Parametrizations of saddle surfaces, with application to the problem of plateau

Introduction. In the study of the properties of rectifiable curves x =x(t), y=y(t), z=z(t), and of integrals of the calculus of variations taken along such curves, the investigator is greatly aided by the fact that the absolute continuity of x(t), y(t) and z(t) is known to be necessary and sufficient in order that the length of the curve be equal to the classical integral fx'2 +y'2?z'21'2dt, and also by the existence of a parametric representation of the curve (in terms of length of arc) in which the defining functions are Lipschitzian. On the other hand, let us suppose that the continuous surface S, represented by the equations x=x(u, v), y=y(u, v), z=z(u, v), has finite area in the sense of Lebesgue. We know no conditions necessary and sufficient to insure that the area of S be equal to ff(EG F2)1/2 dudv, nor can we in general find a parametric representation of S which enjoys any particularly desirable properties. However, in a previous paper? I have found certain conditions on the functions x(u, v), y(u, v), z(u, v) which are sufficient to insure that the area be given by the classical double integral; and I have shown that on the class of all surfaces satisfying these conditions the double integrals of the kind usually considered in the calculus of variations have the property of semi-continuity. It is therefore desirable to show that large classes of surfaces can be given representations satisfying the above mentioned conditions. Certainly it is not true that all continuous surfaces can be given such representations. However, let us restrict our attention to the class of surfaces for which the defining functions x(u, zv), etc., are monotonic in the sense of Lebesgue (including in particular the important class of saddle surfacesll). In the present paper it is shown that for every such surface a representation can be found which satisfies the conditions mentioned, and is in fact almost as advantageous as a Lipschitzian representation would be.