Existence, uniqueness, and regularity theory for local and nonlocal problems

QC 2025-12-02

Abstract

This thesis consists of three papers, an individual summary of each paper, and an introduction. The papers are all related to existence, uniqueness, or regularity theory of local and nonlocal partial differential equations (PDEs).

In Paper A, we establish uniqueness for viscosity solutions of the inhomogeneous nonlocal infinity Laplace equation Lu = f, where the right hand side f is a bounded, continuous, and nonpositive function. Uniqueness is proven through a comparison principle.

In Paper B, we use Perron's method to construct viscosity solutions to the equation ∂u/∂t = L u in Ω, and u = g in the complement.

In Paper C we study regularity of a minimizer of the expression J(u) := ∫ F(∇u) dx, where F(x) is a strongly convex function whose second derivatives might jump at |x| = 1. The specific form of F gives rise to a free boundary Γ, and the resulting Euler-Lagrange equation varies over Γ. In this paper we only consider two-phase flat points. We show that under some regularity and non-degeneracy assumptions the asymptotic expansion of a minimizer u can be written as u(x) = a + ν · x + p(x) + q(x), where a ∈ R, ν ∈  R^n. The function p is a broken polynomial that is defined as a C^1 function consisting of one polynomial in the upper half space and another polynomial in the lower half space, and the function q is a rest term. We derive the PDEs that are satisfied by p and q, respectively, and show many regularity properties for the terms in the expansion. This paper is intended to be the first part of a project that aims at establishing regularity of the free boundary Γ.

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