1,120 research outputs found

### Fields Medals and Nevanlinna Prize Presented at ICM-94 in Zurich

The Notices solicited the following five articles describing the work of the Fields Medalists and Nevanlinna Prize winner

### Strichartz estimates for the water-wave problem with surface tension

Strichartz-type estimates for one-dimensional surface water-waves under
surface tension are studied, based on the formulation of the problem as a
nonlinear dispersive equation. We establish a family of dispersion estimates on
time scales depending on the size of the frequencies. We infer that a solution
$u$ of the dispersive equation we introduce satisfies local-in-time Strichartz
estimates with loss in derivative:
$\| u \|_{L^p([0,T]) W^{s-1/p,q}(\mathbb{R})} \leq C, \qquad \frac{2}{p} +
\frac{1}{q} = {1/2},$ where $C$ depends on $T$ and on the norms of the
initial data in $H^s \times H^{s-3/2}$. The proof uses the frequency analysis
and semiclassical Strichartz estimates for the linealized water-wave operator.Comment: Fixed typos and mistakes. Merged with arXiv:0809.451

### Homogenization of weakly coupled systems of Hamilton--Jacobi equations with fast switching rates

We consider homogenization for weakly coupled systems of Hamilton--Jacobi
equations with fast switching rates. The fast switching rate terms force the
solutions converge to the same limit, which is a solution of the effective
equation. We discover the appearance of the initial layers, which appear
naturally when we consider the systems with different initial data and analyze
them rigorously. In particular, we obtain matched asymptotic solutions of the
systems and rate of convergence. We also investigate properties of the
effective Hamiltonian of weakly coupled systems and show some examples which do
not appear in the context of single equations.Comment: final version, to appear in Arch. Ration. Mech. Ana

### A kinetic theory of diffusion in general relativity with cosmological scalar field

A new model to describe the dynamics of particles undergoing diffusion in
general relativity is proposed. The evolution of the particle system is
described by a Fokker-Planck equation without friction on the tangent bundle of
spacetime. It is shown that the energy-momentum tensor for this matter model is
not divergence-free, which makes it inconsistent to couple the Fokker-Planck
equation to the Einstein equations. This problem can be solved by postulating
the existence of additional matter fields in spacetime or by modifying the
Einstein equations. The case of a cosmological scalar field term added to the
left hand side of the Einstein equations is studied in some details. For the
simplest cosmological model, namely the flat Robertson-Walker spacetime, it is
shown that, depending on the initial value of the cosmological scalar field,
which can be identified with the present observed value of the cosmological
constant, either unlimited expansion or the formation of a singularity in
finite time will occur in the future. Future collapse into a singularity also
takes place for a suitable small but positive present value of the cosmological
constant, in contrast to the standard diffusion-free scenario.Comment: 17 pages, no figures. The present version corrects an erroneous
statement on the physical interpretation of the results made in the original
publicatio

### Topological Change in Mean Convex Mean Curvature Flow

Consider the mean curvature flow of an (n+1)-dimensional, compact, mean
convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We
prove that elements of the m-th homotopy group of the complementary region can
die only if there is a shrinking S^k x R^(n-k) singularity for some k less than
or equal to m. We also prove that for each m from 1 to n, there is a nonempty
open set of compact, mean convex regions K in R^(n+1) with smooth boundary for
which the resulting mean curvature flow has a shrinking S^m x R^(n-m)
singularity.Comment: 19 pages. This version includes a new section proving that certain
kinds of mean curvature flow singularities persist under arbitrary small
perturbations of the initial surface. Newest update (Oct 2013) fixes some
bibliographic reference

### SBV regularity for Hamilton-Jacobi equations in $\mathbb R^n$

In this paper we study the regularity of viscosity solutions to the following
Hamilton-Jacobi equations $\partial_t u + H(D_{x} u)=0 \qquad \textrm{in}
\Omega\subset \mathbb R\times \mathbb R^{n} .$ In particular, under the
assumption that the Hamiltonian $H\in C^2(\mathbb R^n)$ is uniformly convex, we
prove that $D_{x}u$ and $\partial_t u$ belong to the class $SBV_{loc}(\Omega)$.Comment: 15 page

### An adaptive finite element method for the infinity Laplacian

We construct a finite element method (FEM) for the infinity Laplacian. Solutions of this problem are well known to be singular in nature so we have taken the opportunity to conduct an a posteriori analysis of the method deriving residual based estimators to drive an adaptive algorithm. It is numerically shown that optimal convergence rates are regained using the adaptive procedure

### On derivation of Euler-Lagrange Equations for incompressible energy-minimizers

We prove that any distribution $q$ satisfying the equation $\nabla q=\div{\bf
f}$ for some tensor ${\bf f}=(f^i_j), f^i_j\in h^r(U)$ ($1\leq r<\infty$) -the
{\it local Hardy space}, $q$ is in $h^r$, and is locally represented by the sum
of singular integrals of $f^i_j$ with Calder\'on-Zygmund kernel. As a
consequence, we prove the existence and the local representation of the
hydrostatic pressure $p$ (modulo constant) associated with incompressible
elastic energy-minimizing deformation ${\bf u}$ satisfying $|\nabla {\bf u}|^2,
|{\rm cof}\nabla{\bf u}|^2\in h^1$. We also derive the system of Euler-Lagrange
equations for incompressible local minimizers ${\bf u}$ that are in the space
$K^{1,3}_{\rm loc}$; partially resolving a long standing problem. For H\"older
continuous pressure $p$, we obtain partial regularity of area-preserving
minimizers.Comment: 23 page

### Continuity of Optimal Control Costs and its application to Weak KAM Theory

We prove continuity of certain cost functions arising from optimal control of
affine control systems. We give sharp sufficient conditions for this
continuity. As an application, we prove a version of weak KAM theorem and
consider the Aubry-Mather problems corresponding to these systems.Comment: 23 pages, 1 figures, added explanations in the proofs of the main
theorem and the exampl

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