The Dirichlet Problem with Prescribed Asymptotic Singularities

TL;DR

This paper establishes the existence and uniqueness of solutions to the nonlinear Dirichlet problem with prescribed asymptotic singularities, extending previous results to non-uniformly elliptic cases and various types of singularities.

Contribution

It provides new existence and uniqueness results for the Dirichlet problem with prescribed asymptotic singularities, including non-uniform elliptic cases and different singularity types.

Findings

Solutions exist and are unique for prescribed asymptotic singularities.

Applicable to subequations with Riesz characteristic p ≥ 2.

Extends results to non-uniform elliptic cases and finite-type singularities.

Abstract

We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particular, to subequations with a Riesz characteristic $p \geq 2$. In this case it is shown that, without requiring uniform ellipticity, the Dirichlet problem can be solved uniquely for arbitrary continuous boundary data with singularities asymptotic to the Riesz kernel: $\Theta_j K_p(x - x_j)$, where $K_p(x) = - {1\over|x|^{p-2}}$ for $p>2$ and $K_2(x) = \log |x|$, at any prescribed finite set of points $x_1,...,x_k$ in the domain and any finite set of positive real numbers $\Theta_1,..., \Theta_k$. This sharpens a previous result of the authors concerning the discreteness of high-density sets of subsolutions. Uniqueness and existence results are also established for finite-type singularities such as $\Theta_j |x - x_j|^{2-p}$ for $1\leq p<2$. The main results apply similarly with prescribed singularities asymptotic to the fundamental solutions of Armstrong-Sirakov-Smart (in the uniformly elliptic case).