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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815300554&_mathId=si1.gif&_user=111111111&_pii=S0001870815300554&_rdoc=1&_issn=00018708&md5=f66d7e02c92bf962ed0f962362f7b27d" title="Click to view the MathML source">Rⁿclass="mathContainer hidden">class="mathCode">${\mathbf{R}}^n$. The main results apply, in particular, to subequations with a Riesz characteristic class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815300554&_mathId=si190.gif&_user=111111111&_pii=S0001870815300554&_rdoc=1&_issn=00018708&md5=965cab9abc7f822bda5f95b15511499d" title="Click to view the MathML source">p≥2class="mathContainer hidden">class="mathCode">$p\ge 2$. 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815300554&_mathId=si193.gif&_user=111111111&_pii=S0001870815300554&_rdoc=1&_issn=00018708&md5=50f856141dd6f53ed61bc9af76e4b460" title="Click to view the MathML source">Θ_jK_p(x−x_j)class="mathContainer hidden">class="mathCode">${\mathrm{\Theta }}_j{K}_p(x-x_j)$ where at any prescribed finite set of points class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815300554&_mathId=si5.gif&_user=111111111&_pii=S0001870815300554&_rdoc=1&_issn=00018708&md5=7133c7ba92c3537229abd70132cdf436" title="Click to view the MathML source">{x₁,...,x_k}class="mathContainer hidden">class="mathCode">$\{x_1,...,x_k\}$ in the domain and any finite set of positive real numbers class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815300554&_mathId=si235.gif&_user=111111111&_pii=S0001870815300554&_rdoc=1&_issn=00018708&md5=0b4ed55dffc39efbd76385baf50a3953" title="Click to view the MathML source">Θ₁,...,Θ_kclass="mathContainer hidden">class="mathCode">${\mathrm{\Theta }}_1,...,{\mathrm{\Theta }}_k$. 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815300554&_mathId=si7.gif&_user=111111111&_pii=S0001870815300554&_rdoc=1&_issn=00018708&md5=15249c51a4cb566a0615ce63630f5701" title="Click to view the MathML source">Θ_j|x−x_j|^2−pclass="mathContainer hidden">class="mathCode">${\mathrm{\Theta }}_j|x-x_j{|}^{2-p}$ for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815300554&_mathId=si8.gif&_user=111111111&_pii=S0001870815300554&_rdoc=1&_issn=00018708&md5=048ba860a64a078b78ce115cada4eec0" title="Click to view the MathML source">1≤p<2class="mathContainer hidden">class="mathCode">$1\le p<2$.

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