Inequalities in Sobolev Spaces

The aim of this article is to study a variant of the Poincaré inequality in Sobolev spaces $$W^{k,p(\cdot )}(\Omega ) $$ W k , p ( · ) ( Ω ) where $$\Omega $$ Ω is a given open set of $$\mathbb {R}^{n}$$ R n with finite Lebesgue measure and, consequently, the Friedrichs type inequality.

In this paper we consider proper orthogonal decomposition (POD) methods that do not include difference quotients (DQs) of snapshots in the data set. The inclusion of DQs have been shown in the literature to be a key element in obtaining error bounds that do not degrade with the number of snapshots. More recently, the inclusion of DQs has allowed to obtain pointwise (as opposed to averaged) error bounds that decay with the same convergence rate (in terms of the POD singular values) as averaged ones. In the present paper, for POD methods not including DQs in their data set, we obtain error bounds that do not degrade with the number of snapshots if the function from where the snapshots are taken has certain degree of smoothness. Moreover, the rate of convergence is as close as that of methods including DQs as the smoothness of the function providing the snapshots allows. We do this by obtaining discrete counterparts of Agmon and interpolation inequalities in Sobolev spaces. Numerical experiments validating these estimates are also presented.

We estimate the norm of a two-dimensional Hilbert operator in $L_p-spaces. This problen leads to inequalities of the theory of Sobolev spaces. In certain cases, by using methods of the theory of quasiconformal mappings, we get best possible estimates.

We consider parabolic variational inequalities in Sobolev spaces and generalized Lebesgue spaces. Sufficient conditions of existence and uniqueness of solution of these inequalities have been established. We have also proved some comparison theorems.

Observability inequalities in Sobolev spaces for the wave equation with variable coefficients