Inviscid limit for the derivative Ginzburg–Landau equation with small data in modulation and Sobolev spaces

Applying the frequency-uniform decomposition technique, we study the Cauchy problem for derivative Ginzburg–Landau equation $u_t=(\nu +\mathrm{i})\mathrm{\Delta }u+{\overrightarrow{\lambda }}_1?\mathrm{?}({|u|}^{2}u)+({\overrightarrow{\lambda }}_2?\mathrm{?}u){|u|}^2+\alpha {|u|}^{2\delta }u$, where $\delta \in \mathbb{N}$, ${\overrightarrow{\lambda }}_1,{\overrightarrow{\lambda }}_2$ are complex constant vectors, $\nu \in [0,1]$, $\alpha \in \mathbb{C}$. For $n\ge 3$, we show that it is uniformly global well posed for all $\nu \in [0,1]$ if initial data $u_0$ in modulation space $M_{2,1}^s$ and Sobolev spaces $H^{s+n/2}$ ($s>3$) and $6u_0{6}_{L^2}$ is small enough. Moreover, we show that its solution will converge to that of the derivative Schrödinger equation in $C(0,T;L^2)$ if $\nu \to 0$ and $u_0$ in $M_{2,1}^s$ or $H^{s+n/2}$ with $s>4$. For $n=2$, we obtain the local well-posedness results and inviscid limit with the Cauchy data in $M_{1,1}^s$ ($s>3$) and $6u_0{6}_{L^1}?1$.     

Nonlinear Schrödinger equation with unbounded or vanishing potentials: Solutions concentrating on lower dimensional spheres

We study positive bound states for the equation$?{\epsilon }^2\mathrm{\Delta }u+V(x)u=K(x)f(u),x\in {\mathbb{R}}^N,$ where $\epsilon >0$ is a real parameter and V and K are radial positive potentials. We are especially interested in solutions which concentrate on a k-dimensional sphere, $1?k?N?1$, as $\epsilon \to 0$. We adopt a purely variational approach which allows us to consider broader classes of potentials than those treated in previous works. For example, V and K might be singular at the origin or vanish superquadratically at infinity.     

Monochromatic progressions in random colorings

Let $N^{+}(k)=2^{k/2}{k}^{3/2}f(k)$ and $N^{?}(k)=2^{k/2}{k}^{1/2}g(k)$ where $f(k)\to \infty$ and $g(k)\to 0$ arbitrarily slowly as $k\to \infty$. We show that the probability of a random 2-coloring of $\{1,2,\dots ,N^{+}(k)\}$ containing a monochromatic k-term arithmetic progression approaches 1, and the probability of a random 2-coloring of $\{1,2,\dots ,N^{?}(k)\}$ containing a monochromatic k-term arithmetic progression approaches 0, as $k\to \infty$. This improves an upper bound due to Brown, who had established an analogous result for $N^{+}(k)=2^k\log kf(k)$.     

Radial symmetry of positive solutions for semilinear elliptic equations in the unit ball via elliptic and hyperbolic geometry

Let $n\in \mathbb{N}$ with $n?2$, $a\in (?1,0)\cup (0,1]$ and $f:(0,1)\times (0,\infty )\to \mathbb{R}$ such that for each $u\in (0,\infty )$, $r?{(1+a{r}^2)}^{(n+2)/2}f(r,{(1+a{r}^2)}^{?(n?2)/2}u):(0,1)\to \mathbb{R}$ is nonincreasing. We show that each positive solution of$\mathrm{\Delta }u+f(|x|,u)=0\text{in}B,u=0\text{on}?B$ is radially symmetric, where B is the open unit ball in ${\mathbb{R}}^N$.     

Robust local Hölder rigidity of circle maps with breaks

We prove that, for every $\epsilon \in (0,1)$, every two $C^{2+\alpha }$-smooth $(\alpha >0)$ circle diffeomorphisms with a break point, i.e. circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity, with the same irrational rotation number $\rho \in (0,1)$ and the same size of the break $c\in {\mathbb{R}}_{+}\backslash \{1\}$, are conjugate to each other via a conjugacy which is $(1?\epsilon )$-Hölder continuous at the break points. An analogous result does not hold for circle diffeomorphisms even when they are analytic.     

Three-manifolds of constant vector curvature one

A Riemannian manifold has $\mathrm{CVC}(?)$ if its sectional curvatures satisfy $\sec \le \epsilon$ or $\sec \ge \epsilon$ pointwise, and if every tangent vector lies in a tangent plane of curvature ε. We present a construction of an infinite-dimensional family of compact $\mathrm{CVC}(1)$ three-manifolds.     

