A universal H\'older estimate up to dimension 4 for stable solutions to half-Laplacian semilinear equations
| Autor: | Xavier Cabré, Tomás Sanz-Perela |
|---|---|
| Přispěvatelé: | Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. TF-EDP - Grup de Teoria de Funcions i Equacions en Derivades Parcials |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en derivades parcials [Àrees temàtiques de la UPC]
Equacions en derivades parcials Applied Mathematics Differential equations Partial Stable solutions 35 Partial differential equations::35B Qualitative properties of solutions [Classificació AMS] Half-Laplacian Mathematics - Analysis of PDEs 35J61 35R11 35B45 35B35 35B65 Extremal solution FOS: Mathematics Interior estimates Analysis Dirichlet problem Analysis of PDEs (math.AP) |
| Popis: | We study stable solutions to the equation $(-\Delta)^{1/2} u = f(u)$, posed in a bounded domain of $\mathbb{R}^n$. For nonnegative convex nonlinearities, we prove that stable solutions are smooth in dimensions $n\leq 4$. This result, which was known only for $n=1$, follows from a new interior H\"older estimate that is completely independent of the nonlinearity $f$. A main ingredient in our proof is a new geometric form of the stability condition. It is still unknown for other fractions of the Laplacian and, surprisingly, it requires convexity of the nonlinearity. From it, we deduce higher order Sobolev estimates that allow us to extend the techniques developed by Cabr\'e, Figalli, Ros-Oton, and Serra for the Laplacian. In this way we obtain, besides the H\"older bound for $n\leq 4$, a universal $H^{1/2}$ estimate in all dimensions. Our $L^\infty$ bound is expected to hold for $n\leq 8$, but this has been settled only in the radial case or when $f(u) = \lambda e^u$. For other fractions of the Laplacian, the expected optimal dimension for boundedness of stable solutions has been reached only when $f(u) = \lambda e^u$, even in the radial case. Comment: 38 pages, 1 figure. Version 2 contains a new Liouville result (Corollary 1.5) which is a corollary of our main estimate. Version 3 contains a short comment (Remark 3.1) on the difficulties of extending the result to other powers of the Laplacian. Version 4 is the final version after referee report and proofs, we have added a comment to facilitate the interpretation of the colors of Figure 1 |
| Databáze: | OpenAIRE |
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