Abstract
A “Hodge” theorem is proved for a non-linear system of equations which are not uniformly elliptic. The solutions are p-forms which minimize a non-regular energy functional over cohomology classes. The theorem is proved by regularizing the functional and proving weak L2 existence. To obtain regularity, we first show that a scalar function of the solution is a subsolution of an elliptic equation from which it follows that the solution is bounded. Hölder continuity is then proved by comparison with a solution of a simpler system known to be Hölder continuous. The solution of the non-regular problem is then obtained from the Shiffman regularization.
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Research supported in part by NSF grant MCS81-03403 and by Sonderforschungsbereich 72, University of Bonn.
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Sibner, L.M. An existence theorem for a non-regular variational problem. Manuscripta Math 43, 45–72 (1983). https://doi.org/10.1007/BF01169096
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DOI: https://doi.org/10.1007/BF01169096