By Mariano Giaquinta
The description for this booklet, a number of Integrals within the Calculus of diversifications and Nonlinear Elliptic structures. (AM-105), could be forthcoming.
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These methods were extended to the n-dimensional case by many authors (M. Miranda, G. Stampacchia, P. Hartman, D. Gilbarg, among others). Analogous methods were also developed by S. Bernstein mainly in the spirit of studying second order partial differential equations (Euler equations) and exploited deeply by J. Serrin, N. S. Trudinger, Bakel'man, etc. (see [17] [18] [129] [268]). We must also recall that direct methods have been used for the (parametric) Plateau problem by J. Douglas, R. Courant, E.
3) play an important role for example in the linear (and maybe nonlinear) elasticity theory. I. 9. Suppose that L and M are sequentially continuous from X with the weak topology in respectively Lq(fl, Rk) with the strong topology and in Lq(ft, Rm) with the weak topology. c. with respect to the weak topology of X. 10. Suppose that L be sequentially continuous from X with the weak topology in Lq(f1, Rk) with the strong topology and that M be linear and continuous from X with the strong topology in Lq(fl, Rm), q > s, with the strong topology.
I. SEMICONTINUITY, EXISTENCE AND DIFFERENTIABILITY 29 We would like to remark that for m = 1 we cannot carry on the above RN) are not weakly compact. argument, because bounded sets of This is the case for the area problem f/i + IQu dx -. min uon 4 . In this case we can estimate uniformly the HI-I norm of a minimizing sequence but we cannot deduce (in fact it is not true) that any subsequence converges weakly in H1,1 Now we want to state a theorem of existence which is general enough. 1. Let IuhIC H"(0, RN).