16,409 research outputs found

    Lack of compactness in two-scale convergence

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    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 Γ-convergence of integral functionals and its application to homogenisation of nonlinear high-contrast periodic composites

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    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, “stress-strain”, response on different parts of the period cell. The approach of this article is based on the concept of “two-scale Γ-convergence”, which is a kind of “hybrid” of the classical Γ-convergence (De Giorgi and Franzoni in Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur (8)58:842–850, 1975) and the more recent two-scale convergence (Nguetseng in SIAM J Math Anal 20:608–623, 1989). The present study focuses on a basic high-contrast model, where “soft” inclusions are embedded in a “stiff” matrix. It is shown that the standard Γ-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 Γ-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 “non-classical” two-scale extension of the well-known theorem by MĂŒller (Arch Rational Mech Anal 99:189–212, 1987)

    On the lack of compactness in the 2D critical Sobolev embedding

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    This paper is devoted to the description of the lack of compactness of Hrad1(R2)H^1_{rad}(\R^2) 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

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    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

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    In Rd\mathbb{R}^d, the characterization of the \mbox{lack of compactness of the continuous Sobolev injection H˚sâ†ȘLp \mathring{H}^s \hookrightarrow L^p }, with sd+1p=12 \displaystyle{\frac{s}{d} + \frac{1}{p} = \frac{1}{2}} and $\displaystyle{

    Anisotropic stars as ultracompact objects in General Relativity

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    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

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    This paper is devoted to the description of the lack of compactness of the Sobolev embedding of H1(R2)H^1(\R^2) 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 L∞L^ \infty estimate far away from the origin and which is no longer valid in the general framework. Within the general framework of H1(R2)H^1(\R^2), 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

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    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

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    We characterize the lack of compactness in the critical embedding of functions spaces X⊂YX\subset Y having similar scaling properties in the following terms : a sequence (un)n≄0(u_n)_{n\geq 0} bounded in XX has a subsequence that can be expressed as a finite sum of translations and dilations of functions (ϕl)l>0(\phi_l)_{l>0} such that the remainder converges to zero in YY as the number of functions in the sum and nn tend to +∞+\infty. Such a decomposition was established by G\'erard for the embedding of the homogeneous Sobolev space X=H˙sX=\dot H^s into the Y=LpY=L^p in dd dimensions with 0<s=d/2−d/p0<s=d/2-d/p, and then generalized by Jaffard to the case where XX 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 XX and YY that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of XX and YY satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, H\"older and BMO spaces.Comment: 24 page
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