A priori regularity estimates for equations degenerating on nodal sets
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arXiv
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| Format: | Preprint |
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2024
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| author | Terracini, Susanna Tortone, Giorgio Vita, Stefano |
| author_facet | Terracini, Susanna Tortone, Giorgio Vita, Stefano |
| contents | We prove a priori and a posteriori Hölder bounds and Schauder $C^{1,α}$ estimates for continuous solutions of degenerate elliptic equations with variable coefficients of the form $$ \mathrm{div}\left(|u|^a A\nabla w\right)=0\qquad\mathrm{in \ }Ω\subset\mathbb R^2,\quad a\in\mathbb R, $$ where the weight $u$ is itself a solution to an elliptic equation of the type $\mathrm{div}(A \nabla u) = 0$, with $A$ a Lipschitz-continuous, uniformly elliptic matrix. The function $u$ is allowed to have a nontrivial, possibly singular nodal set.
The estimates are uniform with respect to $u$ within a class of normalized solutions having bounded Almgren frequency. In the special case $a = 2$, our results apply to the ratio of two solutions to the same elliptic equation sharing a common zero set. Precisely, we prove higher-order boundary Harnack principles on nodal domains, via the derived Schauder estimates for the associated degenerate equations. The results are based upon a fine blow-up argument, a Liouville theorem, and quasiconformal maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06980 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A priori regularity estimates for equations degenerating on nodal sets Terracini, Susanna Tortone, Giorgio Vita, Stefano Analysis of PDEs We prove a priori and a posteriori Hölder bounds and Schauder $C^{1,α}$ estimates for continuous solutions of degenerate elliptic equations with variable coefficients of the form $$ \mathrm{div}\left(|u|^a A\nabla w\right)=0\qquad\mathrm{in \ }Ω\subset\mathbb R^2,\quad a\in\mathbb R, $$ where the weight $u$ is itself a solution to an elliptic equation of the type $\mathrm{div}(A \nabla u) = 0$, with $A$ a Lipschitz-continuous, uniformly elliptic matrix. The function $u$ is allowed to have a nontrivial, possibly singular nodal set. The estimates are uniform with respect to $u$ within a class of normalized solutions having bounded Almgren frequency. In the special case $a = 2$, our results apply to the ratio of two solutions to the same elliptic equation sharing a common zero set. Precisely, we prove higher-order boundary Harnack principles on nodal domains, via the derived Schauder estimates for the associated degenerate equations. The results are based upon a fine blow-up argument, a Liouville theorem, and quasiconformal maps. |
| title | A priori regularity estimates for equations degenerating on nodal sets |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.06980 |