/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

This paper deals with the fractional Sobolev spaces W s;p . We analyze the relations among some of their possible denitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.

/pdf/hitchhiker-s-guide-to-the-fractional-sobolev-spaces-irm2sao95g.pdf

Hitchhiker's guide to the fractional Sobolev spaces

Metamaterials are typically engineered by arranging a set of small scatterers or apertures in a regular array throughout a region of space, thus obtaining some desirable bulk electromagnetic behavior. The desired property is often one that is not normally found naturally (negative refractive index, near-zero index, etc.). Over the past ten years, metamaterials have moved from being simply a theoretical concept to a field with developed and marketed applications. Three-dimensional metamaterials can be extended by arranging electrically small scatterers or holes into a two-dimensional pattern at a surface or interface. This surface version of a metamaterial has been given the name metasurface (the term metafilm has also been employed for certain structures). For many applications, metasurfaces can be used in place of metamaterials. Metasurfaces have the advantage of taking up less physical space than do full three-dimensional metamaterial structures; consequently, metasurfaces offer the possibility of less-lossy structures. In this overview paper, we discuss the theoretical basis by which metasurfaces should be characterized, and discuss their various applications. We will see how metasurfaces are distinguished from conventional frequency-selective surfaces. Metasurfaces have a wide range of potential applications in electromagnetics (ranging from low microwave to optical frequencies), including: (1) controllable “smart” surfaces, (2) miniaturized cavity resonators, (3) novel wave-guiding structures, (4) angular-independent surfaces, (5) absorbers, (6) biomedical devices, (7) terahertz switches, and (8) fluid-tunable frequency-agile materials, to name only a few. In this review, we will see that the development in recent years of such materials and/or surfaces is bringing us closer to realizing the exciting speculations made over one hundred years ago by the work of Lamb, Schuster, and Pocklington, and later by Mandel'shtam and Veselago.

An Overview of the Theory and Applications of Metasurfaces: The Two-Dimensional Equivalents of Metamaterials

1 Introduction 2 Going variational 2.1 Griffith's theory 2.2 The 1-homogeneous case - A variational equivalence 2.3 Smoothness - The soft belly of Griffith's formulation 2.4 The non 1-homogeneous case - A discrete variational evolution 2.5 Functional framework - A weak variational evolution 2.6 Cohesiveness and the variational evolution 3 Stationarity versus local or global minimality - A comparison 3.1 1d traction 3.1.1 The Griffith case - Soft device 3.1.2 The Griffith case - Hard device 3.1.3 Cohesive case - Soft device 3.1.4 Cohesive case - Hard device 3.2 A tearing experiment 4 Initiation 4.1 Initiation - The Griffith case 4.1.1 Initiation - The Griffith case - Global minimality 4.1.2 Initiation - The Griffith case - Local minimality 4.2 Initiation - The cohesive case 4.2.1 Initiation - The cohesive 1d case - Stationarity 4.2.2 Initiation - The cohesive 3d case - Stationarity 4.2.3 Initiation - The cohesive case - Global minimality 5 Irreversibility 5.1 Irreversibility - The Griffith case - Well-posedness of the variational evolution 5.1.1 Irreversibility - The Griffith case - Discrete evolution 5.1.2 Irreversibility - The Griffith case - Global minimality in the limit 5.1.3 Irreversibility - The Griffith case - Energy balance in the limit 5.1.4 Irreversibility - The Griffith case - The time-continuous evolution 5.2 Irreversibility - The cohesive case 6 Path 7 Griffith vs. Barenblatt 8 Numerics and Griffith 8.1 Numerical approximation of the energy 8.1.1 The first time step 8.1.2 Quasi-static evolution 8.2 Minimization algorithm 8.2.1 The alternate minimization algorithm 8.2.2 The backtracking algorithm 8.3 Numerical experiments 8.3.1 The 1D traction (hard device) 8.3.2 The Tearing experiment 8.3.3 Revisiting the 2D traction experiment on a fiber reinforced matrix 9 Fatigue 9.1 Peeling Evolution 9.2 The limitfatigue law when d tends to 0 9.3 A variational formulation for fatigue 9.3.1 Peeling revisited 9.3.2 Generalization Appendix Glossary References.

The Variational Approach to Fracture

We study a functional with variable exponent, $1\leq p(x)\leq 2$, which provides a model for image denoising, enhancement, and restoration. The diffusion resulting from the proposed model is a combination of total variation (TV)-based regularization and Gaussian smoothing. The existence, uniqueness, and long-time behavior of the proposed model are established. Experimental results illustrate the effectiveness of the model in image restoration.

Variable Exponent, Linear Growth Functionals in Image Restoration

Given a bounded sequence (un) in L 1 ( ; ; IR d ), we describe the weak limits in the sense of measures of f(x;un) for a class of continuous integrands with linear growth at innit y. The defect of uniform integrability of the sequence f(x;un) is described by a measure m and a family of probability measures on S d 1 whereas the classical Young measure is associated with the biting limits in the sense of Chacon’s lemma. Some consequences of this new approach are given in Calculus of Variations.

/pdf/non-uniform-integrability-and-generalized-young-measures-icsgffuqx0.pdf

Non-Uniform Integrability and Generalized Young Measures

A new method for the identification of the integral representation of a class of functionals defined on $BV(\Omega ;{\Bbb R}^d)\times {\cal A}(\Omega)$(where ${\cal A}(\Omega)$ represents the family of open subsets of Ω) is presented. Applications are derived, such as the integral representation of the relaxed energy in $BV(\Omega;{\Bbb R}^d)$ corresponding to a functional defined in $W^{1,1}(\Omega;{\Bbb R}^d)$ with a discontinuous integrand with linear growth; relaxation and homogenization results in $SBV(\Omega;{\Bbb R}^d)$ are recovered in the case where bulk and surface energies are present.

A Global Method for Relaxation

We study some problems of optimal distribution of masses, and we show that they can be characterized by a suitable Monge-Kantorovich equation. In the case of scalar state functions, we show the equivalence with a mass transport problem, emphasizing its geometrical approach through geodesics. The case of elasticity, where the state function is vector valued, is also considered. In both cases some examples are presented.

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=3&iss=2&rank=2

Characterization of optimal shapes and masses through Monge-Kantorovich equation

Homogenization near resonances and artificial magnetism from dielectrics

We make the connection between the geometric model for capillarity with line tension and the Cahn‐Hilliard model of two‐phase fluids. To this aim we consider the energies
 
$$F_\eps(u):=\eps\int_\O |Du|^2 + {1\over\eps} \int_\O W(u) + \lambda\int_{\bd\O} V(u)$$
 where u is a scalar density function and W and V are double‐well potentials. We show that the behaviour of F

 e
 in the limit e→0 and λ→∞ depends on the limit of e log λ. If this limit is finite and strictly positive, then the singular limit of the energies F

 e
 leads to a coupled problem of bulk and surface phase transitions, and under certain assumptions agrees with the relaxation of the capillary energy with line tension. These results were announced in [ABS1] and [ABS2].

Guy Bouchitté

Papers

Non-Uniform Integrability and Generalized Young Measures

A Global Method for Relaxation

Characterization of optimal shapes and masses through Monge-Kantorovich equation

Homogenization near resonances and artificial magnetism from dielectrics

Phase transition with the line-tension effect