Adiabatic limits of closed orbits for some Newtonian systems in R^n

We deal with a Newtonian system like x'' + V'(x) = 0. We suppose that V: \R^n \to \R possesses an (n-1)-dimensional compact manifold M of critical points, and we prove the existence of arbitrarity slow periodic orbits. When the period tends to infinity these orbits, rescaled in time, converge to some closed geodesics on M.Comment: 28 page

Conformal Metrics with Constant Q-Curvature

We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant $Q$-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the manifold.Comment: This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Concentration on minimal submanifolds for a singularly perturbed Neumann problem

We consider the equation - \e^2 \D u + u= u^p in $\Omega \subseteq \R^N$, where $\Omega$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of \partial \O, for $N \geq 3$ and for $k \in \{1, ..., N-2\}$. We impose Neumann boundary conditions, assuming $1<p <\frac{N-k+2}{N-k-2}$ and \e \to 0^+. This result settles in full generality a phenomenon previously considered only in the particular case $N = 3$ and $k = 1$.Comment: 62 pages. To appear in Adv. in Mat

A Moser-Trudinger inequality for the singular Toda system

In this paper we prove a sharp version of the Moser-Trudinger inequality for the Euler-Lagrange functional of a singular Toda system, motivated by the study of models in Chern-Simons theory. Our result extends those for the scalar case, as well as for the regular Toda system. We expect this inequality to be a basic tool to attack variationally the existence problem under general assumptions.Comment: 13 pages, accepted on Bulletin of the Institute of Mathematica Academia Sinic

Existence and non-existence results for the SU(3) singular Toda system on compact surfaces

We consider the SU(3) Toda system on a compact surface. We give both existence and non-existence results under some conditions on the parameters. Existence results are obtained using variational methods, which involve a geometric inequality of new type; non-existence results are obtained using blow-up analysis and localized Pohozaev identities.Comment: 41 pages, 9 figures, accepted on Journal of Functional Analysi

Periodic solutions to the Cahn-Hilliard equation in the plane

In this paper we construct entire solutions to the Cahn-Hilliard equation $-\Delta(-\Delta u+W^{'}(u))+W^{"}(u)(-\Delta u+W^{'}(u))=0$ in the Euclidean plane, where $W(u)$ is the standard double-well potential $\frac{1}{4} (1-u^2)^2$. Such solutions have a non-trivial profile that shadows a Willmore planar curve, and converge uniformly to $\pm 1$ as $x_2 \to \pm \infty$. These solutions give a counterexample to the counterpart of Gibbons' conjecture for the fourth-order counterpart of the Allen-Cahn equation. We also study the $x_2$-derivative of these solutions using the special structure of Willmore's equation