Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus

On the eigenvalue spacing distribution for a point scatterer on the flat torus

We study the level spacing distribution for the spectrum of a point scatterer on a flat torus. In the 2-dimensional case, we show that in the weak coupling regime the eigenvalue spacing distribution coincides with that of the spectrum of the Laplacian (ignoring multiplicties), by showing that the perturbed eigenvalues generically clump with the unperturbed ones on the scale of the mean level spacing. We also study the three dimensional case, where the situation is very different.

Wasserstein Asymptotics for the Empirical Measure of Fractional Brownian Motion on a Flat Torus

We establish asymptotic upper and lower bounds for the Wasserstein distance of any order $p\ge 1$ between the empirical measure of a fractional Brownian motion on a flat torus and the uniform Lebesgue measure. Our inequalities reveal an interesting interaction between the Hurst index $H$ and the dimension $d$ of the state space, with a "phase-transition" in the rates when $d=2+1/H$, akin to the Ajtai-Komlós-Tusnády theorem for the optimal matching of i.i.d. points in two-dimensions. Our proof couples PDE's and probabilistic techniques, and also yields a similar result for discrete-time approxi…

Compactness issues and bubbling phenomena for the prescribed Gaussian curvature equation on the Torus

In the spirit of the paper "Large conformal metrics of prescribed Gauss curvature on surfaces of higher genus" by Borer-Galimberti-Struwe, where we dealt with the case of a closed Riemann surface $(M,g_0)$ of genus greater than one, here we study the behaviour of the conformal metrics $g_λ$ of prescribed Gauss curvature $K_{g_λ} = f_0 + λ$ on the torus, when the parameter $λ$ tends to one of the boundary points of the interval of existence of $g_λ$, and we characterize their "bubbling behaviour".

Long-time behaviour and phase transitions for the McKean--Vlasov equation on the torus

We study the McKean-Vlasov equation \[ \partial_t \varrho= β^{-1} Δ\varrho + κ\nabla \cdot (\varrho \nabla (W \star \varrho)) \, , \] with periodic boundary conditions on the torus. We first study the global asymptotic stability of the homogeneous steady state. We then focus our attention on the stationary system, and prove the existence of nontrivial solutions branching from the homogeneous steady state, through possibly infinitely many bifurcations, under appropriate assumptions on the interaction potential. We also provide sufficient conditions for the existence of continuous and discontinu…

Uniform $L^p$ Resolvent Estimates on the Torus

A new range of uniform $L^p$ resolvent estimates is obtained in the setting of the flat torus, improving previous results of Bourgain, Shao, Sogge and Yao. The arguments rely on the $\ell^2$-decoupling theorem and multidimensional Weyl sum estimates.

Eigenfunction statistics for a point scatterer on a three-dimensional torus

In this paper we study eigenfunction statistics for a point scatterer (the Laplacian perturbed by a delta-potential) on a three-dimensional flat torus. The eigenfunctions of this operator are the eigenfunctions of the Laplacian which vanish at the scatterer, together with a set of new eigenfunctions (perturbed eigenfunctions). We first show that for a point scatterer on the standard torus all of the perturbed eigenfunctions are uniformly distributed in configuration space. Then we investigate the same problem for a point scatterer on a flat torus with some irrationality conditions, and show un…

Non-existence of solutions for a mean field equation on flat tori at critical parameter $16π$

It is known from \cite{LW} that the solvability of the mean field equation $Δu+e^{u}=8nπδ_{0}$ with $n\in\mathbb{N}_{\geq 1}$ on a flat torus $E_τ$ essentially depends on the geometry of $E_τ$. A conjecture is the non-existence of solutions for this equation if $E_τ$ is a rectangular torus, which was proved for $n=1$ in \cite{LW}. For any $n\in \mathbb{N}_{\geq2}$, this conjecture seems challenging from the viewpoint of PDE theory. In this paper, we prove this conjecture for $n=2$ (i.e. at critical parameter $16π$).

环面上两物种聚集-扩散系统的长时间行为与分岔分析

Long-time behaviour and bifurcation analysis of a two-species aggregation-diffusion system on the torus

本文研究了环面上非局部聚集-扩散方程的稳态存在性与稳定性，利用 Crandall & Rabinowitz 分岔理论对两物种系统进行严格分析，分类解分支并推导 Fréchet 导数，揭示了纯吸引作用下细胞分选起始时的稳定分离模式。

Quantum revivals in two degrees of freedom integrable systems : the torus case

The paper deals with the semi-classical behaviour of quantum dynamics for a semi-classical completely integrable system with two degrees of freedom near Liouville regular torus. The phenomomenon of wave packet revivals is demonstrated in this article. The framework of this paper is semi-classical analysis (limit :). For the proofs we use standard tools of real analysis, Fourier analysis and basic analytic number theory.

Length orthospectrum and the correlation function on flat tori

This note presents some of the results obtained in arXiv:2207.05410 and it has beenthe object of a talk of the second author during the Journées "Équations auxDérivées Partielles" (Obernai, june 2022). We study properties of geodesics that are orthogonal to two convex subsets of the flat torus. We discuss meromorphic properties of a geometric Epstein zeta function associated to the set of lengths of such orthogeodesics. We also define the associated length distribution and discuss singularities of its Fourier transform. Our analysis relies on a fine study of the dynamical correlation function …

Courant-sharp eigenvalues of a two-dimensional torus

In this paper, we determine, in the case of the Laplacian on the flat two-dimensional torus (R/Z) 2 , all the eigenvalues having an eigenfunction which satisfies Courant's theorem with equality (Courant-sharp situation). Following the strategy o A. Pleijel (1956), the proof is a combination of a lower bound a la Weyl) of the counting function, with an explicit remainder term, and of a Faber--Krahn inequality for domains on the torus (deduced as in B{é}rard-Meyer from an isoperimetric inequality), with an explicit upper bound on the area.

椭圆方程在环面上的非唯一性

Lack of uniqueness for an elliptic equation with nonlinear and nonlocal drift posed on a torus

研究了在环面上的非线性和非局部椭圆方程的非唯一性，通过分支定理证明了非常数周期解的存在，并构造了特定一维形式的显式多重解。

Two-microlocal regularity of quasimodes on the torus

We study the regularity of stationary and time-dependent solutions to strong perturbations of the free Schrödinger equation on two-dimensional flat tori. This is achieved by performing a second microlocalization related to the size of the perturbation and by analysing concentration and nonconcentration properties at this new scale. In particular, we show that sufficiently accurate quasimodes can only concentrate on the set of critical points of the average of the potential along geodesics.