Related papers: Stability results for Sobolev, logarithmic Sobolev…
We present how entropy estimates and logarithmic Sobolev inequalities on the one hand, and the notion of quasi-stationary distribution on the other hand, are useful tools to analyze metastable overdamped Langevin dynamics, in particular to…
This paper considers the stability problem of a linear time invariant system in feedback with a string equation. A new Lyapunov functional candidate is proposed based on the use of augmented states which enriches and encompasses the…
Functional inequalities such as the Poincar\'e and log-Sobolev inequalities quantify convergence to equilibrium in continuous-time Markov chains by linking generator properties to variance and entropy decay. However, many applications,…
We review recent results on functional inequalities for systems of orthonormal functions. The key finding is that for various operators the orthonormality leads to a gain over a simple application of the triangle inequality. The operators…
We prove that in the context of general Markov semigroups Beckner inequalities with constants separated from zero as $p\to 1^+$ are equivalent to the modified log Sobolev inequality (previously only one implication was known to hold in this…
In the present article, solvability in Sobolev spaces is investigated for a class of degenerate stochastic integro-differential equations of parabolic type. Existence and uniqueness is obtained, and estimates are given for the solution.
In this paper, we investigate an inverse random source problem concerned with recovering the strength of a random, uncorrelated acoustic source from correlation measurements of emitted time-harmonic acoustic waves. Such problems arise in…
We are mainly interested in extending the known results on ob-servability inequalities and stabilization for the Schr{\"o}dinger equation to the magnetic Schr{\"o}dinger equation. That is in presence of a magnetic potential. We establish…
In this article, we provide a modified argument for proving conditional stability for inverse problems of determining spatially varying functions in evolution equations by Carleman estimates. Our method needs not any cut-off procedures and…
In this paper, we consider the Euclidean logarithmic Sobolev inequality \begin{eqnarray*} \int_{\mathbb{R}^d}|u|^2\log|u|dx\leq\frac{d}{4}\log\bigg(\frac{2}{\pi d e}\|\nabla u\|_{L^2(\mathbb{R}^d)}^2\bigg), \end{eqnarray*} where $u\in…
Proceeding the study of local properties of analytic functions started in [Br] we prove new dimensionless inequalities for such functions in terms of their Chebyshev degree. As a consequence, we obtain the reverse Holder inequalities for…
Convergence rates results for Tikhonov regularization of nonlinear ill-posed operator equations in abstract function spaces require the handling of both smoothness conditions imposed on the solution and structural conditions expressing the…
This paper contains two main contributions. First, it provides optimal stability estimates for advection-diffusion equations in a setting in which the velocity field is Sobolev regular in the spatial variable. This estimate is formulated…
We provide several characterizations of convergence to unstable equilibria in nonlinear systems. Our current contribution is three-fold. First we present simple algebraic conditions for establishing local convergence of non-trivial…
The problem of formulating self-consistent local and global stability exponents is shown to require global separation of variables. Posing the separation of variable problem, we see that many such separations are possible, but only one is…
The existence of extremal functions for the Sobolev trace inequalities is studied using the concentration compactness theorem. The conjectured extremal, the function of conformal factor, is considered and is proved to be an actual extremal…
This article carries out a qualitative analysis on a system of integral equations of the Hardy--Sobolev type. Namely, results concerning Liouville type properties and the fast and slow decay rates of positive solutions for the system are…
In this paper, we investigate the problems of Cauchy theory and exponential stability for the inhomogeneous Boltzmann equation without angular cut-off. We only deal with the physical case of hard potentials type interactions (with a…
We study the sharp constant in the Hardy inequality for fractional Sobolev spaces defined on open subsets of the Euclidean space. We first list some properties of such a constant, as well as of the associated variational problem. We then…
This paper proposes an unconditionally stable numerical method for solving a nonlinear Sobolev model with distributed delay. The proposed computational approach approximates the time derivative by interpolation technique whereas the spatial…