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

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

Some Liouville Theorems for the p-Laplacian

We present several Liouville type results for the $p$-Laplacian in $\R^N$. Suppose that $h$ is a nonnegative regular function such that $$h(x) = a|x|^\gamma\ {\rm for}\ |x|\ {\rm large},\ a>0\ {\rm and}\ \gamma> -p.$$ We obtain the following non -existence result: 1) Suppose that $N>p>1$, and $u\in W^{1,p}_{loc} (\R^N)\cap {\cal C} (\R^N)$ is a nonnegative weak solution of - {\rm div} (|\nabla u|^{p-2 }\nabla u) \geq h(x) u^q \;\;\mbox{in }\; \R^N . Suppose that $p-1< q\leq {(N+\gamma)(p-1)\over N-p}$ then $u\equiv 0$. 2) Let $N\leq p$. If $u\in W^{1,p}_{loc} (\R^N)\cap {\cal C} (\R^N)$ is a weak solution bounded below of $-{\rm div} (|\nabla u|^{p-2 }\nabla u)\geq 0$ in $\R^N$ then $u$ is constant. 3) Let $N>p$ if $u$ is bounded from below and $-{\rm div} (|\nabla u|^{p-2 }\nabla u)=0$ in $\R^N$ then $u$ is constant. 4)If $-\Delta_p u+h(x) u^q\leq 0,$. If $q> p-1$, then $u\equiv 0$.Comment: 19 page