## Shape optimization

Shape optimization is a part of the field of optimal control theory. The typical problem is to find the shape which is optimal in that it minimizes a certain cost functional while satisfying given constraints. In many cases, the functional being solved depends on the solution of a given partial differential equation defined on the variable domain.

My first contact with Shape Optimization was during a talk given by Dorin Bucur at my University of Timisoara in November 2010. A presentation of this talk can be found here.

This area of research has known a great development in the last years. I am not an expert in this field, I am a beginner trying to learn more about it. Below are some problems and theorems related to this field.

Some introductory Bibliography for the subject:

1. Henrot Antoine, Michel Pierre, Variation et Optimisation des Formes, Une Analyse Geometrique
2. Bucur Dorin, Buttazzo Giuseppe, Variational Methods in Shape Optimization Problems
3. B. Kawohl, O. Pironneau, L. Tartar, J.-P Zolesio, Optimal Shape Design. Lectures given at a summer school held in Troia, Portugal, 1998
4. Antoine Henrot, Extremum Problems for Eigenvalues of Elliptic Operators
5. J.-P. Zolesio, M.C. Delfour, Shapes and Geometries – Analysis, Differential Calculus, and Optimization
6. Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations
7. Luc Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces
8. Enrico Giusti, Minimal Surfaces and Functions of Bounded Variation
9. Evans, Gariepy, Measure Theory and Fine Properties of Functions

A nice introduction course to shape optimization can be found in the following link: http://www.unilim.fr/pages_perso/noureddine.igbida/   (go to Enseignement / Introduction to Shape Optimiztation)

In the following posts there are some problems concerning convergence of domains of $\Bbb{R}^N$, and in the first post I present some definitions for types of convergence.
Problem 1.
Problem 2.
Problem 3.
Problem 4.
Problem 5.
Problem 6.

I participated in an event held at West University Timisoara this last week(October 31 – November 4 2011):

Intensive Course on Shape Optimization Problems

At this course, the speakers were Giuseppe Buttazzo (University of Pisa, Italy) and Edouard Oudet (University of Grenoble, France). The main organizers were Dorin Bucur (University of Savoie ) and Bogdan Sasu (West University Timisoara). You can find a summary of my notes in the following posts:

Day 1

Day 2

Day 3

Day 4

Day 5

A theorem of great theoretical and numerical importance in the approximation of the perimeter of a set is the Modica-Mortola theorem. In the linked post you can find one proof of this theorem, along with other references and comments on the subject.

Here are some blog posts related to shape optimization problems.

The circle can sustain the largest sandpile

Existence result for the isoperimetric problems

Proof of the Isoperimetric Inequality

Application of the isoperimetric inequality

Proof of the Isoperimetric Inequality 2

Proof of the Isoperimetric Inequality 3

Isoperimetric inequality for polygons

Brunn-Minkowski Inequality

Proof of the Isoperimetric Inequality 4

Proof of the Isoperimetric Inequality 5

The Faber Krahn inequality

The optimal shape for the second eigenvalue of the Laplace operator with Dirichlet boundary conditions

In the following I put the links to a series of blog posts in which I present my master thesis, prepared at Universite de Savoie under the coordination of Dorin Bucur and Edouard Oudet. It concerns the study of the problem of optimal partitions in the sense of finding the partitions which minimize an energy which modelizes the energy of a configuration of immiscible fluids. The study is concentrated on existence results and on some numerical results. You can see the contents of this study in the following linked posts or in the pdf file: Master final paper

Part 1 (Introduction)

Part 2 (Framework of BV functions)

Part 3 (The case of two fluids partition – trace inequalities method)

Part 4 (The case of three fluids partition – trace inequalities method)

Part 5 ($\Gamma$-convergence and the Modica Mortola theorem)

Part 6 (A generalization of the Modica Mortola theorem to multiple phase systems)

Part 7 (The main existence result)

Part 8 (Some numerical results concerning $\Gamma$-convergence applications: the isoperimetric problem, the anisotropic perimeter, partitions of two fluids)

– Relaxation for the Anisotropic Perimeter: Part 1, Part 2

Here are some numerical computations regarding the optimal shapes for the eigenvalues of the Laplace operator with Dirichlet boundary conditions in 2D for the volume constraint and for the perimeter constraint:

Volume Constraint

Perimeter Constraint