Caratheodory theory for the Bernoulli problem

The variational formulation of the interior Bernoulli free boundary problem is considered. The problem is formulated as follows. Choose an arbitrary bounded simply connected domain G ⊂ ℝ² and smooth positive functions g : ∂G → ℝ, Q : G → ℝ. Denote by 𝒞 the totality of all connected compact sets ω ⊂ ., such that the flow domain Ω = G \ ω is double-connected. The notation 𝒞 ⁺ ⊂𝒞 stands for the totality of the set ω ∈𝒞 of positive measure. The cost function 𝒥 (ω) is defined by the equalities 𝒥(ω)=∫Ω(| ∇u |2+Q2)dx,Δu=0 in Ω, u=g on ∂G, u=0 on ∂ω. \matrix{{\mathcal{J}\left( \omega \right) = \int_\Omega {\left( {{{\left| {\nabla u} \right|}^2} + {Q^2}} \right)dx,} } \hfill \cr {\Delta u = 0\,\,{\rm{in}}\,\,\Omega ,\,\,u = g\,\,{\rm{on}}\,\,\partial G,\,\,u = 0\,\,{\rm{on}}\,\,\,\partial \omega .} \hfill \cr } We prove that, under the natural nondegeneracy assumption, the variational problem min minω∈𝒞+ 𝒥(ω) \mathop {\min }\limits_{\omega \in {\mathcal{C}^ + }} \,\,\mathcal{J}\left( \omega \right) has a solution ω ∈ 𝒞 ⁺. The approach is based on the methods of complex variables theory and the potential theory. The key observation is that every subset of 𝒞, separated from ∂G is sequentially compact with respect to the Caratheodory-Hausdor convergence.