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

In this paper, we are interested in shape optimization problems involving the ge ometry (normal, curvatures) of the surfaces. We consider a class of hypersurface s in $\mathbb{R}^{n}$ satisfying a uniform ball condition and we prove the exist ence of a $C^{1,1}$-regular minimizer for general geometric functionals and cons traints involving the first- and second-order properties of surfaces, such as in $\mathbb{R}^{3}$ problems of the form: $$\inf \int_{\partial \Omega} j_0 [ \mathbf{x},\mathbf{n}(\mathbf{x}) ] dA (\mathbf{x}) + \int_{\partial \Omega} j_1 [ \mathbf{x},\mathbf{n}(\mathbf{x}),H(\mathbf{x}) ] dA (\mathbf{x}) + \int_{\partial \Omega} j_2 [\mathbf{x},\mathbf{n}(\mathbf{x}),K(\mathbf{x})] dA (\mathbf{x}),$$ where $\mathbf{n}$, $H$, and $K$ respectively denotes the normal, the scalar mea n curvature and the Gaussian curvature. We gives some various applications in th e modelling of red blood cells such as the Canham-Helfrich energy and the Willmo re functional

On the minimization of total mean curvature

International audienceIn this paper we are interested in possible extensions of an inequality due to Minkowski: $\int_{\partial\Omega} H\,dA \geq \sqrt{4\pi A(\partial\Omega)}$ valid for any regular open set $\Omega\subset\mathbb{R}^3$, where $H$ denotes the scalar mean curvature and $A$ the area. We prove that this inequality holds true for axisymmetric domains which are convex in the direction orthogonal to the axis of symmetry. We also show that this inequality cannot be true in more general situations. However we prove that $\int_{\partial\Omega} |H|\,dA \geq \sqrt{4\pi A(\partial\Omega)}$ remains true for any axisymmetric domain

Uniform ball condition and existence of optimal shapes for geometric functionals involving boundary-value problems

In this article, we are interested in shape optimization problems where the functionals are defined on the boundary of the domain, involving the geometry of the associated surface and the boundary values of the solution to a state equation posed on the inner domain enclosed by the shape. Hence, we pursue here the study initiated in a previous work by considering a specific class admissible shapes. Given $\varepsilon > 0$ and a fixed non-empty large bounded open hold-all $B \subset \mathbb{R}^{n}$, $n \geqslant 2$, we define $\mathcal{O}_{\varepsilon}(B)$ as the class of open sets $\Omega \subset B$ satisfying a $\varepsilon$-ball condition, which has an equivalent characterization in terms of uniform $C^{1,1}$-regularity of the boundary $\partial \Omega$. The main contribution of this paper is to prove the existence of a minimizer in the class $\mathcal{O}_{\varepsilon}(B)$ for problems of the form:$$\inf_{\Omega \in \mathcal{O}_{\varepsilon}(B)} \int_{\partial \Omega} j \left[ u_{\Omega} \left( \mathbf{x} \right), \nabla u_{\Omega} \left( \mathbf{x} \right), \mathbf{x}, \mathbf{n} \left( \mathbf{x} \right), H \left(\mathbf{x} \right) \right] dA \left( \mathbf{x} \right) ,$$ where $u_{\Omega}$ denotes to the solution of the Dirichlet Laplacian posed on the domain $\Omega$ or to the one associated with a Neumann or Robin boundary condition, where $\mathbf{n}$ is the unit outward normal vector, and where $H$ can refer either to the the scalar mean curvature, to the Gaussian curvature, or more generally to any of the symmetric polynomials in the principal curvatures. We only assume here the continuity of $j$ with respect to the set of variables, convexity with respect to the last variable, and quadratic growth regarding the first two variables. We give various applications in the field of partial differential equations such as existence for:$$\inf_{\Omega \in \mathcal{O}_{\varepsilon}(B)} \int_{\Omega} j \left[ \mathbf{x}, u_{\Omega} \left( \mathbf{x} \right) , \nabla u_{\Omega} \left( \mathbf{x} \right) , \mathrm{Hess}~u_{\Omega} \left( \mathbf{x} \right) \right] dV \left( \mathbf{x} \right),$$ and boundary shape identifications in the area of inverse and control problems:\[ \inf_{\substack{\Omega \in \mathcal{O}_{\varepsilon}(B) \\ \Gamma_{0} \subseteq \partial \Omega}} \int_{\Gamma_{0}} \left[ \left( \partial_{n}u_{\Omega} - f_{0} \right)^{2} + \left( u_{\Omega} - g_{0} \right)^{2} \right] dA. \

Optimal Shape of an Underwater Moving Bottom Generating Surface Waves Ruled by a Forced Korteweg-de Vries Equation

© 2018, Springer Science+Business Media, LLC, part of Springer Nature.It is well known since Wu and Wu (in: Proceedings of the 14th symposium on naval hydrodynamics, National Academy Press, Washington, pp 103–125, 1982) that a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, continuously and periodically, a succession of solitary waves propagating ahead of the disturbance in procession. One possible new application of this phenomenon could very well be surfing competitions, where in a controlled environment, such as a pool, waves can be generated with the use of a translating bottom. In this paper, we use the forced Korteweg–de Vries equation to investigate the shape of the moving body capable of generating the highest first upstream-progressing solitary wave. To do so, we study the following optimization problem: maximizing the total energy of the system over the set of non-negative square-integrable bottoms, with uniformly bounded norm

Optimal shape of an underwater moving bottom generating surface waves ruled by a forced Korteweg-de Vries equation

It is well known since Wu & Wu (1982) that a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, continuously and periodically, a succession of solitary waves propagating ahead of the disturbance in procession. One possible new application of this phenomenon could very well be surfing competitions, where in a controlled environment, such as a pool, waves can be generated with the use of a translating bottom. In this paper, we use the forced Korteweg-de Vries equation to investigate the shape of the moving body capable of generating the highest first upstream-progressing solitary wave. To do so, we study the following optimization problem: maximizing the total energy of the system over the set of non-negative square-integrable bottoms, with uniformly bounded norms and compact supports. We establish analytically the existence of a maximizer saturating the norm constraint, derive the gradient of the functional, and then implement numerically an optimization algorithm yielding the desired optimal shape