Free boundary regularity for almost-minimizers

Guy David, Max Engelstein, Tatiana Toro

Research output: Contribution to journalArticle

2 Scopus citations


In this paper we study the free boundary regularity for almost-minimizers of the functional J(u)=∫Ω|∇u(x)| 2 +q + 2 (x)χ {u>0} (x)+q 2 (x)χ {u<0} (x)dx where q ± ∈L (Ω). Almost-minimizers satisfy a variational inequality but not a PDE or a monotonicity formula the way minimizers do (see [4], [5], [9], [37]). Nevertheless, using a novel argument which brings together tools from potential theory and geometric measure theory, we succeed in proving that, under a non-degeneracy assumption on q ± , the free boundary is uniformly rectifiable. Furthermore, when q ≡0, and q + is Hölder continuous we show that the free boundary is almost-everywhere given as the graph of a C 1,α function (thus extending the results of [4] to almost-minimizers).

Original languageEnglish (US)
Pages (from-to)1109-1192
Number of pages84
JournalAdvances in Mathematics
StatePublished - Jul 9 2019
Externally publishedYes


  • Almost-minimizer
  • Free boundary problem
  • Uniform rectifiability

Fingerprint Dive into the research topics of 'Free boundary regularity for almost-minimizers'. Together they form a unique fingerprint.

  • Cite this