Another Theorem of Classical Solvability ‘In Small’ for One-Dimensional Variational Problems

Abstract

In this paper we suggest a direct method for studying local minimizers of one-dimensional variational problems which naturally complements the classical local theory. This method allows us both to recover facts of the classical local theory and to resolve a number of problems which were previously unreachable. The basis of these results is a regularity theory (a priori estimates and compactness in C1) for solutions of obstacle problems with sufficiently close obstacles. In these problems we establish that solutions exist and inherit regularity of the obstacles even under assumptions on integrands that are much weaker than those required in the classical local theory.