RECORD
On a few counterexamples to solvability of the $div$ equation in domains with external cusps
Analysis of PDEs · Submitted 19 Mar 2025
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 $W^{1,p}_0(\Omega)$.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This paper examines the solvability of the equation $\mathrm{div} \ \mathbf{u} = f$ with a zero Dirichlet boundary condition for $\mathbf{u}$. | background 0.80 | 0.64 | 0.04 | asserted 0.87 |
| s1 | 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 $f$. | background 1.00 | 0.04 | 0.04 | asserted 1.00 |
| s2 | The results were extended to John domains and excluded the existence of the solution operator in domains with external cusps. | background 0.97 | 0.10 | 0.09 | asserted 1.00 |
| s3 | 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 $W^{1,p}_0(\Omega)$. | contribution 0.96 | 0.39 | 0.13 | asserted 0.65 |
| s4 | 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. | contribution 0.98 | 0.91 | 0.13 | asserted 0.99 |
| s5 | We then generalize this result to an arbitrary dimension $N$. | contribution 0.94 | 0.91 | 0.08 | asserted 0.96 |
| s6 | 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. | contribution 0.94 | 0.93 | 0.15 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original 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 $\Omega \subset \mathbb{R}^N$ 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.
Human review
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.