相关论文: Eigenvalue estimates and L1 energy on closed manif…

We discuss properties of $L^2$-eigenfunctions of Schr\"odinger operators and elliptic partial differential operators. The focus is set on unique continuation principles and equidistribution properties. We review recent results and announce…

In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell,…

We prove an extension to R^n, endowed with a suitable metric, of the relation between the Einstein-Hilbert action and the Dirac operator which holds on closed spin manifolds. By means of complex powers, we first define the regularised…

In this short survey, we derive some weyl-type universal inequalities of eigenvalues of the Laplacian on a closed Riemannian manifold of nonnegative Ricci curvature. We also give upper bounds for the $L_{\infty}$ norm of eigenfunctions of…

We consider a family of manifolds with a class of degenerating warped product metrics $g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$, with $M$ compact, $\rho$ homogeneous degree one, $a \le -1$ and $b > 0$. We study the…

In this paper, we study a first Dirichlet eigenfunction of the weighted $p$-Laplacian on a bounded domain in a complete weighted Riemannian manifold. By constructing gradient estimates for a first eigenfunction, we obtain some relationships…

Let $x: M\rightarrow \mathbb{R}^{N}$ be an $n$-dimensional compact self-shrinker in $\mathbb{R}^N$ with smooth boundary $\partial\Omega$. In this paper, we study eigenvalues of the operator $\mathcal{L}_r$ on $M$, where $\mathcal{L}_r$ is…

We prove a lower estimate for the first eigenvalue of the Dirac operator on a compact locally reducible Riemannian spin manifold with positive scalar curvature. We determine also the universal covers of the manifolds on which the smallest…

We derive localized and global noncompact versions of Hamilton's gradient estimate for positive solutions to the heat equation on Riemannian manifolds with Ricci curvature bounded below. Our estimates are essentially optimal and…

These notes were prepared in occasion of a mini-course given by the author at the "CIMPA Research School - Hamiltonian and Lagrangian Dynamics" (10-19 March 2015 - Salto, Uruguay). The talks were meant as an introduction to the problem of…

In this paper, we prove the concavity of the Shannon entropy power for the heat equation associated with the Laplacian or the Witten Laplacian on complete Riemannian manifolds with suitable curvature-dimension condition and on compact super…

$\mathfrak{L}_{II}$ operator is introduced by Y.-L. Xin (\emph{Calculus of Variations and Partial Differential Equations. 2015, \textbf{54}(2):1995-2016)}, which is an important extrinsic elliptic differential operator of divergence type…

In this paper, we study the Ricci flow on closed manifolds equipped with warped product metric $(N\times F,g_{N}+f^2 g_{F})$ with $(F,g_{F})$ Ricci flat. Using the framework of monotone formulas, we derive several estimates for the adapted…

We establish effective existence and uniqueness for the heat flow on time-dependent Riemannian manifolds, under minimal assumptions tailored towards the study of Ricci flow through singularities. The main point is that our estimates only…

Let $\Delta_\varphi = \Delta -\nabla \varphi \nabla$ be a symmetric diffusion operator with an invariant weighted volume measure $d\mu = e^{-\varphi} dv$ on an $n$-dimensional compact Riemannian manifold $(M,g)$, where $g=g(t)$ solves the…

For a family of elliptic operators with rapidly oscillating periodic coefficients, we study the convergence rates for Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The results rely on an…

We consider the discrete Laplacian $\Delta$ on the cubic lattice $\mathbb Z^d$, and deduce estimates on the group $e^{it\Delta}$ and the resolvent $(\Delta-z)^{-1}$, weighted by $\ell^q(\mathbb Z^d)$-weights for suitable $q\geq 2$. We apply…

In this paper, we study the drifted Laplacian $\Delta_f$ on a hypersurface $M$ in a Ricci shrinker $(\overline{M},g,f)$. We prove that the spectrum of $\Delta_f$ is discrete for immersed hypersurfaces with bounded weighted mean curvature in…

In the first part of this paper, we get new Li-Yau type gradient estimates for positive solutions of heat equation on Riemmannian manifolds with $Ricci(M)\ge -k$, $k\in \mathbb R$. As applications, several parabolic Harnack inequalities are…

We study linear Klein-Gordon equations with moving potentials motivated by the stability analysis of traveling waves and multi-solitons. In this paper, Strichartz estimates, local energy decay and the scattering theory for these models are…