A mathematical preprint reports a constructed counterexample at the borderline of Dirichlet regularity. On a bounded planar Lipschitz domain, the endpoint Dirichlet Laplace operator has a range that is not closed and is therefore not surjective. In plain language, the paper identifies one domain where the endpoint problem does not cover every target in the stated space. This is an existence result about a particular constructed domain, not a claim that all Lipschitz domains behave this way.
The issue is the endpoint s = 1/2 in the regularity shift studied for Lipschitz domains. A Sobolev norm can be read here as a mathematical score for the amount of regularity a solution has to possess. The central question is whether that target norm can be controlled by the size of the forcing term at the endpoint.
The boundary is built to oscillate
This is a deterministic mathematical construction, not an empirical study. The authors work with explicitly built domains and functions in the plane, including harmonic and inhomogeneous Dirichlet solutions. Their proof is based directly on homogeneity arguments, which track how the relevant objects behave under rescaling. They present this as an elementary, self-contained route that avoids the deep harmonic-analysis results named in the paper.
The geometry uses several constructed families: circular saws, periodic saws and straight saws. The setup includes a limiting plane domain and inner domains whose boundaries converge in Hausdorff distance, a standard way of measuring how close two sets are, while the whole boundary family remains uniformly Lipschitz. The authors emphasize rapid, self-similar boundary oscillations as the important feature of the construction.
The resulting comparison is between a controlled geometric limit and increasingly fine boundary structure. The paper does not rely on sampled cases; it follows explicitly specified domains, intervals and oscillation indices through the Dirichlet calculation. That design makes the claimed failure an exact statement about the displayed sequence rather than a statistical estimate.
Where the estimate breaks
Within the constructed domains, a bounded sequence of Dirichlet right-hand sides, or forcing terms, is paired with solutions whose target Sobolev norms diverge. In the circular-saw construction, the corresponding solution norms grow with the sequence index. At the same time, the sequence vN used as boundary data remains bounded in the relevant Sobolev norm on the unit disk.
To isolate the effect, the saw analysis combines a coordinate mapping called xi with cut-off functions. The resulting cut-off saw functions have a fractional Sobolev seminorm, the part of the regularity measure that tracks fractional smoothness, bounded below by a positive multiple of log log N. Their Laplacians remain uniformly bounded in a weighted square-integrable norm. The same construction therefore places a growing solution measure beside a bounded forcing measure.
The paper then uses the closed graph or Banach open mapping theorem to connect the operator question with an a-priori estimate, meaning a finite bound linking the forcing to the target solution norm. In the glued construction, the relevant function sequence belongs to the zero-trace Sobolev space, its norm ratio tends to infinity, and no such estimate exists.
A specific counterexample, with a wider question left open
The scope of the main theorem is deliberately specific. It establishes the existence of a bounded planar Lipschitz domain with the endpoint failure, while the constructions depend on selected boundary profiles and sequences of intervals and oscillations. The supplied analysis does not turn that example into a universal statement about every Lipschitz domain or every possible profile.
The authors also state that a modified construction gives the same endpoint result for C1 domains. This extends the class of domains addressed by the paper's stated construction, although the main detailed argument concerns the Lipschitz-domain example.
The supplied document is an arXiv version 1 preprint dated 25 August 2026. Its manuscript date line gives 26 August 2026.
Paper data and sources
Original title: End-point non-regularity of the Dirichlet problem on Lipschitz domains -- An elementary proof
Authors: Martin Costabel, Monique Dauge
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text