Gradient estimates for the $p$-Laplacian perfect conductivity problem with partially flat and $C^{1,γ}$ inclusions
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918289033134080 |
|---|---|
| author | Dong, Hongjie Xu, Longjuan |
| author_facet | Dong, Hongjie Xu, Longjuan |
| contents | In this paper, we investigate the gradient estimates for solutions to the perfect conductivity problem with two closely spaced perfect conductors embedded in a homogeneous matrix, modeled by $p$-Laplacian elliptic equations. We first prove that the gradient of the solution remains bounded when the conductors possess partially ``flat" boundaries. This contrasts with the case involving strictly convex inclusions, where the gradient can blow up. Second, for conductors with $C^{1,γ}$ boundaries ($γ\in(0,1)$), we establish both upper and lower bounds on the gradient, with optimal blow-up rates. Furthermore, we provide precise asymptotic expansions in some special cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09435 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gradient estimates for the $p$-Laplacian perfect conductivity problem with partially flat and $C^{1,γ}$ inclusions Dong, Hongjie Xu, Longjuan Analysis of PDEs In this paper, we investigate the gradient estimates for solutions to the perfect conductivity problem with two closely spaced perfect conductors embedded in a homogeneous matrix, modeled by $p$-Laplacian elliptic equations. We first prove that the gradient of the solution remains bounded when the conductors possess partially ``flat" boundaries. This contrasts with the case involving strictly convex inclusions, where the gradient can blow up. Second, for conductors with $C^{1,γ}$ boundaries ($γ\in(0,1)$), we establish both upper and lower bounds on the gradient, with optimal blow-up rates. Furthermore, we provide precise asymptotic expansions in some special cases. |
| title | Gradient estimates for the $p$-Laplacian perfect conductivity problem with partially flat and $C^{1,γ}$ inclusions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.09435 |