We can now complete the proof of Theorem 3. By the G-convergence of Ah to A we obtain, by using Lemma 3. Let us now return to the shifted and perturbed Dirac operator Jih. We will throughout this section assume the hypotheses of Theorem 2. It easily also follows from this mean-value property that. We are now interested in the asymptotic behavior of the operator and the spectrum of the perturbed Dirac operator Hh.
We recall the spectral problem for Hh, that is,. We know, by Theorem 2. This means that the Dirac operator is neither a positive or negative semi-definite operator and thus the G-convergence method introduced in the previous section for positive self-adjoint operators is not directly applicable. In order to use G-convergence methods for the asymptotic analysis of Hh we therefore use spectral projection and study the corresponding asymptotic behavior of projections Hh which are positive so that G-converg-ence methods apply.
Let A be a fixed a-algebra of subsets of r, and let r, A be a measurable space. Consider the spectral measures EH and EHh of the families of Dirac operators Hh and H, respectively, each one of these measures maps A onto px, where px is the set of orthogonal projections on X.
By the spectral theorem. It is clear that for uh e Xh we have. By Lemma 2. Moreover, by Theorem 3. The limit shifted Dirac operator restricted to X is explicitly given by. This follows by standard arguments in homogenization theory, see for example, . We denote by Xh the orthogonal complement in X to the eigenspace Xph. Thus, Xh is the closed subspace invariant with respect to corresponding to the absolutely continuous.
Next we define the restriction H? Therefore by Proposition Finally, by considering the operator -H? De Giorgi and S. Spagnolo, "Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine," Bollettino della Unione Matematica Italiana, vol. Spagnolo, "Sul limite delle soluzioni di problemi di cauchy relativi all'equazione del calore," Annali della Scuola Normale Superiore di Pisa, vol. Spagnolo, "Convergence in energy for elliptic operators," in Proceedings of the 3rd Symposium on the Numerical Solution of Partial Differential Equations, pp.
Birman and M. Weidmann, "Strong operator convergence and spectral theory of ordinary differential operators," Universitatis Iagellonicae. Acta Mathematica, no. Oleinik, A. Shamaev, and G. Bensoussan, J. Lions, and G. However, users may print, download, or email articles for individual use. CC BY. Variational principles for spectral analysis of one Sturm-Liouville problem with transmission conditions. In this work we study self-adjoint projections of Dirac operators which satisfy the positivity so that the theory of G-convergence becomes applicable. Preliminaries Let A be a linear operator on a Hilbert space.
Dirac Operator We recall some basic facts regarding the Dirac operator. In those papers the concept of algebra with mean value, denoted homogenizaton algebra, and non-periodic multiscale convergence, denoted Sigma convergence, are the main tools. Here we show that the general theory also applies to the situation of multiple scales and e. The result of Theorem It is also seen from the framework that the result easily extends to any number of well sep- arated scales.
A typical situation where periodic and random scales occur is the modeling of porous media.
A meso-scale can be modeled as a periodic distribu- tion of solid parts whereas a sub-scale on a finer level can be modeled by a certain random distribution. The homogenization problem for random fields in the linear elliptic case is studied in . The extension to monotone operators in the random setting is studied in  and has been further studied in a series of papers by Efendiev and Pankov, see  and the references therein.
They consider single spatial and temporal scales. For homogenization problem for random fields of monotone parabolic problems we refer to . A prototype application of results in this paper is damped elastic wave propagation in heterogeneous media of e. Voigth type.
We will use the general framework of G-convergence for monotone parabolic operators developed in  to prove a stochastic homogenization result for 1. The result will rely on a number of well-known results for elliptic and parabolic G-convergence.
For the benefit of the reader we review these results with references to the original proofs. This makes the present work more self-contained.
The paper is organized as follows: In Section 2 we recall the basic terminology of G-convergence of parabolic operators, in Section 3 we introduce some basic facts about monotone operators in reflexive Banach Spaces. Sections 5, 6 and 7 review some basic results for elliptic and parabolic G-convergence that will be needed in the proof of con- vergence of hyperbolic problems which is presented in Section 8.
Section 9 is a preparation on multiscale stochastic operators where the framework is based on a dynamical systems setup and in Section 10 we prove a homogenization result for the nonlinearly damped quasilinear hyperbolic problem 1.
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The maps ah are assumed to be monotone and to satisfy certain boundedness and coercivness assumptions uniformly in h. This yields G-convergence of quasilinear parabolic operators. For a complete treatment of G-convergence of monotone parabolic operators we refer to . Some notations Let us introduce some function spaces related to the differential equations studied in this paper. For a nice introduction to partial differential operators in Banach spaces we refer to the monograph  by Barbu.
Then the evolution triples considered above are well-defined with dense embeddings. In order to obtain the existence and uniqueness for 4. Theorem 4. Consider the hyperbolic initial-boundary value problem 4. We refer to  or  for a full proof. By the results of Theorem 4. Proposition 5. The following G-compactness result is proved in . Theorem 5. Elliptic G-convergence For a complete treatment of a large class of possibly multivalued elliptic operators we refer to  and .
Definition 6. See .
[PDF] An Introduction to Homogenization and G-convergence - Semantic Scholar
Parameter-dependent elliptic G-convergence We begin by stating a compactness result with respect to elliptic G-convergence for parameter dependent elliptic problems: Theorem 7. See . Theorem 7. Convergence of hyperbolic problems We begin by stating a general compensated compactness theorem for quasilin- ear monotone hyperbolic problems. Theorem 8. For the proof we refer to . We also recall the following compactness result recently proved in .
We now extend our result to the damped nonlinear wave equation with non- linear damping.
An Introduction to G-Convergence
We first observe that by the existence Theorem 4. By Theorem 8. I: An integral representation and its consequences. Side - Mathematical aspects of the physics of disordered systems. Critical phenomena, random systems, gauge theories, Proc. Summer Sch. Side - Un teorema di passaggio al limite per la somma di funzioni convesse, Boll. Side - Cont re-examples pour divers problemes ou le controle intervient dans les coefficients.