We study elliptic equations with the general nonstandard growth conditions involving Lebesgue measurable functions on . We prove the global regularity of bounded weak solutions of these equations with the Dirichlet boundary condition. Our results generalize the regularity results for the elliptic equations in divergence form not only in the variable exponent case but also in the constant exponent case.
Dans cet article, on considère les opérateurs différentiels , où et sont deux fonctions mesurables, bornées et accrétives, et . Les résultats principaux portent sur les propriétés fonctionnelles de , de sa racine carrée, avec applications à l’équation elliptique sur . On démontre que est un opérateur de Calderón-Zygmund qui dépend analytiquement du couple . Les estimations ponctuelles optimales sur le noyau du semi-groupe et le calcul fonctionnel permettent de développer une théorie...
In this work we introduce a new class of lowest order methods for diffusive problems on general meshes with only one unknown per element. The underlying idea is to construct an incomplete piecewise affine polynomial space with optimal approximation properties starting from values at cell centers. To do so we borrow ideas from multi-point finite volume methods, although we use them in a rather different context. The incomplete polynomial space replaces classical complete polynomial spaces in discrete...
In this work we introduce a new class of lowest order methods for diffusive problems on general meshes with only one unknown per element. The underlying idea is to construct an incomplete piecewise affine polynomial space with optimal approximation properties starting from values at cell centers. To do so we borrow ideas from multi-point finite volume methods, although we use them in a rather different context. The incomplete polynomial space replaces classical complete polynomial spaces...
This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in the two- and three-dimensional quasistatic regimes. CALR associated with negative index materials was discovered by Milton and Nicorovici [21] for constant plasmonic structures in the two-dimensional quasistatic regime. Two key features of this phenomenon are the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others, and the connection between the localized resonance...
There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a -valued function defined on the boundary of a bounded regular domain of . When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within the homology...