Existence Theorems for Double Integral Problems of the Calculus of Variations
E. J. McShane · Transactions of the American Mathematical Society · 1935
For single-integral problems of the calculus of variations there are in the literature a number of existence theorems of considerable generality.Recently Tonelli has established several existence theorems for double integral problems of the form fff(x, y, z, zx, zy)dxdy = mm.But to the best of my knowledge, except for the several discussions of the problem of Plateau the literature contains no proof of any existence theorem for double-integral problems in parametric form, that is, for problems of the form F(S) = fff(x,y,z, X, Y,Z)dudv = min, where the equations x = x{u, v), y = y(u, v), z = z(u, v) represent a surface and X, F, Z are the three jacobians of {x, y, z) with respect to (u, v).The present paper gives the proof of two such theorems, in each of which the integrand function is permitted to be a function of (X, Y, Z) of quite general type, but is required to be independent of the coordinates (x, y, z).The theorems are based on a semi-continuity proof and a convergence theorem.The semi-continuity of quasi-regular functionals F(S) I have already established under conditions of adequate generality.Here I develop the convergence theorem needed.The methods are extensions of those previously used in connection with the problem of Plateau.| 1.Preliminary remarks.The word surface will always be used to mean a continuous surface of the type of the circle, represented by three equations x = x(u, v), y = y(u, v), z = z(u, v), where (u, v) ranges over the interior and boundary of a Jordan region B (i.e., a region bounded by a simple closed curve).In case the six partial derivatives xu, xv, etc., all exist and are finite, we denote the three jacobians of x, y, z with respect to u, v by the symbols X, Y, Z :Let us suppose that f(X, F, Z) is a function positively homogeneous of degree 1 in (X, Y, Z) and continuous together with its first partial derivatives for all (X, Y, Z) ^ (0,0,0).For all arguments (X, F, Z) such that X2 + Y2+Z2 t Presented to the Society, October 28,