Calculus of Variations and Geometric Measure Theory

C. De Lellis - G. De Philippis - B. Kirchheim - R. Tione

Geometric measure theory and differential inclusions

created by dephilipp on 01 Oct 2019
modified by tione on 20 Dec 2022


Published Paper

Inserted: 1 oct 2019
Last Updated: 20 dec 2022

Journal: Annales de la Faculte' des Sciences de Toulouse
Year: 2019

ArXiv: 1910.00335 PDF


In this paper we consider Lipschitz graphs of functions which are stationary points of strictly polyconvex energies. Such graphs can be thought as integral currents, resp. varifolds, which are stationary for some elliptic integrands. The regularity theory for the latter is a widely open problem, in particular no counterpart of the classical Allard's theorem is known. We address the issue from the point of view of differential inclusions and we show that the relevant ones do not contain the class of laminates which are used in \cite{SMVS} and \cite{LSP} to construct nonregular solutions. Our result is thus an indication that an Allard's type result might be valid for general elliptic integrands. We conclude the paper by listing a series of open questions concerning the regularity of stationary points for elliptic integrands.