Higher order boundary Harnack principle via degenerate equations
| Autor: | Terracini, Susanna, Tortone, Giorgio, Vita, Stefano |
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| Rok vydání: | 2022 |
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| Zdroj: | Arch. Ration. Mech. Anal. 248-2 (2024), 1-44 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00205-024-01973-1 |
| Popis: | As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type: \[ -\mathrm{div}\left(\rho^aA\nabla w\right)=\rho^af+\mathrm{div}\left(\rho^aF\right) \quad\textrm{in}\; \Omega \] for exponents $a>-1$, where the weight $\rho$ vanishes in a non degenerate manner on a regular hypersurface $\Gamma$ which can be either a part of the boundary of $\Omega$ 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 $\rho=u$, $a=2$ and $w=v/u$). We prove first $C^{k,\alpha}$-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. Comment: 36 pages, 1 figure |
| Databáze: | arXiv |
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