Relative rearrangement on a finite measure space. Application to weighted spaces and to P.D.E.
In this paper, we consider a Borel measurable function on the space of matrices taking the value , such that its rank-one-convex envelope is finite and satisfies for some fixed : where . Let be a given regular bounded open domain of . We define on the functional Then, under some technical restrictions on , we show that the relaxed functional for the weak topology of has the integral representation: where for a given function , denotes its quasiconvex...
Multidimensional vectorial non-quasiconvex variational problems are relaxed by means of a generalized-Young-functional technique. Selective first-order optimality conditions, having the form of an Euler-Weiestrass condition involving minors, are formulated in a special, rather a model case when the potential has a polyconvex quasiconvexification.
For a function the notion of p-mean variation of order 1, is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space in terms of is directly related to the characterisation of by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.
We present several continuous embeddings of the critical Besov space . We first establish a Gagliardo-Nirenberg type estimate , for 1 < p ≤ q < ∞, 1 ≤ ν < ρ ≤ ∞ and the weight function with 0 < r < n. Next, we prove the corresponding Trudinger type estimate, and obtain it in terms of the embedding , where the function Φ₀ of the weighted Besov-Orlicz space is a Young function of the exponential type. Another point of interest is to embed into the weighted Besov space with...
In compressible Neohookean elasticity one minimizes functionals which are composed by the sum of the norm of the deformation gradient and a nonlinear function of the determinant of the gradient. Non–interpenetrability of matter is then represented by additional invertibility conditions. An existence theory which includes a precise notion of invertibility and allows for cavitation was formulated by Müller and Spector in 1995. It applies, however, only if some -norm of the gradient with is controlled...
Dans ce travail on s’intéresse aux opérateurs de composition sur certains espaces de Besov et de Lizorkin-Triebel à valeurs dans . Dans le but de caractériser les fonctions qui opèrent, on établit que la condition de Lipschitz, locale ou globale suivant que l’espace ou se plonge ou non dans , est nécessaire pour , et que l’appartenance locale au même espace l’est aussi pour . Nous étudions enfin la régularité de l’opérateur .
We derive weighted Poincaré inequalities for vector fields which satisfy the Hörmander condition, including new results in the unweighted case. We also derive a new integral representation formula for a function in terms of the vector fields applied to the function. As a corollary of the versions of Poincaré’s inequality, we obtain relative isoperimetric inequalities.
We give a representation of the spaces as spaces of vector-valued sequences and use it to investigate their topological properties and isomorphic classification. In particular, it is proved that is isomorphic to the sequence space , thereby showing that the isomorphy class does not depend on the dimension N if p=2.
This paper studies analytic aspects of so-called resistance conditions on metric measure spaces with a doubling measure. These conditions are weaker than the usually assumed Poincaré inequality, but however, they are sufficiently strong to imply several useful results in analysis on metric measure spaces. We show that under a perimeter resistance condition, the capacity of order one and the Hausdorff content of codimension one are comparable. Moreover, we have connections to the Sobolev inequality...
L’objet de cet article est de prouver des théorèmes du genre suivant : “Soient un opérateur différentiel sur , une fonction à valeurs réelles, un nombre réel et une distribution à support compact : alors, si , ” ; l’espace est ici l’espace de Sobolev “d’ordre variable” associé à ; bien entendu, il faut des hypothèses sur , et . Les cas traités sont :1) certains opérateurs à coefficients variables déjà considérés dans le chapitre VIII du livre de L. Hörmander ;2) tous les opérateurs...