Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSobolevs inequality under a curvature-dimension condition
196
0
0.0
(
0
)
Authors
Louis Dupaigne
Ask ChatGPT about the research
No Arabic abstract
In this note we present a new proof of Sobolevs inequality under a uniform lower bound of the Ricci curvature. This result was initially obtained in 1983 by Ilias. Our goal is to present a very short proof, to give a review of the famous inequality and to explain how our method, relying on a gradient-flow interpretation, is simple and robust. In particular, we elucidate computations used in numerous previous works, starting with Bidaut-V{e}ron and V{e}rons 1991 classical work.
rate research
Read More
In this paper, we study the extremal problem for the Strichartz inequality for the Schr{o}dinger equation on $mathbb{R}^2$. We show that the solutions to the associated Euler-Lagrange equation are exponentially decaying in the Fourier space and thus can be extended to be complex analytic. Consequently we provide a new proof to the characterization of the extremal functions: the only extremals are Gaussian functions, which was investigated previously by Foschi and Hundertmark-Zharnitsky.
We establish existence and multiplicity of solutions for a elliptic resonant elliptic problem under Dirichlet boundary conditions.
We prove an a priori estimate of type sup*inf on Riemannian manifold of dimension 3 (not necessarily compact).
We consider a class of semilinear nonlocal problems with vanishing exterior condition and establish a Ambrosetti-Prodi type phenomenon when the nonlinear term satisfies certain conditions. Our technique makes use of the probabilistic tools and heat kernel estimates.
Usually Fokker-Planck type partial differential equations (PDEs) are well-posed if the initial condition is specified. In this paper, alternatively, we consider the inverse problem which consists in prescribing final data: in particular we give sufficient conditions for existence and uniqueness. In the second part of the paper we provide a probabilistic representation of those PDEs in the form a solution of a McKean type equation corresponding to the time-reversal dynamics of a diffusion process.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
