On the foundations of calculus of variations

The subject of this paper will be variational problems fF(x, t)dt = min in parameter form with fixed endpoints. The existence of rectifiable minimizing arcs has been proved under exceedingly general conditions. However, as soon as one wants to establish differentiability properties of the solutions one uses the Euler equations and must therefore assume the existence of second partial derivatives of F(x, x). Hence it is not at all clear exactly which differentiability properties of the solutions are due to which properties of F(x, x). The present paper tries to take a first step towards filling this gap. Simple examples show that without continuity or without the strict convexity of the indicatrix of F(x, x) no general statements about the differentiability of the solution will be possible. Also, an example was given(1) to show that even if these two conditions are satisfied, the minimizing curves are not necessarily of class D'. In the example the indicatrix is an ellipse everywhere so that the variation of F(x, x) for fixed x is as smooth as possible. This and the example of the Minkowskian geometry (corresponding to an integrand not depending on the x) suggest investigation of the implications of the Lipschitz condition: