Constrained radial symmetry for the infinity-Laplacian

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.