Explicit examples of Lipschitz, one-homogeneous solutions of log-singular planar elliptic systems

TL;DR

This paper constructs explicit Lipschitz, one-homogeneous solutions for certain singular elliptic systems, revealing new solutions that are minimizers and exploring their uniqueness under specific conditions.

Contribution

It provides explicit examples of non-trivial solutions for singular elliptic systems with logarithmic singularities, expanding understanding of solution structures in such systems.

Findings

Existence of Lipschitz, one-homogeneous solutions in singular elliptic systems.

Conditions under which these solutions are minimizers.

Analysis of the uniqueness of these solutions.

Abstract

We give examples of systems of Partial Differential Equations that admit non-trivial, Lipschitz and one-homogeneous solutions in the form $u(R,\theta) = Rg(\theta)$, where $(R,\theta)$ are plane polar coordinates and $g: \mathbb{R}^{2} \to \mathbb{R}^{m}$, $m \geq 2$. The systems are singular in the sense that they arise as the Euler-Lagrange equations of the functionals $I(u) = \int_{B}W(x,\nabla u(x))\,dx$, where $D_{F}W(x,F)$ behaves like $\frac{1}{|x|}$ as $|x| \to 0$ and $W$ satisfies an ellipticity condition. Such solutions cannot exist when $|x|D_{F}W(x,F) \to 0$ as $|x| \to 0$, so the condition is optimal. The associated analysis exploits the well-known Fefferman-Stein duality. We also discuss conditions for the uniqueness of these one-homogeneous solutions and demonstrate that they are minimizers of certain variational functionals.