Fundamental Solutions and Green Functions for Nonhomogeneous Second Order Elliptic Operators with Discontinuous Coefficients

Abstract

This thesis is devoted to the properties of fundamental solutions, Green functions, and Neumann-Green functions for general non-homogeneous second order elliptic systems with discontinuous coefficients. We establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems. We impose certain non-homogeneous versions of de Giorgi-Nash-Moser bounds on the weak solutions and investigate in detail the assumptions on the lower order terms sufficient to guarantee such conditions. Our results, in particular, establish the existence and fundamental estimates for the Green functions associated to the Schrödinger, magnetic Schrödinger, and generalized Schrödinger operators with bounded measurable real and complex coefficients on arbitrary domains. Most of the results above rely on the construction of the averaged fundamental solutions and Green functions with sharp uniform estimates. We also showcase a different approach to Green and Neumann-Green functions via layer potentials which yields, in addition, certain new mapping properties for the Green operators. A substantial portion of the results of this thesis gave rise [DHM17], submitted for publication.