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.
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