Abstract
In this paper we study the free boundary regularity for almost-minimizers of the functionalJ(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 C1,α function (thus extending the results of [4] to almost-minimizers).
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