JournalsrmiVol. 37, No. 5pp. 1991–2020

The obstacle problem for a class of degenerate fully nonlinear operators

  • João Vítor da Silva

    Universidade Estadual de Campinas, Brazil
  • Hernán Vivas

    Universidad de Buenos Aires and Centro Marplatense de Investigaciones Matemáticas, Mar del Plata, Argentina
The obstacle problem for a class of degenerate fully nonlinear operators cover

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Abstract

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

{min{fDuγF(D2u),uϕ}=0 in Ω,u=g on Ω,\left\{\begin{array}{rll} \min\left\{f-|Du|^\gamma F(D^2u),u-\phi\right\} &= 0 & \textrm{ in } \Omega,\\ u & = g & \textrm{ on } \partial \Omega, \end{array}\right.

for some degeneracy parameter γ0\gamma\geq 0, uniformly elliptic operator FF, bounded source term ff, and suitably smooth obstacle ϕ\phi and boundary datum gg. We obtain existence/uniqueness of solutions and prove sharp regularity estimates at the free boundary points, namely {u>ϕ}Ω\partial\{u>\phi\} \cap \Omega. In particular, for the homogeneous case (f0f\equiv0) we get that solutions are C1,1C^{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 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]=DuγΔu\mathcal{G}[u] = |Du|^\gamma\Delta u, with γ>0\gamma >0 and f1f \equiv 1.

Cite this article

João Vítor da Silva, Hernán Vivas, The obstacle problem for a class of degenerate fully nonlinear operators. Rev. Mat. Iberoam. 37 (2021), no. 5, pp. 1991–2020

DOI 10.4171/RMI/1256