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It remains to show the existence of u, which we present in several steps. First suppose that a(u, v) is symmetric, and define the functional Let d = infiK I(u). Since we see that Letunbeaminimizingsequenceof/in IK such that Applying the parallelogram law, and keeping in mind that IK is convex, w see that We have used Hence the sequence {«„} is Cauchy and the closed set IK contains an element usuchthatu n-u in H and /(«„) -*• /(u). So /(M) = d. Now for any u e IK, u + e(t; — M) e IK, 0 < e < 1, and I(u + e(v — u)) > /(M).

EXERCISES 1. 6. AmappingFfromUNintotRN)'is calledcyclicallymonotone if one has for any set of points (x0,x . . ,xn}(n arbitrary). Show that F(x) = grad /(x) is cyclically monotone if/(x)saC1convex function. 3. 1. (ii) Let F be a continuous mapping of a closed ball "L <=UNintoitself. EXERCISES 19 Assume that the vectorF(x)never has the same direction x forx e dl. Then thereexistsapointx 0ofIwhereF(x0)= x0. 5. we may choosey= F(x therefore F(x0)= x 0 . ) 4. State and solve the complementarity problem when F is a continuous mappingromUNnto(KN)'.

But it is a known variant of Tietze's extension theorem that ueH1'°°(Q) admits an extension to u e HO X(UN)for any bounded open domain Q c UN (cf. the book (Stein [1]), for example). Hence H 1>ao (fl) is dense in //1>S(Q), 1 < s < oo, for any bounded open domain Q. Suppose once again that <3Q is Lipschitz. e. e. in Q0. According to Poincare's inequality, there is a /? 1) for #o(Q). We shall always understand that For 1 < s < oo, Hm's(£l) and //o's(Q) are reflexive Banach spaces and // (Q) and //S(Q) are Hilbert spaces.