Maximally singular solutions of Laplace equations

TL;DR

This paper constructs solutions to Laplace equations with maximal singularity sets of Hausdorff dimension N-4, demonstrating pointwise concentration of singularities and solutions with contrasting regularity and singularity properties.

Contribution

It proves the existence of solutions with maximal singular sets of Hausdorff dimension N-4 at all points and constructs solutions with prescribed regularity in parts of the domain and maximal singularity elsewhere.

Findings

Existence of solutions with Hausdorff dimension N-4 singular sets at all points.

Construction of solutions with regularity in some regions and maximal singularity in others.

Identification of open problems related to singular solutions of Laplace equations.

Abstract

It is known that there exists an explicit function $F$ in $L^2(\Omega)$, where $\Omega$ is a given bounded open subset of $\mathbb{R}^N$, such that the corresponding weak solution of the Laplace BVP $-\Delta u=F(x)$, $u\in H_0^1(\Omega)$, is maximally singular; that is, the singular set of $u$ (defined in the Introduction) has the Hausdorff dimension equal to $(N-4)^+$. This constant is optimal, i.e., the largest possible. Here, we show that much more is true: when $N \geq 5$, there exists $F\in L^2(\Omega)$ such that the corresponding weak solution has the pointwise concentration of singular set of $u$, in the sense of the Hausdorff dimension, equal to $N-4$ at all points of $\Omega$. We also consider the problem of generating weak solutions with the property of contrast; that is, we construct solutions $u$ that are regular (more specifically, of class $C_{loc}^{2,\alpha}$ for arbitrary $\alpha\in(0,1)$) in any prescribed open subset $\Omega_r$ of $\Omega$, while they are maximally singular in its complement $\Omega\setminus\Omega_r$. We indicate several open problems.