Calabi Product Lagrangian Immersions in Complex Projective Space and Complex Hyperbolic Space

Starting from two Lagrangian immersions and a Legendre curve ${\tilde{\gamma}(t)}$ in ${\mathbb{S}^3(1)}$ $({\rm or\,in}\,{\mathbb{H}_1^3(-1)})$ , it is possible to construct a new Lagrangian immersion in ${\mathbb{CP}^n(4)}$ $({\rm or\,in}\,{\mathbb{CH}^n(-4)})$ , which is called a warped product Lagrangian immersion. When ${\tilde{\gamma}(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, \;r_2e^{i(- \frac{r_1}{r_2}at)})}$ $({\rm or}\,{\tilde{\gamma}(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, \;r_2e^{i( \frac{r_1}{r_2}at)})})$ , where r ₁, r ₂, and a are positive constants with ${r_1^2+r_2^2=1}$ $({\rm or}\,{-r_1^2+r_2^2=-1})$ , we call the new Lagrangian immersion a Calabi product Lagrangian immersion. In this paper, we study the inverse problem: how to determine from the properties of the second fundamental form whether a given Lagrangian immersion of ${\mathbb{CP}^n(4)}$ or ${\mathbb{CH}^n(-4)}$ is a Calabi product Lagrangian immersion. When the Calabi product is minimal, or is Hamiltonian minimal, or has parallel second fundamental form, we give some further characterizations.     

1D Schrödinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

For real ${L_\infty(\mathbb{R})}$ -functions ${\Phi}$ and ${\Psi}$ of compact support, we prove the norm resolvent convergence, as ${\varepsilon}$ and ${\nu}$ tend to 0, of a family ${S_{\varepsilon \nu}}$ of one-dimensional Schrödinger operators on the line of the form $$S_{\varepsilon \nu} = -\frac{d^2}{dx^2} + \frac{\alpha}{\varepsilon^2} \Phi \left( \frac{x}{\varepsilon} \right) + \frac{\beta}{\nu} \Psi \left(\frac{x}{\nu} \right),$$ provided the ratio ${\nu/\varepsilon}$ has a finite or infinite limit. The limit operator S ₀ depends on the shape of ${\Phi}$ and ${\Psi}$ as well as on the limit of ratio ${\nu/\varepsilon}$ . If the potential ${\alpha\Phi}$ possesses a zero-energy resonance, then S ₀ describes a non trivial point interaction at the origin. Otherwise S ₀ is the direct sum of the Dirichlet half-line Schrödinger operators.     

Prescribed diagonal Schouten tensor in locally conformally flat manifolds

We consider the pseudo-euclidean space ${(\mathbb{R}^n, g)}$ , with n ≥  3 and ${g_{ij} = \delta_{ij} \varepsilon_i, \varepsilon_i = \pm 1}$ and tensors of the form ${T = \sum \nolimits_i \varepsilon_i f_i (x) dx_i^2}$ . In this paper, we obtain necessary and sufficient conditions for a diagonal tensor to admit a metric ${\bar{g}}$ , conformal to g, so that ${A_{\bar g}=T}$ , where ${A_{\bar g}}$ is the Schouten Tensor of the metric ${\bar g}$ . The solution to this problem is given explicitly for special cases for the tensor T, including a case where the metric ${\bar g}$ is complete on ${\mathbb{R}^n}$ . Similar problems are considered for locally conformally flat manifolds. As an application of these results we consider the problem of finding metrics ${\bar g}$ , conformal to g, such that ${\sigma_2 ({\bar g })}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })}}$ is equal to a given function. We prove that for some functions, f ₁ and f ₂, there exist complete metrics ${\bar{g} = g/{\varphi^2}}$ , such that ${\sigma_2 ({\bar g }) = f_1}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })} = f_2}$ .     

Existence of semiclassical states for a quasilinear Schrödinger equation with critical exponent in {\mathbb{R}^N}

We consider the following perturbed version of quasilinear Schrödinger equation $$\begin{array}{lll}-\varepsilon^2\Delta u +V(x)u-\varepsilon^2\Delta (u^2)u=h(x,u)u+K(x)|u|^{22^*-2}u\end{array}$$ in ${\mathbb{R}^N}$ , where N ≥ 3, 22* = 4N/(N ? 2), V(x) is a nonnegative potential, and K(x) is a bounded positive function. Using minimax methods, we show that this equation has at least one positive solution provided that ${\varepsilon \leq \mathcal{E}}$ ; for any ${m\in\mathbb{N}}$ , it has m pairs of solutions if ${\varepsilon \leq \mathcal{E}_m}$ , where ${\mathcal{E}}$ and ${\mathcal{E}_m}$ are sufficiently small positive numbers. Moreover, these solutions ${u_\varepsilon \to 0}$ in ${H^1(\mathbb{R}^N)}$ as ${\varepsilon \to 0}$ .     

