Uniform ball property and existence of optimal shapes for a wide class of geometric functionals
Résumé
In this paper, we are interested in shape optimization problems involving the geometry (normal, curvatures) of the surfaces. We consider a class of hypersurfaces in R n satisfying a uniform ball condition and we prove the existence of a C 1,1-regular minimizer for general geometric functionals and constraints involving the rst-and second-order properties of surfaces, such as in R 3 problems of the form: inf ∂Ω j0 [x, n (x)] dA (x) + ∂Ω j1 [x, n (x) , H (x)] dA (x) + ∂Ω j2 [x, n (x) , K (x)] dA (x) , where n, H, and K respectively denotes the normal, the scalar mean curvature and the Gaussian curvature. We gives some various applications in the modelling of red blood cells such as the Canham-Helfrich energy and the Willmore functional.
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