2,529 research outputs found
Local behavior of fractional -minimizers
We extend the De Giorgi-Nash-Moser theory to nonlocal, possibly degenerate
integro-differential operators.Comment: 26 pages. To appear in Ann. Inst. H. Poincare Anal. Non Lineaire.
arXiv admin note: text overlap with arXiv:1405.784
Global regularity for the energy-critical NLS on
We establish global existence for the energy-critical nonlinear Schr\"odinger
equation on . This follows similar lines to the work on
but requires new extinction results for linear solutions and
bounds on the first nonlinear iterate at a Euclidean profile that are adapted
to the new geometry.Comment: to appear in the Annales IHP, Analyse non lineaire. arXiv admin note:
text overlap with arXiv:1102.5771, arXiv:1101.452
The Near Field Refractor
We present an abstract method in the setting of compact metric spaces which
is applied to solve a number of problems in geometric optics. In particular, we
solve the one source near field refraction problem. That is, we construct
surfaces separating two homogenous media with different refractive indices that
refract radiation emanating from the origin into a target domain contained in
an n-1 dimensional hypersurface. The input and output energy are prescribed.
This implies the existence of lenses focusing radiation in a prescribed manner.Comment: 39 pages, 4 figures, Annales de l'Institut Henri Poincare (C) Analyse
Non Lineaire (to appear). Geometric optics, lens design, measure equations,
Descartes ovals, Monge-Ampere type equation
Equilibrium states for non-uniformly expanding maps: decay of correlations and strong stability
We study the rate of decay of correlations for equilibrium states associated
to a robust class of non-uniformly expanding maps where no Markov assumption is
required. We show that the Ruelle-Perron-Frobenius operator acting on the space
of Holder continuous observables has a spectral gap and deduce the exponential
decay of correlations and the central limit theorem. In particular, we obtain
an alternative proof for the existence and uniqueness of the equilibrium states
and we prove that the topological pressure varies continuously. Finally, we use
the spectral properties of the transfer operators in space of differentiable
observables to obtain strong stability results under deterministic and random
perturbations.Comment: 29 pages, Annales de l'Institut Henri Poincare - Analyse non lineaire
(to appear
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