Hyperbolic regularization effects for degenerate elliptic equations
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908925818830848 |
|---|---|
| author | Lamy, Xavier Tione, Riccardo |
| author_facet | Lamy, Xavier Tione, Riccardo |
| contents | This paper investigates the regularity of Lipschitz solutions $u$ to the general two-dimensional equation $\text{div}(G(Du))=0$ with highly degenerate ellipticity. Just assuming strict monotonicity of the field $G$ and heavily relying on the differential inclusions point of view, we establish a pointwise gradient localization theorem and we show that the singular set of nondifferentiability points of $u$ is $\mathcal{H}^1$-negligible. As a consequence, we derive new sharp partial $C^1$ regularity results under the assumption that $G$ is degenerate only on curves. This is done by exploiting the hyperbolic structure of the equation along these curves, where the loss of regularity is compensated using tools from the theories of Hamilton-Jacobi equations and scalar conservation laws. Our analysis recovers and extends all the previously known results, where the degeneracy set was required to be zero-dimensional. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04753 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hyperbolic regularization effects for degenerate elliptic equations Lamy, Xavier Tione, Riccardo Analysis of PDEs This paper investigates the regularity of Lipschitz solutions $u$ to the general two-dimensional equation $\text{div}(G(Du))=0$ with highly degenerate ellipticity. Just assuming strict monotonicity of the field $G$ and heavily relying on the differential inclusions point of view, we establish a pointwise gradient localization theorem and we show that the singular set of nondifferentiability points of $u$ is $\mathcal{H}^1$-negligible. As a consequence, we derive new sharp partial $C^1$ regularity results under the assumption that $G$ is degenerate only on curves. This is done by exploiting the hyperbolic structure of the equation along these curves, where the loss of regularity is compensated using tools from the theories of Hamilton-Jacobi equations and scalar conservation laws. Our analysis recovers and extends all the previously known results, where the degeneracy set was required to be zero-dimensional. |
| title | Hyperbolic regularization effects for degenerate elliptic equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.04753 |