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The well-posedness of a phase-field approximation to the Willmore flow witharea and volume constraints is established when the functional approximatingthe area has no critical point satisfying the two constraints. The existenceproof relies on the underlying gradient flow structure of the problem: the timediscrete approximation is solved by a variational minimization principle. Themain difficulty stems from the nonlinearity of the area constraint.

A phase-field approximation of the Willmore flow with volume and area constraints