},
     pages = {1361--1420},
     publisher = {mathdoc},
     volume = {19},
     number = {5},
     year = {2017},
     doi = {10.4171/jems/695},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/695/}
}
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Tadeusz Kulczycki. On concavity of solutions of the Dirichlet problem for the equation $(-\Delta)^{1/2} \varphi = 1$ in convex planar regions. Journal of the European Mathematical Society, Tome 19 (2017) no. 5, pp. 1361-1420. doi : 10.4171/jems/695. http://geodesic.mathdoc.fr/articles/10.4171/jems/695/

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