Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions
| Autor: | Stefano Vita, Susanna Terracini, Yannick Sire |
|---|---|
| Přispěvatelé: | Sire, Y, Terracini, S, Vita, S |
| Rok vydání: | 2020 |
| Předmět: |
Schauder estimates
Class (set theory) Pure mathematics Liouville type theorems Mathematics::Analysis of PDEs Type (model theory) Degenerate and singular elliptic equations 01 natural sciences Mathematics::Numerical Analysis Schauder estimate Matrix (mathematics) Mathematics - Analysis of PDEs Liouville type theorem FOS: Mathematics Fermi coordinates 0101 mathematics Divergence (statistics) blow-up fractional divergence form elliptic operator fractional Laplacian Mathematics Applied Mathematics 010102 general mathematics Degenerate energy levels Degenerate and singular elliptic equation 010101 applied mathematics Singular problems 35J70 35J75 35R11 35B40 35B44 35B53 Analysis Analysis of PDEs (math.AP) |
| Zdroj: | Communications in Partial Differential Equations. 46:310-361 |
| ISSN: | 1532-4133 0360-5302 |
| DOI: | 10.1080/03605302.2020.1840586 |
| Popis: | We consider a class of equations in divergence form with a singular/degenerate weight $$-\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)\; \quad\textrm{or} \ \textrm{div}(|y|^aF(x,y))\;.$$ Under suitable regularity assumptions for the matrix $A$ and $f$ (resp. $F$) we prove H\"older continuity of solutions which are even in $y\in\mathbb{R}$, and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the $C^{0,\alpha}$ and $C^{1,\alpha}$ a priori bounds for approximating problems in the form $$-\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x,y)\nabla u)=(\varepsilon^2+y^2)^{a/2} f(x,y)\; \quad\textrm{or} \ \textrm{div}((\varepsilon^2+y^2)^{a/2}F(x,y))$$ as $\varepsilon\to 0$. Finally, we derive $C^{0,\alpha}$ and $C^{1,\alpha}$ bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems. Comment: 47 pages |
| Databáze: | OpenAIRE |
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