Abstract
This chapter discusses convergence in energy for elliptic operators. The chapter discusses the principal results on G-convergence, that is, the convergence of the Green's operators for Dirichlet's and other boundary problems. The study of G-convergence had its starting point with an example of De Giorgi concerning a sequence of ordinary linear differential operators of the second order whose coefficients rapidly vary. Several phenomena observed in this example can be extended to a suitable class of elliptic or parabolic partial differential equations. The chapter gives a direct proof of the compactness and locality properties of G-convergence, which avoids recourse to parabolic equations and uses elliptic regularization. It also presents various theorems and lemmas.
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