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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.14779 |
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| _version_ | 1866911082754342912 |
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| author | Wang, Jiangwen Jiang, Feida |
| author_facet | Wang, Jiangwen Jiang, Feida |
| contents | We study a series of regularity results for solutions to a degenerate or singular fully nonlinear integro-differential equation of the form
$$- \big( σ_{1}(|Du|) + a(x) σ_{2}(|Du|) \big) \mathcal{I}_τ(u,x) = f(x).$$
In the degenerate case, we establish borderline regularity, provided the inverse of the degeneracy law $ σ_{2}$ is Dini-continuous. In addition, we show Schauder-type higher regularity at local extremum points for a specific non-local degenerate equation. In the singular case, we establish Hölder continuity of the gradient for solutions to a general non-local equation.
It is noteworthy that these results are new even in the case $ a(x) \equiv 0 $. Finally, as a byproduct of the borderline regularity analysis, we demonstrate how our methods can be applied to study of the corresponding regularity for a class of degenerate non-local normalized $ p$-Laplacian equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14779 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularity of solutions for degenerate or singular fully nonlinear integro-differential equations Wang, Jiangwen Jiang, Feida Analysis of PDEs We study a series of regularity results for solutions to a degenerate or singular fully nonlinear integro-differential equation of the form $$- \big( σ_{1}(|Du|) + a(x) σ_{2}(|Du|) \big) \mathcal{I}_τ(u,x) = f(x).$$ In the degenerate case, we establish borderline regularity, provided the inverse of the degeneracy law $ σ_{2}$ is Dini-continuous. In addition, we show Schauder-type higher regularity at local extremum points for a specific non-local degenerate equation. In the singular case, we establish Hölder continuity of the gradient for solutions to a general non-local equation. It is noteworthy that these results are new even in the case $ a(x) \equiv 0 $. Finally, as a byproduct of the borderline regularity analysis, we demonstrate how our methods can be applied to study of the corresponding regularity for a class of degenerate non-local normalized $ p$-Laplacian equations. |
| title | Regularity of solutions for degenerate or singular fully nonlinear integro-differential equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.14779 |