Regularity of the minimizers in the composite membrane problem in R2

AbstractWe 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<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