Maximally singular weak solutions of Laplace equations

Abstract

It is known that there exists an explicit function $F$ in $L^2(\mathrm{\Omega })$, where $\mathrm{\Omega }$ is a given bounded open subset of ${ℝ}^N$, such that the corresponding weak solution of the Poisson boundary value problem $-\mathrm{\Delta }u=F(x)$, $u\in H_0^1(\mathrm{\Omega })$, is maximally singular; that is, the singular set of $u$ has Hausdorff dimension equal to ${(N-4)}^{+}$. This constant is optimal, i.e., the largest possible. Here, we show that much more is true: when $N\ge 5$, there exists $F\in L^2(\mathrm{\Omega })$ 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 $\mathrm{\Omega }$. 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 $C_{\mathrm{loc}}^{2,\alpha }$ for arbitrary $\alpha \in (0,1)$) in any prescribed open subset ${\mathrm{\Omega }}_r$ of $\mathrm{\Omega }$, while they are maximally singular in its complement $\mathrm{\Omega }\setminus {\mathrm{\Omega }}_r$. We indicate several open problems.