The obstacle problem for a class of degenerate fully nonlinear operators

2019/05/15 by Da Silva, João Vitor, Vivas, Hernán · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1905.06146

Abstract

We study the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on the gradient: min \f-|Du|^γF(D²u),u-ϕ\ = 0 \textrm in Ω. We obtain existence of solutions and prove sharp regularity estimates along the free boundary points, namely ∂\u>ϕ\ ∩ Ω. In particular, for the homogeneous case (f≡0) we get that solutions are C^1,1 at free boundary points, in the sense that they detach from the obstacle in a quadratic fashion, thus beating the optimal regularity allowed for such degenerate operators. We also present further features of the solutions and partial results regarding the free boundary. These are the first results for obstacle problems driven by degenerate type operators in non-divergence form and they are a novelty even for the simpler scenario given by an operator of the form G[u] = |Du|^γΔu.

Cited by

• Boundary Hölder gradient estimates for fully nonlinear degenerate or singular parabolic equations
• Comparison Principle, A.B.P.-type estimates for solutions of quasi-linear elliptic equations in non-divergence form and some implications

Related

• Sharp regularity for degenerate obstacle type problems: a geometric approach
• The thin obstacle problem for some variable coefficient degenerate elliptic operators
• The double obstacle problem on non divergence form
• A brief survey on the obstacle problem
• Regularity of the singular set in the fully nonlinear obstacle problem