JournalsifbVol. 20, No. 2pp. 211–260

Uniform ball property and existence of optimal shapes for a wide class of geometric functionals

  • Jérémy Dalphin

    Université de Lorraine, Vandœuvre-Lès-Nancy, France
Uniform ball property and existence of optimal shapes for a wide class of geometric functionals cover
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Abstract

In this article, we study shape optimization problems involving the geometry of surfaces (normal vector, principal curvatures). Given ε>0\varepsilon > 0 and a fixed non-empty large bounded open hold-all BRnB \subset \mathbb{R}^{n}, n2n \geqslant 2, we consider a specific class Oε(B)\mathcal{O}_{\varepsilon}(B) of open sets ΩB\Omega \subset B satisfying a uniform ε\varepsilon-ball condition. First, we recall that this geometrical property ΩOε(B)\Omega \in \mathcal{O}_{\varepsilon}(B) can be equivalently characterized in terms of C1,1C^{1,1}-regularity of the boundary Ω\partial \Omega \neq \emptyset, and thus also in terms of positive reach and oriented distance function. Then, the main contribution of this paper is to prove the existence of a C1,1C^{1,1}-regular minimizer among ΩOε(B)\Omega \in \mathcal{O}_{\varepsilon}(B) for a general range of geometric functionals and constraints defined on the boundary Ω\partial \Omega, involving the first- and second-order properties of surfaces, such as problems of the form:

infΩOε(B)Ω(j0[x,n(x)] + j1[x,n(x),H(x)] + j2[x,n(x),K(x)])dA(x),\inf_{\Omega \in \mathcal{O}_{\varepsilon}(B)} \int_{\partial \Omega} \left( \begin{matrix} \\ \\ \end{matrix} j_{0} \left[ \mathbf{x},\mathbf{n}\left(\mathbf{x}\right) \right] ~+~ j_{1} \left[ \mathbf{x},\mathbf{n}\left(\mathbf{x}\right),H\left( \mathbf{x} \right)\right] ~+~ j_{2}\left[\mathbf{x},\mathbf{n}\left(\mathbf{x}\right),K\left(\mathbf{x}\right)\right] \begin{matrix} \\ \\ \end{matrix} \right) dA \left( \mathbf{x}\right),

where n\mathbf{n}, HH, KK respectively denote the unit outward normal vector, the scalar mean curvature and the Gaussian curvature. We only assume continuity of j0,j1,j2j_{0},j_{1},j_{2} with respect to the set of variables and convexity of j1,j2j_{1},j_{2} with respect to the last variable, but no growth condition on j1,j2j_{1},j_{2} are imposed here regarding the last variable. Finally, we give various applications in the modelling of red blood cells such as the Canham-Helfrich energy and the Willmore functional.

Cite this article

Jérémy Dalphin, Uniform ball property and existence of optimal shapes for a wide class of geometric functionals. Interfaces Free Bound. 20 (2018), no. 2, pp. 211–260

DOI 10.4171/IFB/401