Constrained radial symmetry for the infinity-Laplacian. (October 2017)

• Record Type: Journal Article
• Authors: Greco, Antonio
• Is Part Of: Nonlinear analysis. Volume 37(2017)
• Page Start: 239
• Page End: 248
• Publication Date: 2017-10
• Subjects: Infinity-Laplacian -- Overdetermined problems -- Radial symmetry; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2017.02.016
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: Three main results concerning the infinity-Laplacian are proved.Theorem 1.1 shows that some overdetermined problems associated to an inhomogeneous infinity-Laplace equation are solvable only if the domain is a ball centered at the origin : this is the reason why we speak of constrained radial symmetry.Theorem 1.2 deals with a Dirichlet problem for infinity-harmonic functions in a domain possessing a spherical cavity. The result shows that under suitable control on the boundary data the unknown part of the boundary is relatively close to a sphere. Finally, Theorem 1.4 gives boundary conditions implying that the unknown part of the boundary is exactly a sphere concentric to the cavity. Incidentally, a boundary-point lemma of Hopf's type for the inhomogeneous infinity-Laplace equation is obtained.