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We consider nonnegative minimizers of the functional Jp(u;Ω)=∫ Ω |∇u| p +λ p p Χ {u>0 }, 1 0} ∩Ω has no singularities and is a real analytic hypersurface if p = 2 and n 0 such that the same result holds if p - 2| < ∈ 0 .

On the full regularity of the free boundary in a class of variational problems