An Introduction to G-ConvergenceThe last twentyfive years have seen an increasing interest for variational convergences and for their applications to different fields, like homogenization theory, phase transitions, singular perturbations, boundary value problems in wildly perturbed domains, approximation of variatonal problems, and non smooth analysis. Among variational convergences, De Giorgi's r-convergence plays a cen tral role for its compactness properties and for the large number of results concerning r -limits of integral functionals. Moreover, almost all other varia tional convergences can be easily expressed in the language of r -convergence. This text originates from the notes of the courses on r -convergence held by the author in Trieste at the International School for Advanced Studies (S. I. S. S. A. ) during the academic years 1983-84,1986-87, 1990-91, and in Rome at the Istituto Nazionale di Alta Matematica (I. N. D. A. M. ) during the spring of 1987. This text is far from being a treatise on r -convergence and its appli cations. |
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Page viii
... sequences r - limits of decreasing sequences Convexity and equi - continuity r - limits of sequences of locally equi - bounded convex functions Convergence of integrands and T - convergence of functionals 6. Some properties of r ...
... sequences r - limits of decreasing sequences Convexity and equi - continuity r - limits of sequences of locally equi - bounded convex functions Convergence of integrands and T - convergence of functionals 6. Some properties of r ...
Page x
... sequence of superadditive increasing functionals T - limit of a sequence of increasing local functionals 17. The topology of T - convergence 189 Topologies on the space of lower semicontinuous increasing functionals Relationships with ...
... sequence of superadditive increasing functionals T - limit of a sequence of increasing local functionals 17. The topology of T - convergence 189 Topologies on the space of lower semicontinuous increasing functionals Relationships with ...
Page 1
... sequence ( Fr ) of perturbations of F , which converges to F in a very strong way , then , in general , we can prove ... sequence ( Fr ) is equi - coercive ) , and if each function Fh is lower semicontinuous on X , then it is easy to see ...
... sequence ( Fr ) of perturbations of F , which converges to F in a very strong way , then , in general , we can prove ... sequence ( Fr ) is equi - coercive ) , and if each function Fh is lower semicontinuous on X , then it is easy to see ...
Page 2
... sequence ( Fh ) of functionals of the form ( 0.1 ) , corresponding to a sequence of functions ( fh ) . If the usual coerciveness and growth conditions are satisfied uniformly with respect to h , and if for every ER " the sequence ( fh ...
... sequence ( Fh ) of functionals of the form ( 0.1 ) , corresponding to a sequence of functions ( fh ) . If the usual coerciveness and growth conditions are satisfied uniformly with respect to h , and if for every ER " the sequence ( fh ...
Page 3
... sequence ( Fr ) . This elementary example shows that the " variational limit " of a sequence of integral functionals can be different from the pointwise limit , and that , in the case of strongly oscillating integrands , the ...
... sequence ( Fr ) . This elementary example shows that the " variational limit " of a sequence of integral functionals can be different from the pointwise limit , and that , in the case of strongly oscillating integrands , the ...
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Anal Assume Attouch Banach space Borel bounded calculus of variations Chapter coercive compact subset concludes the proof converges convex functions definition denotes Du(x elliptic operators equi-coercive Example F is lower Fh(y functional defined functional F fundamental estimate Giorgi hence homogenization implies increasing functional inequality inf Fh inf Fr(y inf inf Fr(y inner regular integral functionals integral representation K-lim L²(N Lemma Let F Let F:X Let us fix Let us prove lim inf inf lim sup lim sup inf linear lower semicontinuous LP(N Math metric space minimum point neighbourhood non-negative Nonlinear obtain open subset Paris Sér pointwise problems properties Proposition 8.1 Pures Appl quadratic form real numbers Remark resp satisfies sc-F self-adjoint sequence h sequentially strong topology subadditive T-converges to F T-lim inf T-lim sup T-limits Theorem topological space W¹P(N weak topology νευ
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