SPECTRAL THEORY OF THE SUBELLIPTIC P-LAPLACIAN FOR HÖRMANDER VECTOR FIELDS

Abstract

Subject of this thesis is the spectral theory of the subelliptic p-Laplacian in the context of Hörmander vector fields. In Chapter 2, we determine the first eigenvalue 𝜆_1 through the minimization of the Rayleigh quotient, which also leads to finding the best constant in the L^p Poincaré-Friedrichs inequality for Hörmander vector fields. Also, we prove Hölder continuity of eigenfunctions with respect to the Carnot-Carathéodory metric and positivity of the first eigenfunction, which are applied to obtain the simplicity of the first eigenvalue 𝜆_1. By the end of Chapter 2, we show that all eigenfunctions corresponding to any eigenvalue 𝜆 ≠𝜆_1 change sign in the given domain and the first eigenvalue 𝜆_1 is isolated in the set of all eigenvalues. In Chapter 3, we apply the Lusternik-Schnirelman theory to establish the existence of a sequence of variational eigenvalues for the subelliptic p-Laplacian eigenvalue problem. We use two different kinds of compact, symmetric subsets of some manifold to derive variational eigenvalues of the Lusternik-Schnirelman type. As applications in the context of partial differential equations, we demonstrate blow-up and extinction behavior of solutions to some parabolic equations with the subelliptic p-Laplacian in Chapter 4.