C1,αC^{1,\alpha} regularity of solutions of degenerate fully non-linear elliptic equations

In the present paper, a class of fully non-linear elliptic equations are considered, which are degenerate as the gradient becomes small. H\"older estimates obtained by the first author (2011) are combined with new Lipschitz estimates obtained through the Ishii-Lions method in order to get $C^{1,\alpha}$ estimates for solutions of these equations.Comment: Submitte

The Ginzburg-Landau equation in the Heisenberg group

We consider a functional related with phase transition models in the Heisenberg group framework. We prove that level sets of local minimizers satisfy some density estimates, that is, they behave as "codimension one" sets. We thus deduce a uniform convergence property of these level sets to interfaces with minimal area. These results are then applied in the construction of (quasi)periodic, plane-like minimizers, i.e., minimizers of our functional whose level sets are contained in a spacial slab of universal size in a prescribed direction. As a limiting case, we obtain the existence of hypersurfaces contained in such a slab which minimize the surface area with respect to a given periodic metric.Comment: 49 page

Symmetry for solutions of two-phase semilinear elliptic equations on hyperbolic space

Assume that $f(s) = F'(s)$ where $F$ is a double-well potential. Under certain conditions on the Lipschitz constant of $f$ on $[-1,1]$, we prove that arbitrary bounded global solutions of the semilinear equation $\Delta u = f(u)$ on hyperbolic space \HH^n must reduce to functions of one variable provided they admit asymptotic boundary values on the infinite boundary of \HH^n which are invariant under a cohomogeneity one subgroup of the group of isometries of \HH^n. We also prove existence of these one-dimensional solutions.Comment: 24 page

Eigenfunctions for singular fully non linear equations in unbounded domains

In this paper we prove some Harnack inequality for fully non linear degenerate elliptic equations, in the two dimensional case, extending the results of Davila Felmer and Quaas in the singular case but in all dimensions. We then apply this result for the existence of an eigenfunction in smooth bounded domain.Comment: 30 pages 2 figure

The Dirichlet problem for singular fully nonlinear operators

In this paper we prove existence of (viscosity) solutions of Dirichlet problems concerning fully nonlinear elliptic operator, which are either degenerate or singular when the gradient of the solution is zero. For this class of operators it is possible to extend the concept of eigenvalue, this paper concerns the cases when the inf of the principal eigenvalues is positive i.e. when both the maximum and the minimum principle holds.Comment: 10 pages, 0 figure

Symmetry minimizes the principal eigenvalue: an example for the Pucci's sup operator

We explicitly evaluate the principal eigenvalue of the extremal Pucci's sup--operator for a class of special plane domains, and we prove that, for fixed area, the eigenvalue is minimal for the most symmetric set.Comment: 11 pages, 7 figure

A Neumann eigenvalue problem for fully nonlinear operators

In this paper we study the asymptotic behavior of the principal eigenvalues associated to the Pucci operator in bounded domain $\Omega$ with Neumann/Robin boundary condition i.e. $\partial_n u=\alpha u$ when $\alpha$ tends to infinity. This study requires Lipschitz estimates up to the boundary that are interesting in their own rights.Comment: 19 page

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.Comment: 13 page