Minimal Surfaces with an Elastic Boundary

AbstractLetD:= {γ ∈ C3 ( $$\mathbb{R},{\text{ }}\mathbb{R}$$ 3) ∣ γ (s) = γ(s+1), ∣ $$\dot \gamma $$ ∣ ≡ 1 γ ([0,1]) is simple closed curve}.In this paper we show that there is γ ∈ D which minimizes the functional $$E_{\gamma 0} \left( \gamma \right): = \int_0^1 {\left| {\ddot \gamma \left( s \right) - \ddot \gamma _0 \left( s \right)} \right|^2 } ds + $$ + a(area minimizing surface with boundary γ([0,1])), γ0 ∈ D if a ∈ (0,∞) is suitably chosen.where γ0 ∈ D if a ∈ (0, ∞) is suitably chosen.