What happens to the conclusion of the Ekeland variational principle (briefly, EVP) if a considered function f : X → R ∪ {+∞} is lower semicontinuous not on a whole metric space X but only on its domain? We provide a straightforward proof showing that it still holds but only for varying in some interval ]0, β − infX f [, where β is a quantity expressing quantitatively the violation of the lower ...

In this paper we study the minimax control of systems governed by a nonlinear evolution inclusion of the subdifferential type. Using some continuity and lower semicontinuity results for the solution map and the cost functional respectively, we are able to establish the existence of an optimal control. The abstract results are then applied to obstacle problems, semilinear systems with weakly var...

A result of Larsen concerning the structure of the approximate gradient of certain sequences of functions with Bounded Variation is used to present a short proof of Ambrosio’s lower semicontinuity theorem for quasiconvex bulk energies in SBV . It enables to generalize to the SBV setting the decomposition lemma for scaled gradients in dimension reduction and also to show that, from the point of ...

We study the asymptotic behavior of the solution of a Stokes flow in a thin domain, with a thickness of order ε, and a rough surface. The roughness is defined by a quasi-periodic function with period ε. We suppose that the flow is subject to a Tresca fluid-solid interface condition. We prove a new result on the lower-semicontinuity for the two-scale convergence, which allows us to obtain rigoro...

*Correspondence: [email protected] Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, P.R. China Abstract The purpose of this paper is to establish the general theory of the direct methods to functionals I defined on the Grassmann algebra employing the classical approaches. In this paper, various notions of convexity conditions for weak lower semicontinuity of I are di...