On a few counterexamples to solvability of the $div$ equation in domains with external cusps

TL;DR

The paper investigates solvability of the divergence equation $\mathrm{div}\ \mathbf u=f$ with zero Dirichlet data for $f\in \overline{L^p(\Omega)}$ in domains with external cusps. It extends Luc Tartar's 2D counterexample to general $p$ and to higher dimensions, and further to Hölder domains with both fixed and non-fixed exponents, by constructing explicit cusp geometries and right-hand side profiles that prevent $\mathbf u$ from belonging to $ (W^{1,p}_0(\Omega))^N$. The main contribution is a precise nonexistence criterion that depends on cusp shape, namely power type cusps $|y|<x^m$ with $m>1$ and logarithmic cusps $|y|<x(-\ln x)^{-r}$, along with their higher-dimensional analogues, together with the corresponding parameter ranges for $\alpha$ or logarithmic exponents. These results delineate the limits of extending the Bogovskii operator to domains with external cusps and sharpen understanding of how boundary singularities govern solvability in $W^{1,p}_0$ spaces.

Abstract

This paper examines the solvability of the equation $\mathrm{div} \ \mathbf{u} = f$ with a zero Dirichlet boundary condition for $\mathbf{u}$. A classical result establishes that for a bounded domain $Ω\subset \mathbb{R}^N$ with a Lipschitz boundary and for $f \in L^p(Ω)$ with zero mean value there exists a solution $\mathbf{u} \in (W_0^{1, p}(Ω))^N$ for $1 < p < \infty$ with the $W^{1,p}$ norm controlled by the $L^p$ norm of the right-hand side $f$. The results were extended to John domains and excluded the existence of the solution operator in domains with external cusps. Our aim is to specify at least some classes of the right-hand sides for which the problem cannot have a solution in the space $W^{1,p}_0(Ω)$. We first extend the counterexample by Luc Tartar originally formulated for right-hand side functions in $\overline{L^2}$ in two space dimensions to a more general class of functions in $\overline{L^p}$ spaces and a more general type of singular domains. We then generalize this result to an arbitrary dimension $N$. Returning to two space dimensions, we investigate domains with boundary properties superior to those of previously studied Hölder continuous domains and construct counterexamples also in this situation.

Figures (4)

• Figure 1: Domain $\Omega$ used in Tartar's counterexample $(m = 2)$
• Figure 2: Domain $\Omega$ for $m = \frac{1}{2}$
• Figure 3: Domain $\Omega$ for $N=3$
• Figure 4: Domain $\Omega$ for $r =\frac{3}{2}$

Theorems & Definitions (13)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• proof
• proof
• Example 3.1
• proof
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• proof