Publicación: Gradient Bounds for Elliptic Problems Singular at the Boundary
No hay miniatura disponible
Fecha
2011-07-05
Autores
Editor/a
Director/a
Tutor/a
Coordinador/a
Prologuista
Revisor/a
Ilustrador/a
Derechos de acceso
info:eu-repo/semantics/openAccess
Título de la revista
ISSN de la revista
Título del volumen
Editor
Springer Nature
Resumen
Let Ω be a bounded smooth domain in RN, N ≧ 2, and let us denote by d(x) the distance function d(x, ∂Ω). We study a class of singular Hamilton-Jacobi equations, arising from stochastic control problems, whose simplest model is where f belongs to W 1,∞ loc (Ω) and is (possibly) singular at ∂Ω, C ε W1,∞ (Ω)(with no sign condition) and the field B ε W1,∞ (Ω)N has an outward direction and satisfies B · v ≧ α at ∂Ω (ν is the outward normal). Despite the singularity in the equation, we prove gradient bounds up to the boundary and the existence of a (globally) Lipschitz solution. We show that in some cases this is the unique bounded solution. We also discuss the stability of such estimates with respect to α, as α vanishes, obtaining Lipschitz solutions for first order problems with similar features. The main tool is a refined weighted version of the classical Bernstein method to get gradient bounds; the key role is played here by the orthogonal transport component of the Hamiltonian.
Descripción
Categorías UNESCO
Palabras clave
viscosity solution, elliptic problem, neumann condition, transport term, elliptic regularity
Citación
Leonori, T., Porretta, A. Gradient Bounds for Elliptic Problems Singular at the Boundary. Arch Rational Mech Anal 202, 663–705 (2011). https://doi.org/10.1007/s00205-011-0436-9
Centro
Facultades y escuelas::Facultad de Ciencias
Departamento
Matemáticas Fundamentales