Analysis & PDE最新文献

We employ functional analysis techniques in order to deduce some versions of classical and recent interpolation results in Fourier analysis with perturbed nodes. As an application of our techniques, we obtain generalizations of Kadec’s $\frac{1}{4}$-theorem for interpolation formulae in the Paley–Wiener space both in the real and complex cases, as well as versions of the recent interpolation result of Radchenko and Viazovska (Publ. Math. Inst. Hautes Études Sci. 129 (2019), 51–81) and the result of Cohn, Kumar, Miller, Radchenko and Viazovska (Ann. Math$(2)$196:3 (2022), 983–1082) for Fourier interpolation with derivatives in dimensions $8$ and $24$ with suitable perturbations of the interpolation nodes. We also provide several applications of the main results and techniques, relating to recent contributions in interpolation formulae and uniqueness sets for the Fourier transform.

This work concerns $L^p$ norms of high energy Laplace eigenfunctions: $(-{\mathrm{\Delta }}_g�-{\lambda }^2){\varphi }_{\lambda }�=0$, $\parallel {\varphi }_{\lambda }{\parallel }_{L^2}�=1$. Sogge (1988) gave optimal estimates on the growth of $\parallel {\varphi }_{\lambda }{\parallel }_{L^p}$ for a general compact Riemannian manifold. Here we give general dynamical conditions guaranteeing quantitative improvements in $L^p$ estimates for $p�>p_c$, where $p_c$ is the critical exponent. We also apply results of an earlier paper (Canzani and Galkowski 2018) to obtain quantitative improvements in concrete geometric settings including all product manifolds. These are the first results giving quantitative improvements for estimates on the $L^p$ growth of eigenfunctions that only require dynamical assumptions. In contrast with previous improvements, our assumptions are local in the sense that they depend only on the geodesics passing through a shrinking neighborhood of a given set in $M$. Moreover, we give a structure theorem for eigenfunctions which saturate the quantitatively improved $L^p$ bound. Modulo an error, the theorem describes these eigenfunctions as finite sums of quasimodes which, roughly, approximate zonal harmonics on the sphere scaled by , as well as sequences with similar convexity properties. We utilize the wave packet structure of functions with frequency support near an arithmetic progression.

Using a combination of direct geometric methods and an analysis of the linearization of the flow about the horizontal bisector, we prove that there exists a unique (modulo rotations about the origin) convex ancient curve-shortening flow in the disc with free boundary on the circle. This appears to be the first result of its kind in the free-boundary setting.

We introduce a new space IDA of locally integrable functions whose integral distance to holomorphic functions is finite, and use it to completely characterize boundedness and compactness of Hankel operators on weighted Fock spaces. As an application, for bounded symbols, we show that the Hankel operator $H_f$ is compact if and only if $H_{\overline{f}}$ is compact, which complements the classical compactness result of Berger and Coburn. Motivated by recent work of Bauer, Coburn, and Hagger, we also apply our results to the Berezin–Toeplitz quantization.

For the nonlinear Schrödinger equation in dimension 2, the existence of a global minimizer of the energy at fixed momentum has been established by Bethuel, Gravejat and Saut (2009) (see also work of Chiron and Mariş (2017)). This minimizer is a traveling wave for the nonlinear Schrödinger equation. For large momenta, the propagation speed is small and the minimizer behaves like two well-separated vortices. In that limit, we show the uniqueness of this minimizer, up to the invariances of the problem, hence proving the orbital stability of this traveling wave. This work is a follow up to two previous papers, where we constructed and studied a particular traveling wave of the equation. We show a uniqueness result on this traveling wave in a class of functions that contains in particular all possible minimizers of the energy.

This paper proves the stability, with respect to the evolution determined by the vacuum Einstein equations, of the Cartesian product of higher-dimensional Minkowski space with a compact, Ricci-flat Riemannian manifold that admits a spin structure and a nonzero parallel spinor. Such a product includes the example of Calabi–Yau and other special holonomy compactifications, which play a central role in supergravity and string theory. The stability result proved in this paper shows that Penrose’s instability argument [2003] does not apply to localised perturbations.

Monge–Ampère gravitation is a modification of the classical Newtonian gravitation where the linear Poisson equation is replaced by the nonlinear Monge–Ampère equation. This paper is concerned with the rigorous derivation of Monge–Ampère gravitation for a finite number of particles from the stochastic model of a Brownian point cloud, following the formal ideas of a recent work by Brenier (Bull. Inst. Math.Acad. Sin. 11:1(2016), 23–41). This is done in two steps. First, we compute the good rate function corresponding to a large deviation problem related to the Brownian point cloud at fixed positive diffusivity. Second, we study the $\Gamma$-convergence of this good rate function, as the diffusivity tends to zero, toward a (nonsmooth) Lagrangian encoding the Monge–Ampère dynamic. Surprisingly, the singularities of the limiting Lagrangian correspond to dissipative phenomena. As an illustration, we show that they lead to sticky collisions in one space dimension.

We consider smooth solutions of the Burgers–Hilbert equation that are a small perturbation $\delta$ from a global periodic traveling wave with small amplitude $𝜖$. We use a modified energy method to prove the existence time of smooth solutions on a time scale of $1∕(𝜖\delta )$, with $0�<�\delta �\ll �𝜖�\ll 1$, and on a time scale of $𝜖∕{\delta }^2$, with $0�<�\delta �\ll {𝜖}^2�\ll 1$. Moreover, we show that the traveling wave exists for an amplitude $𝜖$ in the range $(0,{𝜖}^{\ast })$, with ${𝜖}^{\ast }\sim 0.23$, and fails to exist for $𝜖�>2∕e$.

A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains, a.k.a. a Morse–Smale complex. This partition is generated by gradient flow lines of the eigenfunction, which bound the so-called Neumann domains. We prove that the Neumann Laplacian defined on a Neumann domain is self-adjoint and has a purely discrete spectrum. In addition, we prove that the restriction of an eigenfunction to any one of its Neumann domains is an eigenfunction of the Neumann Laplacian. By comparison, similar statements about the Dirichlet Laplacian on a nodal domain of an eigenfunction are basic and well-known. The difficulty here is that the boundary of a Neumann domain may have cusps and cracks, so standard results about Sobolev spaces are not available. Another very useful common fact is that the restricted eigenfunction on a nodal domain is the first eigenfunction of the Dirichlet Laplacian. This is no longer true for a Neumann domain. Our results enable the investigation of the resulting spectral position problem for Neumann domains, which is much more involved than its nodal analogue.