We study the asymptotic behavior of a quadratic NLS equation with small, time-dependent potential and small spatially localized initial data. We prove global existence and scattering of solutions. The two main ingredients of the proof are the space-time resonance method and the boundedness of wave operators for the linear Schrödinger equation with potential.

This paper is concerned with the $L^p$ integrability of $N$-harmonic functions with respect to the standard weights ${\left(1-|x{|}^2\right)}^{\alpha }$ on the unit ball $\mathbb{𝔹}$ of ${ℝ}^n$, $n\ge 2$. More precisely, our goal is to determine the real (negative) parameters $\alpha$ for which ${\left(1-|x{|}^2\right)}^{\alpha ∕p}u\left(x\right)\in L^p\left(\mathbb{𝔹}\right)$ implies that $u\equiv 0$ whenever $u$ is a solution of the $N$-Laplace equation on $\mathbb{𝔹}$. This question is motivated by the uniqueness considerations of the Dirichlet problem for the $N$-Laplacian ${\mathrm{\Delta }}^N$.

Our study is inspired by a recent work of Borichev and Hedenmalm (Adv. Math. 264 (2014), 464–505), where a complete answer to the above question in the case $n=2$ is given for the full scale . When $n\ge 3$, we obtain an analogous characterization for and remark that the remaining case can be genuinely more difficult. Also, we extend the remarkable cellular decomposition theorem of Borichev and Hedenmalm to all dimensions.

We prove an isoperimetric inequality for lower-order eigenvalues of the Dirichlet Laplacian on bounded domains of a Euclidean space which strengthens the celebrated Ashbaugh–Benguria inequality conjectured by Payne, Pólya and Weinberger on the ratio of the first two Dirichlet eigenvalues and makes an important step toward the proof of a conjecture by Ashbaugh and Benguria.

We prove an adiabatic theorem for the Landau–Pekar equations. This allows us to derive new results on the accuracy of their use as effective equations for the time evolution generated by the Fröhlich Hamiltonian with large coupling constant $\alpha$. In particular, we show that the time evolution of Pekar product states with coherent phonon field and the electron being trapped by the phonons is well approximated by the Landau–Pekar equations until times short compared to ${\alpha }^2$.

We study existence of the outgoing/incoming resolvents of repulsive Schrödinger operators which may not be essentially self-adjoint on the Schwartz space. As a consequence, we construct $L^2$-eigenfunctions associated with complex eigenvalues by a standard technique of scattering theory. In particular, we give another proof of the classical result via microlocal analysis: the repulsive Schrödinger operators with large repulsive exponent are not essentially self-adjoint on the Schwartz space.

We study the dynamics of corotational wave maps from ${ℝ}^{1+2}\to {\mathbb{𝕊}}^2$ at threshold energy. It is known that topologically trivial wave maps with energy are global and scatter to a constant map. We prove that a corotational wave map with energy equal to $8\pi$ is globally defined and scatters in one time direction, and in the other time direction, either the map is globally defined and scatters, or the map breaks down in finite time and converges to a superposition of two harmonic maps. The latter behavior stands in stark contrast to higher equivariant wave maps with threshold energy, which have been proven to be globally defined for all time. Using techniques developed in this paper, we also construct a corotational wave map with energy $=8\pi$ which blows up in finite time. The blow-up solution we construct provides the first example of a minimal topologically trivial nondispersing solution to the full wave map evolution.

We study sequences of conformal deformations of a smooth closed Riemannian manifold of dimension $n$, assuming uniform volume bounds and $L^{n∕2}$ bounds on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we show that under such bounds the underlying metric spaces are precompact in the Gromov–Hausdorff topology. Our study is based on the use of $A_{\infty }$-weights from harmonic analysis and the geometric controls that this property induces on the limit spaces thus obtained. Our techniques also show that any conformal deformation of the Euclidean metric on ${ℝ}^n$ with infinite volume and finite $L^{n∕2}$ norm of the scalar curvature satisfies the Euclidean isoperimetric inequality.

Let ${\mathcal{𝒦}}_{x_0}^{r_0}$ be a (nondegenerate) truncated corner in ${ℝ}^3$, with $x_0\in {ℝ}^3$ being its apex, and ${\mathit{F}}_j\in C^{\alpha }\left(\overline{{\mathcal{𝒦}}_{x_0}^{r_0}};{ℂ}^3\right)$, $j=1,2$, where $\alpha$ is the positive Hölder index. Consider the electromagnetic problem

where $\nu$ denotes the exterior unit normal vector of $\partial {\mathcal{𝒦}}_{x_0}^{r_0}$. We prove that ${\mathit{F}}_1$ and ${\mathit{F}}_2$ must vanish at the apex $x_0$. There is a series of interesting consequences of this vanishing property in several separate but intriguingly connected topics in electromagnetism. First, we can geometrically characterize nonradiating sources in time-harmonic electromagnetic scattering. Secondly, we consider the inverse source scattering problem for time-harmonic electromagnetic waves and establish the uniqueness result in determining the polyhedral support of a source by a single far-field measurement. Thirdly, we derive a property of the geometric structure of electromagnetic interior transmission eigenfunctions near corners. Finally, we also discuss its implication to invisibility cloaking and inverse medium scattering.

We consider the Cauchy initial value problem for the defocusing quintic nonlinear Schrödinger equation in two dimensions with general data in the critical space ${\dot{H}}^{1∕2}\left({ℝ}^2\right)$. We show that if a solution remains bounded in ${\dot{H}}^{1∕2}\left({ℝ}^2\right)$ in its maximal interval of existence, then the interval is infinite and the solution scatters.

In the past two decades, Sorin Popa’s breakthrough deformation/rigidity theory has produced remarkable rigidity results for von Neumann algebras $M$ which can be deformed inside a larger algebra $\tilde{M}\supseteq M$ by an action $\alpha :ℝ\to Aut\left(\tilde{M}\right)$, while simultaneously containing subalgebras $Q$ rigid with respect to that deformation, that is, such that ${\alpha }_t\to id$ uniformly on the unit ball of $Q$ as $t\to 0$. However, it has remained unclear how to exploit the interplay between distinct rigid subalgebras not in specified relative position.

We show that in fact, any diffuse subalgebra which is rigid with respect to a mixing s-malleable deformation is contained in a subalgebra which is uniquely maximal with respect to being rigid. In particular, the algebra generated by any family of rigid subalgebras that intersect diffusely must itself be rigid with respect to that deformation. The case where this family has two members was the motivation for this work, showing for example that if $G$ is a countable group with , then $L\left(G\right)$ cannot be generated by two property-(T) subalgebras with diffuse intersection; however, the result is most striking when the family is infinite.

Relying on the recent work of Liu and Székelyhidi, we give a weak asymptotic estimate for the Bergman kernels of polarized Kähler manifolds with Ricci lower bound and Sobolev constant upper bound. We will also give a simple proof for the partial $C^0$ estimate along the (generalized) Kähler–Ricci flow on Fano manifolds.