Regularity of the minimizers in the composite membrane problem in R2 ✩

We study the regularity of the minimizers to the problem: λ(α, A) = inf u∈H1 0 (Ω), u 2=1, |D|=A Ω |Du|2 +α D u2. We prove that in the physical case α <λ in R2, any minimizer u is locally C1,1 and the boundary of the set {u > c} is analytic where c is the constant such that D = {u < c} (up to a zero measure set). © 2008 Elsevier Inc. All rights reserved.