Regularity of the minimizers in the composite membrane problem in R2R2

We study the regularity of the minimizers to the problem:λ(α,A)=infu∈H01(Ω),‖u‖2=1,|D|=A∫Ω|Du|2+α∫Du2.We prove that in the physical case α<λα<λ in R2R2, any minimizer u   is locally C1,1C1,1 and the boundary of the set {u>c}{u>c} is analytic where c   is the constant such that D={u