1,570 research outputs found
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 -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 -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 ,
on , where is a
smooth and bounded domain in , the outer unit normal to
\pa\Omega, and a smooth function satisfying in \ov{\Omega}.
We set , and to be respectively the zero-level set of
, {a>0} and {a<0}. Assuming on and on
, 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 on the compact sets of \O_{\pm} as .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 ,
where is open, smooth and bounded, and we prove concentration of
solutions along -dimensional minimal submanifolds of \partial \O, for and for . We impose Neumann boundary conditions,
assuming and \e \to 0^+. This result settles in
full generality a phenomenon previously considered only in the particular case
and .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
- …