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

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 $\Omega \subset \mathbb{R}^N$ with a Lipschitz boundary and for $f \in L^p(\Omega)$ with zero mean value there exists a solution $\mathbf{u} \in (W_0^{1, p}(\Omega))^N$ 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 $W^{1,p}_0(\Omega)$. 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 H\"older continuous domains and construct counterexamples also in this situation.