Higher order boundary Harnack principles in Dini type domains
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914942086545408 |
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| author | Jeon, Seongmin Vita, Stefano |
| author_facet | Jeon, Seongmin Vita, Stefano |
| contents | Aim of this paper is to provide higher order boundary Harnack principles [De Silva-Savin 15] for elliptic equations in divergence form under Dini type regularity assumptions on boundaries, coefficients and forcing terms. As it was proven in [Terracini-Tortone-Vita 22], the ratio $v/u$ of two solutions vanishing on a common portion $Γ$ of a regular boundary solves a degenerate elliptic equation whose coefficients behave as $u^2$ at $Γ$. Hence, for any $k\geq 1$ we provide $C^k$ estimates for solutions to the auxiliary degenerate equation under double Dini conditions, actually for general powers of the weight $a>-1$, and we imply $C^k$ estimates for the ratio $v/u$ under triple Dini conditions, as a corollary in the case $a=2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_05535 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Higher order boundary Harnack principles in Dini type domains Jeon, Seongmin Vita, Stefano Analysis of PDEs 35B45, 35B65, 35J70, 35J75 Aim of this paper is to provide higher order boundary Harnack principles [De Silva-Savin 15] for elliptic equations in divergence form under Dini type regularity assumptions on boundaries, coefficients and forcing terms. As it was proven in [Terracini-Tortone-Vita 22], the ratio $v/u$ of two solutions vanishing on a common portion $Γ$ of a regular boundary solves a degenerate elliptic equation whose coefficients behave as $u^2$ at $Γ$. Hence, for any $k\geq 1$ we provide $C^k$ estimates for solutions to the auxiliary degenerate equation under double Dini conditions, actually for general powers of the weight $a>-1$, and we imply $C^k$ estimates for the ratio $v/u$ under triple Dini conditions, as a corollary in the case $a=2$. |
| title | Higher order boundary Harnack principles in Dini type domains |
| topic | Analysis of PDEs 35B45, 35B65, 35J70, 35J75 |
| url | https://arxiv.org/abs/2305.05535 |