16,409 research outputs found
Lack of compactness in two-scale convergence
This article deals with the links between compensated compactness and two-scale convergence. More precisely, we ask the following question: Is the div-curl compactness assumption sufficient to pass to the limit in a product of two sequences which two-scale converge with respect to the pair of variables (x, x/Δ)? We reply in the negative. Indeed, the div-curl assumption allows us to control oscillations which are faster than 1/Δ but not the slower ones
Two-scale Î-convergence of integral functionals and its application to homogenisation of nonlinear high-contrast periodic composites
An analytical framework is developed for passing to the homogenisation limit in (not necessarily convex) variational problems for composites whose material properties oscillate with a small period Δ and that exhibit high contrast of order 1/Δ between the constitutive, âstress-strainâ, response on different parts of the period cell. The approach of this article is based on the concept of âtwo-scale Î-convergenceâ, which is a kind of âhybridâ of the classical Î-convergence (De Giorgi and Franzoni in Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur (8)58:842â850, 1975) and the more recent two-scale convergence (Nguetseng in SIAM J Math Anal 20:608â623, 1989). The present study focuses on a basic high-contrast model, where âsoftâ inclusions are embedded in a âstiffâ matrix. It is shown that the standard Î-convergence in the L^p -space fails to yield the correct limit problem as Δ tends to 0, due to the underlying lack of L^p -compactness for minimising sequences. Using an appropriate two-scale compactness statement as an alternative starting point, the two-scale Î-limit of the original family of functionals is determined via a combination of techniques from classical homogenisation, the theory of quasiconvex functions and multiscale analysis. The related result can be thought of as a ânon-classicalâ two-scale extension of the well-known theorem by MĂŒller (Arch Rational Mech Anal 99:189â212, 1987)
On the lack of compactness in the 2D critical Sobolev embedding
This paper is devoted to the description of the lack of compactness of
in the Orlicz space. Our result is expressed in terms of the
concentration-type examples derived by P. -L. Lions. The approach that we adopt
to establish this characterization is completely different from the methods
used in the study of the lack of compactness of Sobolev embedding in Lebesgue
spaces and take into account the variational aspect of Orlicz spaces. We also
investigate the feature of the solutions of non linear wave equation with
exponential growth, where the Orlicz norm plays a decisive role.Comment: 38 page
Stochastic homogenization of subdifferential inclusions via scale integration
We study the stochastic homogenization of the system -div \sigma^\epsilon =
f^\epsilon \sigma^\epsilon \in \partial \phi^\epsilon (\nabla u^\epsilon),
where (\phi^\epsilon) is a sequence of convex stationary random fields, with
p-growth. We prove that sequences of solutions (\sigma^\epsilon,u^\epsilon)
converge to the solutions of a deterministic system having the same
subdifferential structure. The proof relies on Birkhoff's ergodic theorem, on
the maximal monotonicity of the subdifferential of a convex function, and on a
new idea of scale integration, recently introduced by A. Visintin.Comment: 23 page
On the lack of compactness on stratified Lie groups
In , the characterization of the \mbox{lack of compactness of
the continuous Sobolev injection }, with
and $\displaystyle{
Anisotropic stars as ultracompact objects in General Relativity
Anisotropic stresses are ubiquitous in nature, but their modeling in General
Relativity is poorly understood and frame dependent. We introduce the first
study on the dynamical properties of anisotropic self-gravitating fluids in a
covariant framework. Our description is particularly useful in the context of
tests of the black hole paradigm, wherein ultracompact objects are used as
black hole mimickers but otherwise lack a proper theoretical framework. We show
that: (i) anisotropic stars can be as compact and as massive as black holes,
even for very small anisotropy parameters; (ii) the nonlinear dynamics of the
1+1 system is in good agreement with linearized calculations, and shows that
configurations below the maximum mass are nonlinearly stable; (iii) strongly
anisotropic stars have vanishing tidal Love numbers in the black-hole limit;
(iv) their formation will usually be accompanied by gravitational-wave echoes
at late times.Comment: 7+2 pages, 6 figures; v2: include extra material (general covariant
framework for anisotropic fluids in General Relativity without symmetries and
code validation); to appear in PR
Lack of compactness in the 2D critical Sobolev embedding, the general case
This paper is devoted to the description of the lack of compactness of the
Sobolev embedding of in the critical Orlicz space {\cL}(\R^2). It
turns out that up to cores our result is expressed in terms of the
concentration-type examples derived by J. Moser in \cite{M} as in the radial
setting investigated in \cite{BMM}. However, the analysis we used in this work
is strikingly different from the one conducted in the radial case which is
based on an estimate far away from the origin and which is no
longer valid in the general framework. Within the general framework of
, the strategy we adopted to build the profile decomposition in
terms of examples by Moser concentrated around cores is based on capacity
arguments and relies on an extraction process of mass concentrations. The
essential ingredient to extract cores consists in proving by contradiction that
if the mass responsible for the lack of compactness of the Sobolev embedding in
the Orlicz space is scattered, then the energy used would exceed that of the
starting sequence.Comment: Submitte
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces
We obtain an improved Sobolev inequality in H^s spaces involving Morrey
norms. This refinement yields a direct proof of the existence of optimizers and
the compactness up to symmetry of optimizing sequences for the usual Sobolev
embedding. More generally, it allows to derive an alternative, more transparent
proof of the profile decomposition in H^s obtained in [P. Gerard, ESAIM 1998]
using the abstract approach of dislocation spaces developed in [K. Tintarev &
K. H. Fieseler, Imperial College Press 2007]. We also analyze directly the
local defect of compactness of the Sobolev embedding in terms of measures in
the spirit of [P. L. Lions, Rev. Mat. Iberoamericana 1985]. As a model
application, we study the asymptotic limit of a family of subcritical problems,
obtaining concentration results for the corresponding optimizers which are well
known when s is an integer ([O. Rey, Manuscripta math. 1989; Z.-C. Han, Ann.
Inst. H. Poincare Anal. Non Lineaire 1991], [K. S. Chou & D. Geng, Differential
Integral Equations 2000]).Comment: 33 page
A general wavelet-based profile decomposition in the critical embedding of function spaces
We characterize the lack of compactness in the critical embedding of
functions spaces having similar scaling properties in the
following terms : a sequence bounded in has a subsequence
that can be expressed as a finite sum of translations and dilations of
functions such that the remainder converges to zero in as
the number of functions in the sum and tend to . Such a
decomposition was established by G\'erard for the embedding of the homogeneous
Sobolev space into the in dimensions with
, and then generalized by Jaffard to the case where is a Riesz
potential space, using wavelet expansions. In this paper, we revisit the
wavelet-based profile decomposition, in order to treat a larger range of
examples of critical embedding in a hopefully simplified way. In particular we
identify two generic properties on the spaces and that are of key use
in building the profile decomposition. These properties may then easily be
checked for typical choices of and satisfying critical embedding
properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, H\"older
and BMO spaces.Comment: 24 page
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