An Existence Theorem in the Calculus of Variations Based on Sobolev's Imbedding Theorems

where l is an arbitrary positive integer, where x tin) denotes the vector x tin) = (x~ ml . . . . . x ~ ) of the derivatives of x of order m taken in an arbitrary but fixed order, and where f2 is a bounded open domain in the n-dimensional Euclidean space E" of points t = ( t x . . . . , t,). In certain basic aspects (e.g. in the use of convexity considerations and of the reflexitivity of the Sobolev spaces) the method of the present paper is the same as the one used in a recent paper by F. E. BROWOER [3] x, while in other aspects the treatment is different. This will be clear from the following outline of the existence proof given in the present paper: The Sobolev space W~=W~(f2) is a reflexive Banach space. Therefore, the closed ball B R c WJ with radius R and center 0 is weakly compact, i.e. compact in the relative topology of BR induced by the weak topology of WJ. Consequently for the proof of the existence of an x o e B R minimizing I ( x ) in BR it will be sufficient to show that l ( x ) is weakly lower semi-continuous (see [10]). To do this, we use the notation (1.2) f (x ; y) = f (t, x (t) . . . . , x ( t 1), y(t)),