On the Lichnerowicz conjecture for CR manifolds with mixed signature

We construct examples of nondegenerate CR manifolds with Levi form of signature $(p,q)$, $2\le p\le q$, which are compact, not locally CR flat, and admit essential CR vector fields. We also construct an example of a noncompact nondegenerate CR manifold with signature $(1,n?1)$ that is not locally CR flat and admits an essential CR vector field. These provide counterexamples to the analogue of the Lichnerowicz conjecture for CR manifolds with mixed signature.     

A nonlinear Korn inequality on a surface

Let ω be a domain in ${\mathbb{R}}^2$ and let $\mathit{\theta }:\overline{\omega }\to {\mathbb{R}}^3$ be a smooth immersion. The main purpose of this paper is to establish a “nonlinear Korn inequality on the surface $\mathit{\theta }(\overline{\omega })$”, asserting that, under ad hoc assumptions, the ${\mathit{H}}^1(\omega )$-distance between the surface $\mathit{\theta }(\overline{\omega })$ and a deformed surface is “controlled” by the ${\mathit{L}}^1(\omega )$-distance between their fundamental forms. Naturally, the ${\mathit{H}}^1(\omega )$-distance between the two surfaces is only measured up to proper isometries of ${\mathbb{R}}^3$.This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let ${\mathit{\theta }}^k:\omega \to {\mathbb{R}}^3$, $k⩾1$, be mappings with the following properties: They belong to the space ${\mathit{H}}^1(\omega )$; the vector fields normal to the surfaces ${\mathit{\theta }}^k(\omega )$, $k⩾1$, are well defined a.e. in ω and they also belong to the space ${\mathit{H}}^1(\omega )$; the principal radii of curvature of the surfaces ${\mathit{\theta }}^k(\omega )$, $k⩾1$, stay uniformly away from zero; and finally, the fundamental forms of the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{L}}^1(\omega )$ toward the fundamental forms of the surface $\mathit{\theta }(\overline{\omega })$ as $k\to \infty$. Then, up to proper isometries of ${\mathbb{R}}^3$, the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{H}}^1(\omega )$ toward the surface $\mathit{\theta }(\overline{\omega })$ as $k\to \infty$.Such results have potential applications to nonlinear shell theory, the surface $\mathit{\theta }(\overline{\omega })$ being then the middle surface of the reference configuration of a nonlinearly elastic shell.     

Biharmonic ideal hypersurfaces in Euclidean spaces

Let $x:M\to {\mathbb{E}}^m$ be an isometric immersion from a Riemannian n-manifold into a Euclidean m-space. Denote by Δ and $\overrightarrow{x}$ the Laplace operator and the position vector of M, respectively. Then M is called biharmonic if ${\mathrm{\Delta }}^2\overrightarrow{x}=0$. The following Chen?s Biharmonic Conjecture made in 1991 is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper we prove that the biharmonic conjecture is true for $\delta (2)$-ideal and $\delta (3)$-ideal hypersurfaces of a Euclidean space of arbitrary dimension.     

Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes

In the present paper we perform the homogenization of the semilinear elliptic problem

$\{\begin{array}{cc}{u}^{\epsilon }\ge 0\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ ?divA(x)D{u}^{\epsilon }=F(x,u^{\epsilon })\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ u^{\epsilon }=0\hfill & \text{on}?{\mathrm{\Omega }}^{\epsilon }.\hfill \end{array}$

In this problem $F(x,s)$ is a Carathéodory function such that $0\le F(x,s)\le h(x)/\mathrm{\Gamma }(s)$ a.e. $x\in \mathrm{\Omega }$ for every $s>0$, with h in some $L^r(\mathrm{\Omega })$ and Γ a $C^1([0,+\infty [)$ function such that $\mathrm{\Gamma }(0)=0$ and ${\mathrm{\Gamma }}^{\prime }(s)>0$ for every $s>0$. On the other hand the open sets ${\mathrm{\Omega }}^{\epsilon }$ are obtained by removing many small holes from a fixed open set Ω in such a way that a “strange term” $\mu u^0$ appears in the limit equation in the case where the function $F(x,s)$ depends only on x.We already treated this problem in the case of a “mild singularity”, namely in the case where the function $F(x,s)$ satisfies $0\le F(x,s)\le h(x)(\frac{1}{s}+1)$. In this case the solution $u^{\epsilon }$ to the problem belongs to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ and its definition is a “natural” and rather usual one.In the general case where $F(x,s)$ exhibits a “strong singularity” at $u=0$, which is the purpose of the present paper, the solution $u^{\epsilon }$ to the problem only belongs to $H_{\text{loc}}^1({\mathrm{\Omega }}^{\epsilon })$ but in general does not belong to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ anymore, even if $u^{\epsilon }$ vanishes on $?{\mathrm{\Omega }}^{\epsilon }$ in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results.In the present paper, using this definition, we perform the homogenization of the above semilinear problem and we prove that in the homogenized problem, the “strange term” $\mu u^0$ still appears in the left-hand side while the source term $F(x,u^0)$ is not modified in the right-hand side.