The Dirichlet Problem with Prescribed Asymptotic 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 $p\ge 2$. 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 $K_p(x)=-\frac{1}{|x{|}^{p-2}}$ 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 $1\le p<2$. The main results apply similarly with prescribed singularities asymptotic to the fundamental solutions of Armstrong-Sirakov-Smart (in the uniformly elliptic case).

Material has been added concerning the Dirichlet Problem with prescribed values at marked interior points when the Riesz characteristic is strictly between 1 and 2

References in corpus (1)

• Removable singularities for nonlinear subequations