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Prescribing singularities of weak solutions of a nonlinear elliptic system in the plane

Bogdan Petraszczuk

Abstract

Inspired by Frehse's [1] 1973 work, we show that his elliptic system $Δu = F(u, \nabla u)$ in the plane has bounded weak solutions $u$ with arbitrarily prescribed singular sets.

Prescribing singularities of weak solutions of a nonlinear elliptic system in the plane

Abstract

Inspired by Frehse's [1] 1973 work, we show that his elliptic system in the plane has bounded weak solutions with arbitrarily prescribed singular sets.
Paper Structure (3 sections, 3 theorems, 57 equations)

This paper contains 3 sections, 3 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Proof of Theorem \ref{['Main theorem']}

Key Result

Theorem 1.1

Fix a small radius $0 < r < \frac{1}{e}$ and consider the ball $B := B(0, r) \subset \mathbb{R}^2$. For every compact subset $K$ within the ball $B$, there exists a solution $u \in W^{1,2}(B, \, \mathbb{R}^2) \cap L^{\infty}$ to a nonlinear elliptic system: where $F$ defined in (Frehse diff equation). This solution $u$ is singular on $K$ and smooth elsewhere.

Theorems & Definitions (5)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof