The Dirichlet problem with prescribed interior singularities

F. Reese Harvey; H. Blaine Lawson

doi:10.1016/j.aim.2016.08.024

The Dirichlet problem with prescribed interior singularities

F. Reese Harvey
, H. Blaine Lawson

Mathematics

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

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. The main results apply, in particular, to subequations with a Riesz characteristic p≥2.

[Formula

at any prescribed finite set of points {x₁,...,x_k} in the domain and any finite set of positive real numbers Θ₁,...,Θ_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 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).

Original language	English
Pages (from-to)	1319-1357
Number of pages	39
Journal	Advances in Mathematics
Volume	303
DOIs	https://doi.org/10.1016/j.aim.2016.08.024
State	Published - Nov 5 2016

Keywords

Dirichlet problem
Nonlinear elliptic equations
Nonlinear Green's function
Prescribed asymptotic singularities
Riesz characteristic
Riesz kernel

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10.1016/j.aim.2016.08.024

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title = "The Dirichlet problem with prescribed interior singularities",

abstract = "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. The main results apply, in particular, to subequations with a Riesz characteristic p≥2.[Formulaat 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 finite-type 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).",

keywords = "Dirichlet problem, Nonlinear elliptic equations, Nonlinear Green's function, Prescribed asymptotic singularities, Riesz characteristic, Riesz kernel",

author = "Harvey, \{F. Reese\} and Lawson, \{H. Blaine\}",

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T1 - The Dirichlet problem with prescribed interior singularities

AU - Harvey, F. Reese

AU - Lawson, H. Blaine

PY - 2016/11/5

Y1 - 2016/11/5

N2 - 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. The main results apply, in particular, to subequations with a Riesz characteristic p≥2.[Formulaat 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 finite-type 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).

AB - 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. The main results apply, in particular, to subequations with a Riesz characteristic p≥2.[Formulaat 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 finite-type 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).

KW - Dirichlet problem

KW - Nonlinear elliptic equations

KW - Nonlinear Green's function

KW - Prescribed asymptotic singularities

KW - Riesz characteristic

KW - Riesz kernel

UR - https://www.scopus.com/pages/publications/84988024276

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DO - 10.1016/j.aim.2016.08.024

M3 - Article

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SN - 0001-8708

VL - 303

SP - 1319

EP - 1357

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

The Dirichlet problem with prescribed interior singularities

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Keywords

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