The Dirichlet problem with prescribed interior singularities

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• 作者：F. Reese Harvey ; H. Blaine Lawson Jr. ; ¹ ; ^{blaine@math.sunysb.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Nonlinear elliptic equations ; Dirichlet problem ; Prescribed asymptotic singularities ; Riesz kernel ; Riesz characteristic ; Nonlinear Green's function
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：5 November 2016
• 年：2016
• 卷：303
• 期：Complete
• 页码：1319-1357
• 全文大小：636 K

文摘

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 Rⁿ${\mathbf{R}}^n$. The main results apply, in particular, to subequations with a Riesz characteristic p≥2$p\ge 2$. 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 Θ_jK_p(x−x_j)${\mathrm{\Theta }}_j{K}_p(x-x_j)$ where at any prescribed finite set of points {x₁,...,x_k}$\{x_1,...,x_k\}$ in the domain and any finite set of positive real numbers Θ₁,...,Θ_k${\mathrm{\Theta }}_1,...,{\mathrm{\Theta }}_k$. 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 Θ_j|x−x_j|^2−p${\mathrm{\Theta }}_j|x-x_j{|}^{2-p}$ for 1≤p<2$1\le p<2$.

The main results apply similarly with prescribed singularities asymptotic to the fundamental solutions of Armstrong–Sirakov–Smart (in the uniformly elliptic case).