𝒞^1,βregularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations

Abstract

We prove Hölder regularity of the gradient, up to the boundary for solutions of some fully-nonlinear, degenerate elliptic equations, with degeneracy coming from the gradient.

Ergodic pairs for singular or degenerate fully nonlinear operators

“…The case α < 0. As proved in [7], if α < 0 a sub solution u and super solution v of equation (2.1) satisfy respectively in the viscosity sense…”

Optimal C1,α regularity for degenerate fully nonlinear elliptic equations with Neumann boundary condition

“…For Dirichlet boundary data, the regularity up to the boundary is fairly well understood. To know, Berindelli and Demengel [7] proved C 1,α (Ω) estimates in the presence of a regular boundary datum. However, for the Neumann problem, there are still not many results.…”

“…7) for universal constant µ > 0, M F > 0 and C. Then applying (3.7) to translations of u we obtain a C α modulus of continuity for u at the bottom ϒ by a standard iterative argument. Then, (3.6) follows by interior regularity.…”

Regularity properties of viscosity solutions for fully nonlinear equations on the model of the anisotropic p →-Laplacian

“…A further question we ask is : does the C 1 or C 1,β regularity holds, as in the case of the classical pLaplacian, [18], [12]? A first step would consist in proving the C 1 regularity when the right hand side is zero and then deduce from it the case f = 0 by methods as in [15], [3], but even in the case f ≡ 0 i have no intuition about the truthfullness of this result. The usual methods in the theory of viscosity solutions, ( [8], [17]), cannot directly be applied to the present case, one of the key argument of their proofs being the uniform ellipticity of the operator.…”