Tommaso, LeonoriPorretta, Alessio2024-11-212024-11-212011-07-05Leonori, 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/s00205-011-0436-90003-9527; e-ISSN: 1432-0673https://doi.org/10.1007/s00205-011-0436-9https://hdl.handle.net/20.500.14468/24467Let 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.eninfo:eu-repo/semantics/openAccess12 Matemáticasartículoviscosity solutionelliptic problemneumann conditiontransport termelliptic regularity