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

Compactness of solutions to some geometric fourth-order equations

We prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds. The proof is based on a refined bubbling analysis, for which the main estimates are given in integral form. Our result is used in a subsequent paper to find critical points (via minimax arguments) of some geometric functional, which give rise to conformal metrics of constant $Q$-curvature. As a byproduct of our method, we also obtain compactness of such metrics.Comment: 32 pages, fixed some bug in the previous versio

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

Transition Layer for the Heterogeneous Allen-Cahn Equation

We consider the equation \e^{2}\Delta u=(u-a(x))(u^2-1) in $\Omega$, $\frac{\partial u}{\partial \nu} =0$ on $\partial \Omega$, where $\Omega$ is a smooth and bounded domain in $\R^n$, $\nu$ the outer unit normal to \pa\Omega, and $a$ a smooth function satisfying $-1<a(x)<1$ in \ov{\Omega}. We set $K$, $\Omega_+$ and $\Omega_-$ to be respectively the zero-level set of $a$, {a>0} and {a<0}. Assuming $\nabla a \neq 0$ on $K$ and $a\ne 0$ on $\partial \Omega$, we show that there exists a sequence \e_j \to 0 such that the above equation has a solution u_{\e_j} which converges uniformly to $\pm 1$ on the compact sets of \O_{\pm} as $j \to + \infty$.Comment: 25 page

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

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