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Gianni Dal Maso (auth.)'s An Introduction to Γ-Convergence PDF

By Gianni Dal Maso (auth.)

ISBN-10: 1461203279

ISBN-13: 9781461203278

ISBN-10: 1461267099

ISBN-13: 9781461267096

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Additional resources for An Introduction to Γ-Convergence

Example text

14). 6. Assume that p > 1 and that f and 9 satisfy (i), (ii), (iii), (iv), with Co > 0 and CI > O. Let K be a sequentially weakly closed subset of WI,p(n). 4) min ( ( f(x, Du(x)) dx + ( g(x, u(x» dx) uEK in in has a solution. If, in addition, K is convex and g(x,·) is strictly convex on R for a. e. 4) has exactly one solution. Proof. 6). 2). 9 the functional F + G + XK is sequentially lower semicontinuous in the weak topology of WI,p(n). 3). 5. 15. 4). e. 20. 4) has exactly one solution. 4) has a solution, even if the lower bound (iv) for 9 holds only with a negative constant CI, provided ICII is small enough.

2). 5) can be obtained by taking the limit as c tends to O. 0 Chapter 6 Some Properties of r-limits In this chapter we study some properties of r -limits and K-limits which hold on every topological space X. Let (Fh) be a sequence of functions from X into R and let (Eh) be a sequence of subsets of X . 1. If (Fhk) is a subsequence of (Fh), then r-liminf Fh :s; r-liminf Fhk , h-+oo r-limsupFh;::: r-limsupFhk' k-+oo In particular, if (Fh) in X. 1) and from the properties of the ordinary lower and upper limits.

F) If Fh(X) = _e- hx2 , then (Fh) r-converges in R to the function -I, ifx=O, F(x) = { 0, if x =I 0, which is the pointwise limit of (Fh)' while (-Fh) r -converges to 0 in R. (g) If Fh(x) ={ 0, if h(x - e h ) is integer, 1, otherwise, then for every x E R and every hEN there exists Y E R such that Iy - xl < l/h and Fh(Y) = O. This shows that (Fh) r-converges to 0 in R. Since e is transcendental, for every x E R there exists at most one index hEN such that Fh(X) = 0, and this shows that (Fh) converges pointwise to 1.

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An Introduction to Γ-Convergence by Gianni Dal Maso (auth.)


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