# Regularity for almost minimizers with free boundary

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In recent work with Guy David we introduce the notion of almost

minimizer for a series of functionals previously studied by Alt-Caffarelli

and Alt-Caffarelli-Friedman.

We prove regularity results for these almost minimizers and explore the

structure of the corresponding free boundary. A key ingredient in the study

of the 2-phase problem is the existence of almost monotone quantities. The

goal of this talk is to present these results in a self-contained manner,

emphasizing both the similarities and differences between minimizers and

almost minimizers.