G-convergence and homogenization of sequences of linear and nonlinear partial differential operators

Abstract: In this thesis we study G-convergence for sequences of linear and nonlinear elliptic and parabolic problems. We also study the special case of homogenization for such problems. In this connection we also develop homogenization procedures and numerical algorithms and present some numerical results for the homogenization of an equilibrium stress problem. The main results in the thesis are the following: Stability results, continuity results, G-convergence results and asymptotic formulas for parameter dependent nonlinear elliptic problems (Theorem 4.2, Lemma 4.1 and Propositions 4.1 and 4.2). A compensated compactness result and a G-convergence result for nonlinear parabolic problems (Theorems 5.1 and 5.2). Stability results and continuity results for G-convergence of nonlinear parabolic problems and comparison results between elliptic and parabolic G-limits (Theorems 6.1-6.6). Homogenization procedures and numerical algorithms, well suited for implementation on work stations and numerical modeling (Section 7). A numerical modeling with numerical computations for an equilibrium stress problem (Section 8). For completeness we also present the results in the linear case and for the convenience of the reader we include an independent proof of the G-convergence in the linear elliptic case (Theorem 2.1). In Section 9 we discuss some further results and open questions in connection to this thesis. We also include a variety of examples from e.g. mechanics where G-convergence and homogenization have been applied.