Homotopy classes of harmonic maps of the stratified 2-spheres and applications to geometric flows

We show that the set of harmonic maps from the 2-dimensional stratified spheres with uniformly bounded energies contains only finitely many homotopy classes. We apply this result to construct infinitely many harmonic map flows and mean curvature flows of 2-sphere in the connected sum of two closed 3-dimensional manifolds _M1≠S³$M_1\ne S^3$ and _M2≠S³,RP³$M_2\ne S^3,\mathbb{R}{\mathbb{P}}^3$, which must develop finite time singularity.     

Harness processes and harmonic crystals

In the Hammersley harness processes the R$\mathbb{R}$-valued height at each site i∈Z^d$i\in {\mathbb{Z}}^d$ is updated at rate 1 to an average of the neighboring heights plus a centered random variable (the noise). We construct the process “a la Harris” simultaneously for all times and boxes contained in Z^d${\mathbb{Z}}^d$. With this representation we compute covariances and show L²$L^2$ and almost sure time and space convergence of the process. In particular, the process started from the flat configuration and viewed from the height at the origin converges to an invariant measure. In dimension three and higher, the process itself converges to an invariant measure in L²$L^2$ at speed t^1−d/2$t^{1-d/2}$ (this extends the convergence established by Hsiao). When the noise is Gaussian the limiting measures are Gaussian fields (harmonic crystals) and are also reversible for the process.     

Low lying eigenvalues of randomly curved quantum waveguides

We consider the negative Dirichlet Laplacian on an infinite waveguide embedded in R²${\mathbb{R}}^2$, and finite segments thereof. The waveguide is a perturbation of a periodic strip in terms of a sequence of independent identically distributed random variables which influence the curvature. We derive explicit lower bounds on the first eigenvalue of finite segments of the randomly curved waveguide in the small coupling (i.e. weak disorder) regime. This allows us to estimate the probability of low lying eigenvalues, a tool which is relevant in the context of Anderson localization for random Schrödinger operators.     

Strong solution for a stochastic model of two-dimensional second grade fluids: Existence,uniqueness and asymptotic behavior

We investigate a stochastic evolution equation for the motion of a second grade fluid filling a bounded domain of R²${\mathbb{R}}^2$. Global existence and uniqueness of strong probabilistic solution is established. In contrast to previous results on this model we show that the sequence of Galerkin approximation converges in mean square to the exact strong probabilistic solution of the problem. We also give two results on the long time behavior of the solution. Mainly we prove that the strong solution of our stochastic model converges exponentially in mean square to the stationary solution of the time-independent second grade fluids equations if the deterministic part of the external force does not depend on time. If the deterministic forcing term explicitly depends on time, then the strong probabilistic solution decays exponentially in mean square.     

The stationary Oseen equations in an exterior domain: An approach in weighted Sobolev spaces

In this work, we study the linearized Navier–Stokes equations in an exterior domain of R³${\mathbb{R}}^3$ at the steady state, that is, the Oseen equations. We are interested in the existence and the uniqueness of weak, strong and very weak solutions in L^p$L^p$-theory which makes our work more difficult. Our analysis is based on the principle that linear exterior problems can be solved by combining their properties in the whole space R³${\mathbb{R}}^3$ and the properties in bounded domains. Our approach rests on the use of weighted Sobolev spaces.     

Maximum principles and isoperimetric inequalities for some Monge–Ampère-type problems

In this note we derive a maximum principle for an appropriate functional combination of u(x)$u(\mathbf{x})$ and |∇u|²${|\mathrm{\nabla }u|}^2$, where u(x)$u(\mathbf{x})$ is a strictly convex classical solution to a general class of Monge–Ampère equations. This maximum principle is then employed to establish some isoperimetric inequalities of interest in the theory of surfaces of constant Gauss curvature in R^N+1${\mathbb{R}}^{N+1}$.     

Global existence of null-form wave equations on small asymptotically Euclidean manifolds

We prove the global existence of the small solutions to the Cauchy problem for quasilinear wave equations satisfying the null condition on (R³,g)$({\mathbb{R}}^3,\mathfrak{g})$, where the metric g$\mathfrak{g}$ is a small perturbation of the flat metric and approaches the Euclidean metric like (1+|x|²)^−ρ/2${(1+{|x|}^2)}^{-\rho /2}$ with ρ>1$\rho >1$. Global and almost global existence for systems without the null condition are also discussed for certain small time-dependent perturbations of the flat metric in Appendix?A.     

A reduction method for semilinear elliptic equations and solutions concentrating on spheres

We show that any general semilinear elliptic problem with Dirichlet or Neumann boundary conditions in an annulus A⊆R^2m$A\subseteq {\mathbb{R}}^{2m}$, m?2$m?2$, invariant by the action of a certain symmetry group can be reduced to a nonhomogeneous similar problem in an annulus D⊂R^m+1$D\subset {\mathbb{R}}^{m+1}$, invariant by another related symmetry. We apply this result to prove the existence of positive and sign changing solutions of a singularly perturbed elliptic problem in A   which concentrate on one or two (m−1)$(m-1)$ dimensional spheres. We also prove that the Morse indices of these solutions tend to infinity as the parameter of concentration tends to infinity.     

