A geometria de domínios extremais para o p-Laplaciano

Abstract

In this Thesis work, our main objective is to characterize (f, p)-extremal domains of the over- determined type, for the p-Laplacian, which admits a positive solution, under some special condition imposed on the function f. To achieve this goal we investigate geometric and topological properties of (f, p)-extremal domains related to problems about the p-Laplacian operator in domains of both the Euclidean space R N and the hyperbolic space Hn . We show that when a function f is Lipschitz and satisfies the condition that f(t) ≥ λtp−1 for all t > 0 and for some constant λ > 0 in the Euclidean case or λ > n – 1 p p in the hyperbolic case, such domains have narrowing properties. The main tools used to obtain our results are an application of the Moving Planes Method (MPM) together with some versions of the maximum, strong maximum and strong comparison principles and an adaptation of the techniques used in (ROS; SICBALDI, 2013) and (ESPINAR; MAO, 2018) for the case p = 2, (COLESANTI, 1994) and (CHORWADWALA et al., 2015) for the general case. We also present some results on estimates of the first eigenvalue for the p-Laplacian with Dirichlet conditions in geodesic balls of Riemannian manifolds that admit a function f that satisfies |∇f| ≤ b and ∆pf ≥ a for positive constants a and b. In this sense, we generalize results obtained previously in (VEERAVALLI, 2003).