Ground states for nonlinear Kirchhoff equations with critical growth

This paper is concerned with the existence, multiplicity and concentration behavior of positive solutions for the critical Kirchhoff-type problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\left(\varepsilon ^2a+\varepsilon b\int _{\mathbb{R }^{3}}|\nabla u|^2\right)\Delta u+V(x)u=u^{2^*-1}+\lambda f(u)&\text{ in}~{\mathbb{R }^{3}},\\ u\in H^1({\mathbb{R }^{3}}), ~u(x)>0&\text{ in}~{\mathbb{R }^{3}}, \end{array}\right. \end{aligned}$$ where $\varepsilon $ and $\lambda $ are positive parameters, and $a,b>0$ are constants, $2^*(=6)$ is the critical Sobolev exponent in dimension three, $V$ is a positive continuous potential satisfying some conditions, and $f$ is a subcritical nonlinear term. We use the variational methods to relate the number of solutions with the topology of the set where $V$ attains its minimum, for all sufficiently large $\lambda $ and small $\varepsilon $ .     

Interface foliation for an inhomogeneous Allen–Cahn equation in Riemannian manifolds

Let ${(\mathcal{M}, \tilde{g})}$ be an N-dimensional smooth compact Riemannian manifold. We consider the problem ${\varepsilon^2 \triangle_{\tilde{g}} \tilde{u} + V(\tilde{z})\tilde{u}(1-\tilde{u}^2)=0\; {\rm in}\; \mathcal{M}}$ , where ${\varepsilon > 0}$ is a small parameter and V is a positive, smooth function in ${\mathcal{M}}$ . Let ${\kappa \subset \mathcal{M}}$ be an (N ? 1)-dimensional smooth submanifold that divides ${\mathcal{M}}$ into two disjoint components ${\mathcal{M}_{\pm}}$ . We assume κ is stationary and non-degenerate relative to the weighted area functional ${\int_{\kappa}V^{\frac{1}{2}}}$ . For each integer m ≥ 2, we prove the existence of a sequence ${\varepsilon = \varepsilon_\ell \rightarrow 0}$ , and two opposite directional solutions with m-transition layers near κ, whose mutual distance is ${{\rm O}(\varepsilon | \log \varepsilon | )}$ . Moreover, the interaction between neighboring layers is governed by a type of Jacobi–Toda system.     

Real Clifford Algebras as Tensor Products over Centers

Denote by ${\mathcal{C}\ell_{p,q}}$ the Clifford algebra on the real vector space ${\mathbb{R}^{p,q}}$ . This paper gives a unified tensor product expression of ${\mathcal{C}\ell_{p,q}}$ by using the center of ${\mathcal{C}\ell_{p,q}}$ . The main result states that for nonnegative integers p, q, ${\mathcal{C}\ell_{p,q} \simeq \otimes^{\kappa-\delta}\mathcal{C}_{1,1} \otimes Cen(\mathcal{C}\ell_{p,q}) \otimes^{\delta} \mathcal{C}\ell_{0,2},}$ where ${p + q \equiv \varepsilon}$ mod 2, ${\kappa = ((p + q) - \varepsilon)/2, p - |q - \varepsilon| \equiv i}$ mod 8 and ${\delta = \lfloor i / 4 \rfloor}$ .     

Infinitesimal Carleson Property for Weighted Measures Induced by Analytic Self-Maps of the Unit Disk

We prove that, for every $\alpha > -1$ , the pull-back measure $\varphi ({\mathcal A }_\alpha )$ of the measure $d{\mathcal A }_\alpha (z) = (\alpha + 1) (1 - |z|^2)^\alpha \, d{\mathcal A } (z)$ , where ${\mathcal A }$ is the normalized area measure on the unit disk $\mathbb D $ , by every analytic self-map $\varphi :\mathbb D \rightarrow \mathbb D $ is not only an $(\alpha \,{+}\, 2)$ -Carleson measure, but that the measure of the Carleson windows of size $\varepsilon h$ is controlled by $\varepsilon ^{\alpha + 2}$ times the measure of the corresponding window of size $h$ . This means that the property of being an $(\alpha + 2)$ -Carleson measure is true at all infinitesimal scales. We give an application by characterizing the compactness of composition operators on weighted Bergman–Orlicz spaces.     

