445 research outputs found
A threshold phenomenon for embeddings of into Orlicz spaces
We consider a sequence of positive smooth critical points of the
Adams-Moser-Trudinger embedding of into Orlicz spaces. We study its
concentration-compactness behavior and show that if the sequence is not
precompact, then the liminf of the -norms of the functions is greater
than or equal to a positive geometric constant.Comment: 14 Page
K+ conductance of mechanically dissociated rat sensory neurones in long lerm cultures in a defined medium
Multiple solutions of the quasirelativistic Choquard equation
We prove existence of multiple solutions to the quasirelativistic Choquard equation with a scalar potential
Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system
We describe the asymptotic behavior as time goes to infinity of solutions of
the 2 dimensional corotational wave map system and of solutions to the 4
dimensional, radially symmetric Yang-Mills equation, in the critical energy
space, with data of energy smaller than or equal to a harmonic map of minimal
energy. An alternative holds: either the data is the harmonic map and the
soltuion is constant in time, or the solution scatters in infinite time
On Singularity formation for the L^2-critical Boson star equation
We prove a general, non-perturbative result about finite-time blowup
solutions for the -critical boson star equation in 3 space dimensions. Under
the sole assumption that the solution blows up in at finite time, we
show that has a unique weak limit in and that has a
unique weak limit in the sense of measures. Moreover, we prove that the
limiting measure exhibits minimal mass concentration. A central ingredient used
in the proof is a "finite speed of propagation" property, which puts a strong
rigidity on the blowup behavior of .
As the second main result, we prove that any radial finite-time blowup
solution converges strongly in away from the origin. For radial
solutions, this result establishes a large data blowup conjecture for the
-critical boson star equation, similar to a conjecture which was
originally formulated by F. Merle and P. Raphael for the -critical
nonlinear Schr\"odinger equation in [CMP 253 (2005), 675-704].
We also discuss some extensions of our results to other -critical
theories of gravitational collapse, in particular to critical Hartree-type
equations.Comment: 24 pages. Accepted in Nonlinearit
On a functional satisfying a weak Palais-Smale condition
In this paper we study a quasilinear elliptic problem whose functional
satisfies a weak version of the well known Palais-Smale condition. An existence
result is proved under general assumptions on the nonlinearities.Comment: 18 page
On completeness of orbits of Killing vector fields
A Theorem is proved which reduces the problem of completeness of orbits of
Killing vector fields in maximal globally hyperbolic, say vacuum, space--times
to some properties of the orbits near the Cauchy surface. In particular it is
shown that all Killing orbits are complete in maximal developements of
asymptotically flat Cauchy data, or of Cauchy data prescribed on a compact
manifold. This result gives a significant strengthening of the uniqueness
theorems for black holes.Comment: 16 pages, Latex, preprint NSF-ITP-93-4
Rotational symmetry of self-similar solutions to the Ricci flow
Let (M,g) be a three-dimensional steady gradient Ricci soliton which is
non-flat and \kappa-noncollapsed. We prove that (M,g) is isometric to the
Bryant soliton up to scaling. This solves a problem mentioned in Perelman's
first paper.Comment: Final version, to appear in Invent. Mat
Surface Gap Soliton Ground States for the Nonlinear Schr\"{o}dinger Equation
We consider the nonlinear Schr\"{o}dinger equation , with and and with periodic in each coordinate direction. This problem
describes the interface of two periodic media, e.g. photonic crystals. We study
the existence of ground state solutions (surface gap soliton ground
states) for . Using a concentration compactness
argument, we provide an abstract criterion for the existence based on ground
state energies of each periodic problem (with and ) as well as a more practical
criterion based on ground states themselves. Examples of interfaces satisfying
these criteria are provided. In 1D it is shown that, surprisingly, the criteria
can be reduced to conditions on the linear Bloch waves of the operators
and .Comment: definition of ground and bound states added, assumption (H2) weakened
(sign changing nonlinearity is now allowed); 33 pages, 4 figure
Analysis of Nematic Liquid Crystals with Disclination Lines
We investigate the structure of nematic liquid crystal thin films described
by the Landau--de Gennes tensor-valued order parameter with Dirichlet boundary
conditions of nonzero degree. We prove that as the elasticity constant goes to
zero a limiting uniaxial texture forms with disclination lines corresponding to
a finite number of defects, all of degree 1/2 or all of degree -1/2. We also
state a result on the limiting behavior of minimizers of the Chern-Simons-Higgs
model without magnetic field that follows from a similar proof.Comment: 40 pages, 1 figur
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