The obstacle problem for a class of degenerate fully nonlinear operators

• João Vítor da Silva ^[1] ; Hernán Vivas ^[2]
  1. [1] Universidade Estadual de Campinas

     Universidade Estadual de Campinas

     Brasil

     
  2. [2] Universidad de Buenos Aires

     Universidad de Buenos Aires

     Argentina

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 37, Nº 5, 2021, págs. 1991-2020
• Idioma: inglés
• DOI: 10.4171/rmi/1256
• Texto completo no disponible (Saber más ...)
• Resumen

  • We study the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on the gradient:

    {min{f−|Du|γF(D2u),u−ϕ}u=0=g in Ω, on ∂Ω, for some degeneracy parameter γ≥0, uniformly elliptic operator F, bounded source term f, and suitably smooth obstacle ϕ and boundary datum g. We obtain existence/uniqueness of solutions and prove sharp regularity estimates at the free boundary points, namely ∂{u>ϕ}∩Ω. In particular, for the homogeneous case (f≡0) we get that solutions are C1,1 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 prove several non-degeneracy properties of 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 prototype given by an operator of the form G[u]=|Du|γΔu, with γ>0 and f≡1.