Non-existence of patterns and gradient estimates in semilinear elliptic equations with Neumann boundary conditions
| Autor: | Samuel Nordmann |
|---|---|
| Přispěvatelé: | Centre d'Analyse et de Mathématique sociales (CAMS), École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École des hautes études en sciences sociales (EHESS) |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Neumann boundary conditions
Pure mathematics Semilinear elliptic equations Robin-curvature Laplacian Poincaré inequality In Flatness Estimate 01 natural sciences Domain (mathematical analysis) Convexity symbols.namesake Symmetry Neumann boundary condition Liou- ville type results [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] 0101 mathematics Mathematical Physics Mathematics Conjecture Spectral Gap Applied Mathematics 010102 general mathematics Liouville type results Cacciopoli Inequality 010101 applied mathematics Elliptic curve Norm (mathematics) symbols De Giorgi's conjecture Laplace operator Stability Analysis |
| Zdroj: | Annales de l'Institut Henri Poincaré (C) Non Linear Analysis Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, Elsevier, In press |
| ISSN: | 0294-1449 |
| Popis: | International audience; We call pattern any non-constant stable solution of a semilinear elliptic equation with Neumann boundary conditions. A classical theorem of Casten, Holland [19] and Matano [49] states that stable patterns do not exist in convex domains. In this article, we show that the assumptions of convexity of the domain and stability of the pattern in this theorem can be relaxed in several directions. In particular, we propose a general criterion for the non-existence of patterns, dealing with possibly non-convex domains and unstable patterns. Our results unfold the interplay between the geometry of the domain, the magnitude of the nonlinearity, and the stability of patterns. We propose several applications, for example, we prove that (under a geometric assumption) there exist no patterns if the domain is shrunk or if the nonlinearity has a small magnitude. We also refine the result of Casten Holland and Matano and show that it is robust under smooth perturbations of the domain and the nonlinearity. In addition, we establish several gradient estimates for the patterns of (1). We prove a general nonlinear Cacciopoli inequality (or an inverse Poincaré inequality), stating that the L2-norm of the gradient of a solution is controlled by the L2-norm of f(u), with a constant that only depends on the domain. This inequality holds for non-homogeneous equations. We also give several flatness estimates. Our approach relies on the introduction of what we call the Robin-curvature Laplacian. This operator is intrinsic to the domain and contains much information on how the geometry of the domain affects the shape of the solutions. Finally, we extend our results to unbounded domains. It allows us to improve the results of our previous paper [53] and to extend some results on De Giorgi's conjecture to a larger class of domains. |
| Databáze: | OpenAIRE |
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