Maximally 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.