Direct methods in variational field theory
In this short paper, the authors show that the classical Weierstras-Hilbert field theory can be extended. From the fact that for each extremal field it is true that if an extremal is an element of the field, then a minimum is attained in the class of Sobolev functions with the same boundary data as for the extremal and with graphs in the set covered by the field, the authors derive that this result remains valid if one of the extremals is singular. On the other hand, it is shown that if there is a field containing more than one singular extremal, then each of these extremals defines the minimization problem with having no solution in the class of Lipschitz functions with graphs in the set covered by the field. The new results are collected in the first section and the rest of the paper is dedicated to the proofs of the main theorem, what makes it to an easy to read contribution to this field.
scientific article; zbMATH DE number 7596647