The Dirichlet problem with a volume constraint

For B the open unit disk in R², let W₁(B) denote the Sobolev space of vector functions x: B→R³ such that x and its first partial derivatives are square integrable. For any y∈W₁(B), S(y) is the set of all x in W₁(B) for which x-y∈W₁₀(B), the closure in W₁(B) of C ₀ ^∞ (B). Assume that for all x ∈ S(y) the area functional A(x)>0. For a given constant K, we show that there is an x_o∈S(y) minimizing the “Dirichlet Integral” $$D(x) = \iint_B {(|x_u |^2 } + |x_v |^2 )dudv$$ in the subset of all x ∈ S(y) for which the oriented volume enclosed by y and x, V(y,x)=K. x_o is analytic on B and is a solution to the differential equation Δx=2H(x_u∧x_v) for some constant H.