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We study the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on the gradient:
for some degeneracy parameter , uniformly elliptic operator , bounded source term , and suitably smooth obstacle and boundary datum . We obtain existence/uniqueness of solutions and prove sharp regularity estimates at the free boundary points, namely . In particular, for the homogeneous case () we get that solutions are 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 , with and .
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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. 1991–2020DOI 10.4171/RMI/1256