Related papers: Sharp local smoothing for manifolds with smooth in…
We consider sharp interface asymptotics for a phase field model of two phase near spherical biomembranes involving a coupling between the local mean curvature and the local composition proposed by the first and second authors. The model is…
We establish new Strichartz estimates for orthonormal systems on compact Riemannian manifolds in the case of wave, Klein-Gordon and fractional Schr\"odinger equations. Our results generalize the classical (single-function) Strichartz…
A comprehensive approach to Sobolev-type embeddings, involving arbitrary rearrangement- invariant norms on the entire Euclidean space R^n, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in…
We prove a family of identities that involve the solutions to the free Schreodinger equation. As a consequence of these identities we shall deduce a lower bound for the local smoothing estimate and a uniqueness criterion.
We establish Strichartz estimates for the Schr\"odinger equation on Riemannian manifolds $(\Omega,\g)$ with boundary, for both the compact case and the case that $\Omega$ is the exterior of a smooth, non-trapping obstacle in Euclidean…
We study the nonlinear Schr\"odinger equation (NLS) with the quadratic nonlinearity $|u|^2$, posed on the two-dimensional torus $\mathbb{T}^2$. While the relevant $L^3$-Strichartz estimate is known only with a derivative loss, we prove…
We integrate the one-dimensional nonlinear Schr\"odinger equation numerically for solitons moving in external potentials. In particular, we study the scattering off an interface separating two regions of constant potential modeled by a…
We prove that the cubic nonlinear Schr\"odinger equation (both focusing and defocusing) is globally well-posed in $H^s(\mathbb R)$ for any regularity $s>-\frac12$. Well-posedness has long been known for $s\geq 0$, see [55], but not…
We prove two new results about the Cauchy problem for nonlinear Schroedinger equations on four-dimensional compact manifolds. The first one concerns global wellposedness for Hartree-type nonlinearities and includes approximations of cubic…
We introduce an information-theoretic framework for smooth structures on topological manifolds, replacing coordinate charts with small-scale entropy data of local probability probes. A concise set of axioms identifies admissible coordinate…
In this note we obtain semiclassical resolvent estimates for non-trapping long range perturbations of the Laplacian on asymptotically Euclidean manifolds. Our proof is based on a positive commutator argument which differs from Mourre-type…
We study a notion of strict pseudoconvexity in the context of topologically (often unsmoothably) embedded 3-manifolds in complex surfaces. Topologically pseudoconvex (TPC) 3-manifolds behave similarly to their smooth analogues, cutting out…
The purpose of this short article is to prove some potential estimates that naturally arise in the study of subelliptic Sobolev inequalites for functions. This will allow us to prove a local subelliptic Sobolev inequality with the optimal…
This paper is concerned with the inverse problem of time-harmonic acoustic scattering by an unbounded, locally rough interface which is assumed to be a local perturbation of a plane. The purpose of this paper is to recover the local…
Numerous dimensionality reduction problems in data analysis involve the recovery of low-dimensional models or the learning of manifolds underlying sets of data. Many manifold learning methods require the estimation of the tangent space of…
We prove the higher-dimensional analogue of Wolff's local smoothing estimate (Geom. Funct. Anal. 2001) for large p. As in the 2+1-dimensional case, the estimate is sharp for any given value of p, but it is likely that the range of p can be…
We prove new local well-posedness results for nonlinear Schr\"odinger equations posed on a general product of spheres and tori, by the standard approach of multi-linear Strichartz estimates. To prove these estimates, we establish and…
We present a new numerical method for the isometric embedding of 2-geometries specified by their 2-metrics in three dimensional Euclidean space. Our approach is to directly solve the fundamental embedding equation supplemented by six…
We study standing periodic waves modeled by the nonlinear Schrodinger equation with the intensity-dependent dispersion coefficient. Spatial periodic profiles are smooth if the frequency of the standing waves is below the limiting frequency,…
In this paper we study maps (curved flats) into symmetric spaces which are tangent at each point to a flat of the symmetric space. Important examples of such maps arise from isometric immersions of space forms into space forms via their…