Ricerc@Sapienza

In this note, we prove C1,γ regularity for solutions of some fully nonlinear degenerate elliptic equations with “superlinear” and “subquadratic” Hamiltonian terms. As an application, we complete the results of Birindelli et al. (ESAIM Control Optim Calc Var, 2019. https://doi.org/10.1051/cocv/2018070) concerning the associated ergodic problem, proving, among other facts, the uniqueness, up to constants, of the ergodic function.

We investigate positivity sets of nonnegative supersolutions of the
fully nonlinear elliptic equations F(x, u,Du,D2u) = 0 in Ω, where Ω is an
open subset of RN, and the validity of the strong maximum principle for
F(x, u,Du,D2u) = f in Ω, with f ∈ C(Ω) being nonpositive. We obtain
geometric characterizations of positivity sets {x ∈ Ω : u(x) > 0} of nonnegative
supersolutions u and establish the strong maximum principle under some
geometric assumption on the set {x ∈ Ω : f(x) = 0}.

We study obstacle problems involving p-Laplace-type operators in non-convex polygons. We establish regularity results in terms of weighted Sobolev spaces. As applications, we obtain estimates for the FEM approximation for obstacle problems in pre-fractal Koch Islands.