caa a22 4500 605515891 CHVBK 20210128100711.0 cr unu---uuuuu 210128e20151201xx s 000 0 eng 10.1007/s00205-015-0888-4 doi (NATIONALLICENCE)springer-10.1007/s00205-015-0888-4 On the Dirichlet and Serrin Problems for the Inhomogeneous Infinity Laplacian in Convex Domains: Regularity and Geometric Results [Elektronische Daten] [Graziano Crasta, Ilaria Fragalà] Given an open bounded subset Ω of $${\mathbb{R}^n}$$ R n , which is convex and satisfies an interior sphere condition, we consider the pde $${-\Delta_{\infty} u = 1}$$ - Δ ∞ u = 1 in Ω, subject to the homogeneous boundary condition u = 0 on ∂Ω. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class C 1(Ω). We then investigate the overdetermined Serrin-type problem, formerly considered in Buttazzo and Kawohl (Int Math Res Not, pp 237-247, 2011), obtained by adding the extra boundary condition $${|\nabla u| = a}$$ | ∇ u | = a on ∂Ω; by using a suitable P-function we prove that, if Ω satisfies the same assumptions as above and in addition contains a ball which touches ∂Ω at two diametral points, then the existence of a solution to this Serrin-type problem implies that necessarily the cut locus and the high ridge of Ω coincide. In turn, in dimension n=2, this entails that Ω must be a stadium-like domain, and in particular it must be a ball in case its boundary is of class C 2. Springer-Verlag Berlin Heidelberg, 2015 Crasta Graziano Dipartimento di Matematica "G. Castelnuovo”, Univ. di Roma I, P.le A. Moro 2, 00185, Rome, Italy aut Fragalà Ilaria Dipartimento di Matematica, Politecnico, Piazza Leonardo da Vinci, 32, 20133, Milan, Italy aut Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 218/3(2015-12-01), 1577-1607 0003-9527 218:3<1577 2015 218 205 https://doi.org/10.1007/s00205-015-0888-4 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-015-0888-4 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Crasta Graziano Dipartimento di Matematica "G. Castelnuovo”, Univ. di Roma I, P.le A. Moro 2, 00185, Rome, Italy aut NATIONALLICENCE 700 1- Fragalà Ilaria Dipartimento di Matematica, Politecnico, Piazza Leonardo da Vinci, 32, 20133, Milan, Italy aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 218/3(2015-12-01), 1577-1607 0003-9527 218:3<1577 2015 218 205