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

By Gianni Dal Maso (auth.)

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10 to the case of a normed vector space. 11. Let X be a normed vector space, let B R = BR(XO) be an open ball with mdius R > 0 centred at a point Xo EX, and let F: BR ..... R be a convex function. Suppose that sup F(x) = M < +00 and inf F(x) = xEBR xEBR m> -00. Let 0 < r < R and let K = (M - m)/(R - r). 3) IF(x) - F(y)1 ::; Kllx - yll for every x, y in the closure Br of B r . Proof. Let x, y E Br with x =I y, let I all t E R such that the point x(t) = tx = la, b[ be the open interval of + (1 - t)y belongs to B R , and An Introduction to 50 let G: I ~ r -convergence R be the convex function defined by G(t) = F(x(t)).

16. Let (Fh) be a sequence of functions from X into R, and let F' = r-liminf Fh, h-+oo F" = r-limsupFh' h-+oo Then epi(F') = K-limsup epi(Fh) ' h-+oo epi(F") = K-liminf epi(Fh) ' h-+oo where the K-limits are taken in the product topology of X x R. In particular (Fh) T-converges to F in X if and only if (epi(Fh)) K-converges to epi(F) in X x R. Proof. We shall prove only the first equality, the other one being analogous. A point (x,t) E X x R belongs to epi(F') if and only if F'(x) ::; t. By the definition of F', this happens if and only if for every c > 0, and for every U E N(x) we have liminf inf Fh(Y) < t h-+oo yEU + c, and this is equivalent to say that for every c > 0, U E N (x), kEN there exists h ;::: k such that inf Fh(Y) < t + c.

10 the functional H is lower semicontinuous in the strong topology of LP(O). Since H ~ F, we have H ~ sc- F. 4). 12. Let 0, p, /, /** be as in the previous example, and let G: W1,P(O) -+ R be the functional defined by G(u) = k/(x,DU(X))dX. 9) sc-G(u) = kf**(x,DU(X))dX. To prove this fact, let us consider the functional H: LP(O) H(u) ={ -+ R defined by (sc-G)(u) , if u E W1,P(O), +00, otherwise. 24), the lower bound in (c) implies that In Cl IDulPdx ~ (scG)(u) for every u E W1,P(O). 10, the functional H is lower semicontinuous in the strong topology 35 Relaxation of LP(O).

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