The tangential variation of a localized flux-type eigenvalue problem

In this work the differentiability of the principal eigenvalue λ=λ1(Γ) to the localized Steklov problem −Δu+qu=0 in Ω, on ∂Ω, where Γ⊂∂Ω is a smooth subdomain of ∂Ω and χΓ is its characteristic function relative to ∂Ω, is shown. As a key point, the flux subdomain Γ is regarded here as the variable with respect to which such differentiation is performed. An explicit formula for the derivative of λ1(Γ) with respect to Γ is obtained. The lack of regularity up to the boundary of the first derivative of the principal eigenfunctions is a further intrinsic feature of the problem. Therefore, the whole analysis must be done in the weak sense of H1(Ω). The study is of interest in mathematical models in morphogenesis.