By Prof. Dr. Ludwig von Bertalanffy (auth.)

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**Example text**

E. 47). Proof. We shall prove only the first statement; the others are proved similarly. , fp^pi^ix, z, p)} dt formulas. It is clear that if Zn -^2: in -^fz W strongly mLr^,r = kl{k~ \), and fzz[Zn] -^fzz[^] strongly in Ls, - ^ 0 strongly in Ls. Thus we see that i{z + 0= l(z) + ii(z,C) + Ihiz, 0 + R(z, C) 1 To see this, let {q} be any subsequence of {n}. Then there is a subsequence {s} of {q} such that Zs{x) -> z{x) and V ^s{^) -> V ^(^) almost everywhere so t h a t / p M - > / 3 ? M for almost all x.

As has already been said, H I L B E R T [1] and L E B E S G U E [2] h a d solved the Dirichlet problem b y essentially direct methods. These methods were exploited and popularized b y TONELLI in a series of papers and a book ([1], [2], [3], [4], [5], [7], [8]), and have been and still are being used b y many others. e. a sequence {zn] of admissible functions for which I[zny G) tends to its infimum in t h e class) which converges in the sense required to some admissible function. Tonelli applied these methods t o many single integral problems and some double integral problems.

Thus the family i^* defines, so to speak, the class of boundary values being allowed. Of course, F * could consist of a single function ^*. Proof of the theorem. Let {zn} be a minimizing sequence; we may assume t h a t I{zny G)