By Augusto Visintin (auth.)

"The booklet is easily prepared, concise and obviously written with a strict interaction among physics and mathematics... mostly self-contained ... hugely urged to all graduate scholars and reserachers in utilized mathematics."

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**Example text**

M, if (n - l)k < t :S nk, for n = 1, ... 28) then read oWm um, f~n _ .. e. in ]0, T[, -",- +Au m = 1m IIWm I UX'(O,T;L2([2» similarly. 29) :S Constant (independent of m). 3 I) we get (\. TY') :S Constant. (iii) Limit Procedure. By the preceding estimates and by classical compactness results, cf. l, there exist'Ll, w such that, possibly taking m -+ 00 along a subsequence, weakly in L 2 (O, T; V), weakly star in L oo (0, T; L\[2)) n HI(O, T; V'). 35) (8) This sort of notation is uscd systematically in the following, to transform tlnite difference schemes into approximate continuous equations.

Analogies and Differences. Generally speaking, for equations continuity properties are crucial for existence of a solution, and so is strict monotonicity for uniqueness. On the other hand, lower semicontinuity properties playa key role for existence of a solution of minimization problems, and so does strict convexity for uniqueness. The solutions of variational problems of the forms (v) and (vi) also fulfill the corresponding Euler equations, which, respectively, are of the classes (iii) and (iv).

32) for the time discretization of this problem. J(n-I)k f'nk . H Int. e. In discretized equation by u~ - g~ (E V). n ,an d H, 39 . 1. In particular, further a priori estimates are derived multiplying the approximate equation by the time derivative of the approximate solution. Some Generalizations. 2 we used the compactness of the injection V --t £2(n), cf. 36), this property is not really needed. 10) by means of a different technique, which is based on the use of the structure of the equation in the limit procedure.