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

Pierluigi Colli (1) , Philippe Laurencot (2)

1 Dipartimento di matematica F. Casorati
2 IMT - Institut de Mathématiques de Toulouse UMR5219

The well-posedness of a phase-field approximation to the Willmore flow with area and volume constraints is established when the functional approximating the area has no critical point satisfying the two constraints. The existence proof relies on the underlying gradient flow structure of the problem: the time discrete approximation is solved by a variational minimization principle. The main difficulty stems from the nonlinearity of the area constraint.

Mots clés: phase-field approximation; gradient flow; minimization principle; well-posedness

Domaines: Equations aux dérivées partielles [math.AP]

Pierluigi Colli, Philippe Laurencot. A phase-field approximation of the Willmore flow with volume and area constraints. SIAM Journal on Mathematical Analysis, 2012, 44 (6), pp.3734--3754. ⟨10.1137/120874126⟩. ⟨hal-00688953⟩