Geometric analysis of optimal shapes

GAOS is a research project financed by Agence Nationale de la Recherche, 2009-2012.
LAMA, Unité Mixte de Recherche 5127 CNRS
Université de Savoie Bâtiment Chablais,
Campus Scientifique 73376 Le Bourget-du-Lac Cedex, France

Phone (+33) 4 79 75 86 29
Fax (+33) 4 79 75 81 42
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The question of optimizing shapes has applications in several domains like acoustics, quantum mechanics, visualization, solid or fluid mechanics and bio-mathematics. Those questions have also a specific mathematical interest since they melt geometrical questions with analysis of partial differential equations and calculus of variations. Despite (or because) their false simplicity, several problems are still open, although formulated hundreds year ago (see for instance the recent books by Bucur and Buttazzo [BB05], Henrot [He06], Henrot and Pierre [HP05]) and give rise to important debates within the international scientific community. Among them, we can cite
  • optimal shapes in structural mechanics,
  • isoperimetric problems,
  • Newton's problem of minimal resistance,
  • optimization of the eigenvalues of differential operators, ...
Each of these problems is associated to one or several constraints for the geometries, which may be local or non local (convexity, connectedness, ...). On the one hand the standard way of dealing with these problems is to search explicit solutions by means of direct estimates, symmetrizations, rearrangements. On the other hand recent techniques of variational type have been developed to prove the existence of an optimal shape and intensive research is carried out to prove the regularity of the free boundaries and to analyse the optimality conditions. Understating these points is crucial for developing new efficient numerical methods to approximate optimal shapes. It is around these last points that our project is oriented.
The main scientific challenge is the qualitative study of optimal shapes for some classes of shape functionals involving partial differential equations and/or geometric quantities associated naturally with shapes: measure, perimeter, curvature. We are particularly interested to handle non smooth and singular shapes (with or without constraints), to understand the minimal regularity under which one can extract information from the optimality conditions, to prove this regularity and finally to obtain qualitative information about the optimal shape.
Precisely, in our project we will concentrate on the following topics:
  • Regularity of optimal shapes,
  • The class of convex bodies,
  • Numerical aspects of singular shapes,
  • Geometric inequalities,
  • Nodal domains and spectral minimal partitions.
Active research is done both to understand analytical aspects and to develop efficient numerical algorithms. Since the first existence results due to D. Chenais in the seventies [Ch75], there was a major step forward in the study of both the existence question and the much more difficult question of regularity of optimal shapes (free boundaries). Alt and Caffarelli [AC81] were the first to prove in 1981 that for a class of energy minimizing free boundary problems, the solution has smoothness properties in relationship with the regularity of minimal surfaces. Concerning Dirichlet boundary conditions, general existence of solution was proved by Buttazzo and Dal Maso in 1993 [BD93], who exhibited a class of shape functionals for which the relaxation does not occur, and for which an optimal shape (a quasi-open set) is a minimizer. Finally, the most surprising result was due to Sverak in 1993 [Sv93] who proved in two space dimensions that a topological constraint, on the number of holes, is crucial for the existence. This last result opened the way to the intensive use of potential theory and of fine regularity properties of solutions of partial differential equations on non-smooth domains, in the study of shape optimisation problems. Homogeneous Neumann conditions on the free part of the boundary are studied in a different framework, which is not directly related to our project (see for instance the book of Allaire [Al02]).
Concerning numerical methods, together with the classical descent algorithms by shape gradient and the relaxation/homogenization methods, several new tools were developed in the last decade for the computation of optimal shapes, among which we refer to fictitious materials, phase field, global stochastic optimisation by using genetic algorithms, the bubble method, or the topological gradient, the level set method, material distribution optimisation where the mass is allowed to concentrate on lower dimensional structures, etc. (see for instance [Al02, HP05, SZ92]).
Our project is intended to take a new direction, precisely to make an exhaustive qualitative analysis of optimal shapes associated with Dirichlet boundary conditions in the variational framework. We particularly concentrate on the study of non-smooth and singular shapes in relationship with geometric and/or topological constraints.

[Al02] Allaire, Grégoire Shape optimization by the homogenization method. Applied Mathematical Sciences, 146. Springer-Verlag, New York, 2002.
[AC81] Alt, H. W.; Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325 (1981), 105-144.
[BB05] Bucur, D., Buttazzo, G. Variational methods in shape optimisation problems. Progress in Nonlinear Differential Equations and their Applications, 65. Birkhauser Boston, Inc., Boston, MA, 2005.
[BD93] Buttazzo, Giuseppe; Dal Maso, Gianni An existence result for a class of shape optimization problems. Arch. Rational Mech. Anal. 122 (1993), no. 2, 183-195.
[Ch75] Chenais, Denise On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975), no. 2, 189-219.
[He06] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Birkhauser Verlag, Basel, 2006.
[HP05] A. Henrot, M. Pierre, Variation et optimisation de formes, Mathématiques et Applications 48, Springer, 2005.
[SZ92] Sokolowski, Jan; Zolésio, Jean-Paul Introduction to shape optimization. Shape sensitivity analysis. Springer Series in Computational Mathematics, 16. Springer-Verlag, Berlin, 1992.
[Sv93] Sverak, V. On optimal shape design. J. Math. Pures Appl. (9) 72 (1993), no. 6, 537--551.

Last update 24/07/2012.