By Mariano Giaquinta

The description for this ebook, a number of Integrals within the Calculus of adaptations and Nonlinear Elliptic platforms. (AM-105), should be forthcoming.

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Rado [256]) was able to show the existence of a unique solution in C°'1(f) of the variational problem 15)LPt us quote the attempt of C. Arzela' [15]. 16)See also D. Hilbert [155], B. Levi [201], G. Fubini [100], J. Hadamard [147] among others. 34 MULTIPLE INTEGRALS AND NONLINEAR ELLIPTIC SYSTEMS assuming that n = 2 , N = 1 , F(p) be strictly convex, 0 be strictly convex and 0 satisfy the `three points condition'. These methods were extended to the n-dimensional case by many authors (M. Miranda, G.

K92 ye; v> 0 . 1) is analytic. During the years 1959-60 C. B. A. Ladyzhenskaya, N. N. Ural'tseva, G. Stampacchia stated analogous results for general linear equations with noncontinuous coefficients and even for nonlinear equations, proving that weak solutions (in the sense of Chapter I) to elliptic nonlinear second order equations, N = 1 (under controllable or natural growth conditions) are smooth (see [189][191], [23111283]). This way the regularity problem for one single equation can be considered as solved.

9), where Ai and Bi satisfy the controllable growth conditions. 10) n verify r [Aap] (x, u, Vu)DPDsul + Aau](x, u, Vu)Dsul + Aaxs(x, U, Vu) + 16 n +& Bi(x, u, Vu)] Daq6ldx = 0 for all ¢( H01(1,RN), spt¢Cl). The proof goes on as before, only a few technical complications will appear: we refer to (231]. 5), it can be seen as a fourth order quasilinear system in u. 1' does not hold in the case of natural growth conditions, even if B = 0 and Ai = Aaj(u)D9u), as we have already stated. 2. Let u r H1'2 n L°°(Q, RN) be a weak solution to system (1.

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