A theorem on measures in dimension 2 and applications to vortex sheets

We find conditions under which measures belong to H⁻¹(R²)$H^{-1}({\mathbb{R}}^2)$. Next we show that measures generated by the Prandtl, Kaden as well as Pullin spirals, objects considered by physicists as incompressible flows generating vorticity, satisfy assumptions of our theorem, thus they are (locally) elements of H⁻¹(R²)$H^{-1}({\mathbb{R}}^2)$. Moreover, as a by-product, we prove an embedding of the space of Morrey type measures in H⁻¹$H^{-1}$.     

Asymptotic profiles for wave equations with strong damping

We consider the Cauchy problem in Rⁿ${\mathbf{R}}^n$ for strongly damped wave equations. We derive asymptotic profiles of these solutions with weighted L^1,1(Rⁿ)$L^{1,1}({\mathbf{R}}^n)$ data by using a method introduced in [9] and/or [10].     

Remark on the stability of the large stationary solutions to the Navier–Stokes equations under the general flux condition

Consider stationary weak solutions of the Navier–Stokes equations in a bounded domain in R³${\mathbb{R}}^3$ under the nonhomogeneous boundary condition. We give a new approach for the stability of the stationary flow in the L²$L^2$-framework. Furthermore, we give some examples of stable solutions which may be large in L³(Ω)$L^3\left(\Omega \right)$ or W^1,3/2(Ω)$W^{1,3/2}\left(\Omega \right)$.     

On removable sets for convex functions

In the present article we provide a sufficient condition for a closed set F∈R^d$F\in {\mathbb{R}}^d$ to have the following property which we call c  -removability: Whenever a continuous function f:R^d→R$f:{\mathbb{R}}^d\to \mathbb{R}$ is locally convex on the complement of F  , it is convex on the whole R^d${\mathbb{R}}^d$. We also prove that no generalized rectangle of positive Lebesgue measure in R²${\mathbb{R}}^2$ is c-removable. Our results also answer the following question asked in an article by Jacek Tabor and Józef Tabor (2010) [5]: Assume the closed set F⊂R^d$F\subset {\mathbb{R}}^d$ is such that any locally convex function defined on R^d?F${\mathbb{R}}^d?F$ has a unique convex extension on R^d${\mathbb{R}}^d$. Is F   necessarily intervally thin (a notion of smallness of sets defined by their “essential transparency” in every direction)? We prove the answer is negative by finding a counterexample in R²${\mathbb{R}}^2$.     

Anisotropic curvature-driven flow of convex sets

We study in this paper the mean curvature evolution, and in particular the anisotropic mean curvature evolution, of convex sets in R^N${\mathbb{R}}^N$ (without driving forces). If the anisotropy is smooth, we show that the evolution remains convex. If the anisotropy is crystalline, we build a convex evolution which satisfies an equation which is a weak form of the crystalline curvature motion equation.     

Decompositions of polyhedral products for shifted complexes

The conjecture of Bahri, Bendersky, Cohen and Gitler (2010)  [3] on wedge decompositions of polyhedral products of shifted complexes is settled affirmatively. As a corollary, it is proved that the homotopy type of the complement of a coordinate subspace arrangement associated with a shifted complex, tensored with R^r${\mathbb{R}}^r$ for any r≥1$r\ge 1$, has the homotopy type of a wedge of spheres.     

Lieb–Thirring type inequalities for non-self-adjoint perturbations of magnetic Schrödinger operators

Let H:=_H0+V$H:=H_0+V$ and _H⊥:=_H0,⊥+V$H_{\perp }:=H_{0,\perp }+V$ be respectively perturbations of the unperturbed Schrödinger operators _H0$H_0$ on L²(R³)$L^2({\mathbb{R}}^3)$ and _H0,⊥$H_{0,\perp }$ on L²(R²)$L^2({\mathbb{R}}^2)$ with constant magnetic field of strength b>0$b>0$, and V a complex relatively compact perturbation. We prove Lieb–Thirring type inequalities on the discrete spectrum of H   and _H⊥$H_{\perp }$. In particular, these estimates give a priori information on the distribution of eigenvalues around the Landau levels of the operator, and describe how fast sequences of eigenvalues converge.     

The topological structure of fuzzy sets with sendograph metric

For a non-degenerate convex subset Y of the n  -dimensional Euclidean space Rⁿ${\mathbb{R}}^n$, let F(Y)$\mathbb{F}(Y)$ be the family of all fuzzy sets of Rⁿ${\mathbb{R}}^n$ which are upper semicontinuous, fuzzy convex and normal with compact supports contained in Y  . We show that the space F(Y)$\mathbb{F}(Y)$ with the topology of sendograph metric is homeomorphic to the separable Hilbert space ?²${?}^2$ if Y   is compact; and the space F(Rⁿ)$\mathbb{F}({\mathbb{R}}^n)$ is also homeomorphic to ?²${?}^2$.     

Basin of attraction of a cusp–fold singularity in 3D piecewise smooth vector fields

Generic one-parameter families of piecewise smooth vector fields on R³${\mathbb{R}}^3$ presenting the so-called cusp–fold singularity are studied. The bifurcation diagrams are exhibited and the asymptotic and structural stabilities are discussed.