Singular perturbation method for inhomogeneous nonlinear free boundary problems

In this paper, we study solutions of one phase inhomogeneous singular perturbation problems of the type: $ F(D^2u,x)=\beta _{\varepsilon }(u) + f_{\varepsilon }(x) $ and $ \Delta _{p}u=\beta _{\varepsilon }(u) + f_{\varepsilon }(x)$ , where $\beta _{\varepsilon }$ approaches Dirac $\delta _{0}$ as $\varepsilon \rightarrow 0$ and $f_{\varepsilon }$ has a uniform control in $L^{q}, q>N.$ Uniform local Lipschitz regularity is obtained for these solutions. The existence theory for variational (minimizers) and non variational (least supersolutions) solutions for these problems is developed. Uniform linear growth rate with respect to the distance from the $\varepsilon -$ level surfaces are established for these variational and nonvaritional solutions. Finally, letting $\varepsilon \rightarrow 0$ basic properties such as local Lipschitz regularity and non-degeneracy property are proven for the limit and a Hausdorff measure estimate for its free boundary is obtained.     

The universal Glivenko–Cantelli property

Let ${\mathcal{F}}$ be a separable uniformly bounded family of measurable functions on a standard measurable space ${(X, \mathcal{X})}$ , and let ${N_{[]}(\mathcal{F}, \varepsilon, \mu)}$ be the smallest number of ${\varepsilon}$ -brackets in L ¹(μ) needed to cover ${\mathcal{F}}$ . The following are equivalent:

1. ${\mathcal{F}}$ is a universal Glivenko–Cantelli class.
2. ${N_{[]}(\mathcal{F},\varepsilon,\mu) < \infty}$ for every ${\varepsilon > 0}$ and every probability measure μ.
3. ${\mathcal{F}}$ is totally bounded in L ¹(μ) for every probability measure μ.
4. ${\mathcal{F}}$ does not contain a Boolean σ-independent sequence.

It follows that universal Glivenko–Cantelli classes are uniformity classes for general sequences of almost surely convergent random measures.     

Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C _?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.     

Singular limits for Liouville-type equations on the flat two-torus

We consider the problem $$\begin{aligned} -\Delta u=\varepsilon ^{2}e^{u}- \frac{1}{|\Omega |}\int _\Omega \varepsilon ^{2} e^{u}+ {4\pi N\over |\Omega |} - 4 \pi N\delta _p, \quad \text{ in} {\Omega }, \quad \int _\Omega u=0 \end{aligned}$$ in a flat two-torus $\Omega $ with periodic boundary conditions, where $\varepsilon >0,\,|\Omega |$ is the area of the $\Omega $ , $N>0$ and $\delta _p$ is a Dirac mass at $p\in \Omega $ . We prove that if $1\le m<N+1$ then there exists a family of solutions $\{u_\varepsilon \}_{\varepsilon }$ such that $\varepsilon ^{2}e^{u_\varepsilon }\rightharpoonup 8\pi \sum _{i=1}^m\delta _{q_i}$ as $\varepsilon \rightarrow 0$ in measure sense for some different points $q_{1}, \ldots , q_{m}$ . Furthermore, points $q_i$ , $i=1,\dots ,m$ are different from $p$ .     

Profile of the least energy solution of a singular perturbed Neumann problem with mixed powers

We consider the problem ${\varepsilon^{2}\Delta u - u^q + u^p = 0\,{\rm in}\,\Omega,\,u > 0\,{\rm in}\,\Omega,\,\frac{\partial u}{\partial \nu} = 0\,{\rm on}\,\partial\Omega }$ where Ω is a smooth bounded domain in ${\mathbb{R}^N}$ , ${1 < q < p < {N+2\over N-2}}$ if N ≥ 2 and ${\varepsilon}$ is a small positive parameter. We determine the location and shape of the least energy solution when ${\varepsilon \rightarrow 0.}$      

On the boundary of the attainable set of the dirichlet spectrum

Denoting by ${\varepsilon\subseteq\mathbb{R}^2}$ the set of the pairs ${(\lambda_1(\Omega),\,\lambda_2(\Omega))}$ for all the open sets ${\Omega\subseteq\mathbb{R}^N}$ with unit measure, and by ${\Theta\subseteq\mathbb{R}^N}$ the union of two disjoint balls of half measure, we give an elementary proof of the fact that ${\partial\varepsilon}$ has horizontal tangent at its lowest point ${(\lambda_1(\Theta),\,\lambda_2(\Theta))}$ .     

