478 research outputs found
Global pointwise decay estimates for defocusing radial nonlinear wave equations
We prove global pointwise decay estimates for a class of defocusing
semilinear wave equations in dimensions restricted to spherical symmetry.
The technique is based on a conformal transformation and a suitable choice of
the mapping adjusted to the nonlinearity. As a result we obtain a pointwise
bound on the solutions for arbitrarily large Cauchy data, provided the
solutions exist globally. The decay rates are identical with those for small
data and hence seem to be optimal. A generalization beyond the spherical
symmetry is suggested.Comment: 9 pages, 1 figur
Quantization for an elliptic equation of order 2m with critical exponential non-linearity
On a smoothly bounded domain we consider a sequence of
positive solutions in to
the equation subject to Dirichlet
boundary conditions, where . Assuming that
we
prove that is an integer multiple of
\Lambda_1:=(2m-1)!\vol(S^{2m}), the total -curvature of the standard
-dimensional sphere.Comment: 33 page
Existence of solutions to a higher dimensional mean-field equation on manifolds
For we prove an existence result for the equation on a closed Riemannian
manifold of dimension for certain values of .Comment: 15 Page
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
State-of-the-art glycosaminoglycan characterization
Glycosaminoglycans (GAGs) are heterogeneous acidic polysaccharides involved in a range of biological functions. They have a significant influence on the regulation of cellular processes and the development of various diseases and infections. To fully understand the functional roles that GAGs play in mammalian systems, including disease processes, it is essential to understand their structural features. Despite having a linear structure and a repetitive disaccharide backbone, their structural analysis is challenging and requires elaborate preparative and analytical techniques. In particular, the extent to which GAGs are sulfated, as well as variation in sulfate position across the entire oligosaccharide or on individual monosaccharides, represents a major obstacle. Here, we summarize the current state-of-the-art methodologies used for GAG sample preparation and analysis, discussing in detail liquid chromatograpy and mass spectrometry-based approaches, including advanced ion activation methods, ion mobility separations and infrared action spectroscopy of mass-selected species
Singular kernels, multiscale decomposition of microstructure, and dislocation models
We consider a model for dislocations in crystals introduced by Koslowski,
Cuiti\~no and Ortiz, which includes elastic interactions via a singular kernel
behaving as the norm of the slip. We obtain a sharp-interface limit
of the model within the framework of -convergence. From an analytical
point of view, our functional is a vector-valued generalization of the one
studied by Alberti, Bouchitt\'e and Seppecher to which their rearrangement
argument no longer applies. Instead we show that the microstructure must be
approximately one-dimensional on most length scales and exploit this property
to derive a sharp lower bound
Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres
Using mixed analytical and numerical methods we investigate the development
of singularities in the heat flow for corotational harmonic maps from the
-dimensional sphere to itself for . By gluing together
shrinking and expanding asymptotically self-similar solutions we construct
global weak solutions which are smooth everywhere except for a sequence of
times at which there occurs the type I blow-up at one
of the poles of the sphere. We show that in the generic case the continuation
beyond blow-up is unique, the topological degree of the map changes by one at
each blow-up time , and eventually the solution comes to rest at the zero
energy constant map.Comment: 24 pages, 8 figures, minor corrections, matches published versio
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
Diffeomorphism-invariant properties for quasi-linear elliptic operators
For quasi-linear elliptic equations we detect relevant properties which
remain invariant under the action of a suitable class of diffeomorphisms. This
yields a connection between existence theories for equations with degenerate
and non-degenerate coerciveness.Comment: 16 page
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
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