The Dirichlet Problem with Prescribed Interior Singularities

In this paper 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 R. The main results apply, in particular, to subequations with a Riesz characteristic p ≥ 2. 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 ΘjKp(x− xj) where Kp(x) = { − 1 |x|p−2 for 2 < p <∞, log|x| if p = 2. at any prescribed finite set of points {x1, ..., xk} in the domain and any finite set of positive real numbers Θ1, ...,Θk. 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 finitetype singularities such as Θj |x− xj |2−p for 1 ≤ p < 2. The main results apply similarly with prescribed singularities asymptotic to the fundamental solutions of Armstrong-Sirakov-Smart (in the uniformly elliptic case).