Higher order boundary Harnack principle via degenerate equations
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866929300764098560 |
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| author | Terracini, Susanna Tortone, Giorgio Vita, Stefano |
| author_facet | Terracini, Susanna Tortone, Giorgio Vita, Stefano |
| contents | As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type: \[ -\mathrm{div}\left(ρ^aA\nabla w\right)=ρ^af+\mathrm{div}\left(ρ^aF\right) \quad\textrm{in}\; Ω\] for exponents $a>-1$, where the weight $ρ$ vanishes in a non degenerate manner on a regular hypersurface $Γ$ which can be either a part of the boundary of $Ω$ or mostly contained in its interior. As an application, we extend such estimates to the ratio $v/u$ of two solutions to a second order elliptic equation in divergence form when the zero set of $v$ includes the zero set of $u$ which is not singular in the domain (in this case $ρ=u$, $a=2$ and $w=v/u$). We prove first $C^{k,α}$-regularity of the ratio from one side of the regular part of the nodal set of $u$ in the spirit of the higher order boundary Harnack principle established by De Silva and Savin. Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension $n=2$, we provide local gradient estimates for the ratio which hold also across the singular set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_00227 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Higher order boundary Harnack principle via degenerate equations Terracini, Susanna Tortone, Giorgio Vita, Stefano Analysis of PDEs 35B45, 35B65, 35B53, 35J70, 35J75 As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type: \[ -\mathrm{div}\left(ρ^aA\nabla w\right)=ρ^af+\mathrm{div}\left(ρ^aF\right) \quad\textrm{in}\; Ω\] for exponents $a>-1$, where the weight $ρ$ vanishes in a non degenerate manner on a regular hypersurface $Γ$ which can be either a part of the boundary of $Ω$ or mostly contained in its interior. As an application, we extend such estimates to the ratio $v/u$ of two solutions to a second order elliptic equation in divergence form when the zero set of $v$ includes the zero set of $u$ which is not singular in the domain (in this case $ρ=u$, $a=2$ and $w=v/u$). We prove first $C^{k,α}$-regularity of the ratio from one side of the regular part of the nodal set of $u$ in the spirit of the higher order boundary Harnack principle established by De Silva and Savin. Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension $n=2$, we provide local gradient estimates for the ratio which hold also across the singular set. |
| title | Higher order boundary Harnack principle via degenerate equations |
| topic | Analysis of PDEs 35B45, 35B65, 35B53, 35J70, 35J75 |
| url | https://arxiv.org/abs/2301.00227 |