Congruence Properties of Binary Partition Functions

Let ${\mathcal{A}}$ be a finite subset of ${\mathbb{N}}$ containing 0, and let f (n) denote the number of ways to write n in the form ${\sum \varepsilon _{j}2^{j}}$ , where ${\varepsilon _{j} \epsilon \mathcal{A}}$ . We show that there exists a computable ${T = T (\mathcal{A})}$ so that the sequence (f (n) mod 2) is periodic with period T. Variations and generalizations of this problem are also discussed.     

Multiplicity and concentration of positive solutions for the Schr?dinger�CPoisson equations

This paper is concerned with the multiplicity and concentration of positive solutions for the nonlinear Schr?dinger?CPoisson equations $$ \left\{ \begin{array}{l@{\quad}l} -\varepsilon^2\triangle u+V(x)u+\phi(x) u=f(u)& {\rm in}\,{\mathbb R}^3, \\ -\varepsilon^2\triangle \phi=u^2 & {\rm in}\,{\mathbb R}^3, \\ u\in H^1({\mathbb R}^3), u(x) > 0,& \forall x\in{\mathbb R}^3, \\ \end{array} \right. $$ where ???>?0 is a parameter, ${V: {\mathbb R}^3\rightarrow{\mathbb R}}$ is a continuous function and ${f: {\mathbb R}\rightarrow {\mathbb R}}$ is a C ¹ function having subcritical growth. The proof of the main result is based on minimax theorems and the Ljusternik?CSchnirelmann theory.     

On Deformations of Lie Algebroids

For any Lie algebroid A, its 1-jet bundle ${\mathfrak{J} A}$ is a Lie algebroid naturally and there is a representation ${\pi:\mathfrak{J} A\longrightarrow\mathfrak{D} A}$ . Denote by ${{\rm d}_{\mathfrak{J}}}$ the corresponding coboundary operator. In this paper, we realize the deformation cohomology of a Lie algebroid A introduced by M. Crainic and I. Moerdijk as the cohomology of a subcomplex ${(\Gamma({\rm Hom}(\wedge^\bullet\mathfrak{J} A,A)_{{\mathfrak{D}} A}),{\rm d}_{\mathfrak{J}})}$ of the cochain complex ${(\Gamma({\rm Hom}(\wedge^\bullet\mathfrak{J} A, A)),{\rm d}_\mathfrak{J})}$ .     

Donsker-type theorems for nonparametric maximum likelihood estimators

Let ${\mathcal{P}}$ be a nonparametric probability model consisting of smooth probability densities and let ${\hat{p}_{n}}$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law ${\mathbb{P}}$ . With $\hat{\mathbb{P}}_{n}$ denoting the measure induced by the density ${\hat{p}_{n}}$ , define the stochastic process ${\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P})$ where f ranges over some function class ${\mathcal{F}}$ . We give a general condition for Donsker classes ${\mathcal{F}}$ implying that the stochastic process $\hat{\nu}_{n}$ is asymptotically equivalent to the empirical process in the space ${\ell ^{\infty }(\mathcal{F})}$ of bounded functions on ${ \mathcal{F}}$ . This implies in particular that $\hat{\nu}_{n}$ converges in law in ${\ell ^{\infty }(\mathcal{F})}$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes ${\mathcal{ F}}$ . We give a number of applications: convergence of the probability measure ${\hat{\mathbb{P}}_{n}}$ to ${\mathbb{P}}$ at rate ${\sqrt{n}}$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; ${\sqrt{n}}$ -efficient estimation of nonlinear functionals defined on ${\mathcal{P}}$ ; limit theorems at rate ${\sqrt{n}}$ for the maximum likelihood estimator of the convolution product ${\mathbb{P\ast P}}$ .