Singular elliptic equations

C1,γ regularity for singular or degenerate fully nonlinear equations and applications

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.

The Dirichlet problem for fully nonlinear degenerate elliptic equations with a singular nonlinearity

We investigate the homogeneous Dirichlet problem in uniformly convex domains for a large class of degenerate elliptic equations with singular zero order term. In particular we establish sharp existence and uniqueness results of positive viscosity solutions.

The Dirichlet problem for the $1$-Laplacian with a general singular term and $L^1$-data

see attached file

Finite and Infinite energy solutions of singular elliptic problems: existence and uniqueness

We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by
$$
-\Delta u= h(u){f} \ \ \text{in}\,\ \Omega,
$$
where $f$ is an irregular datum, possibly a measure, and $h$ is a continuous function that may blow up at zero. We also provide regularity results on both the solution and the lower order term depending on the regularity of the data, and we discuss their optimality.