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Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals. (English) Zbl 0999.49024
A fundamental result within the framework of geometric measure theory is Allard’s famous small excess regularity theorem for varifolds with controlled first variation. Here, the authors give a new proof an an analogous result for integer multiplicity rectifiable currents of arbitrary dimension and codimension minimizing an elliptic parametric variational integral. Their proof works in the interior case as well as at the boundary. The main difference to most existing proofs in similar situations is the fact that the authors succeed in avoiding any indirect blow-up arguments. In fact, their method not only provides explicit but often optimal estimates. These optimal regularity theorems are new even in the interior case, and it seems they cannot be deduced from the existing literature.

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
49N60 Regularity of solutions in optimal control
49Q05 Minimal surfaces and optimization
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