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The Dirichlet problem with a volume constraint

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Abstract

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 xo∈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. xo is analytic on B and is a solution to the differential equation Δx=2H(xu∧xv) for some constant H.

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This work was done at the Sonderforschungsbereich Theoretische Mathematik at the University of Bonn while the author was a visiting member.

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Wente, H.C. The Dirichlet problem with a volume constraint. Manuscripta Math 11, 141–157 (1974). https://doi.org/10.1007/BF01184954

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  • DOI: https://doi.org/10.1007/BF01184954

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