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

By Gianni Dal Maso (auth.)

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E. x E f2 and for every °such that ~ ERn. 12 shows that the pointwise convergence of a sequence of integral functionals satisfying (i), (ii), (iii) implies the r -convergence in the strong topology of W 1,P(f2). The following theorem shows that the pointwise convergence of the integrands implies the r -convergence of the corresponding integral functionals in the weak topology of W 1,P(f2). 14. h be the corresponding integrands. e. on f2. Then (Fh) r-converyes to F in the weak topology of W 1,P(f2).

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. ) denotes the scalar product in Rn, vERn\{O},and -oo 0 and k2 2 0 such that C < cp,n' Then there exist two for every u E W~,p(n). The constant kl depends only on c and cp,n, while k2 depends on c, cp,n, and IIcpil Wl,p(n) . Proof. Let us fix u E W~'P(fl). -pc l - p 10 IDcplPdx + P +cpn cl - P flcplPdx , for every c E ]0, 1[.

Proof. If (Uh) converges to a function u weakly in W1,p(n), then, by Rellich's compactness theorem, it converges to u strongly in Lfoc(n). 21). 3. If n is bounded and has a Lipschitz boundary, then G is sequentially lower semicontinuous in the weak topology of W1,p(n) for every Cl E R. 21). 4. In general, the functional G is not lower semicontinuous in the weak topology of W1,p(n). A simple counterexample is given by g(x,s) = -\s\P, which satisfies condition (iv) with Cl = -1. Suppose, by contradiction, that the corresponding functional G is lower semicontinuous in the weak topology of W1,p(n).

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