## Numerical Results – Optimal Shapes – Dirichlet Eigenvalues – Volume Constraint

The only known exact shapes which minimize the eigenvalues of the Laplace operator with Dirichlet condition are those for and . Nothing is proved for higher eigenvalues, but there are some numerical tests which were performed to find what the optimal shapes look like. Such tests were made first by Edouard Oudet for and recently by P. Freitas, P. Antunes for .

## Master 2

In the following, we propose a mathematical model for the energy of configurations of different immiscible fluids situated in a container , which is thought as an open subset of , . We consider fluids modeled by measurable sets and since we are going to study functionals which deal with the surface tension energies we will need a good framework for studying the perimeters of such sets. For this we consider the space of functions with bounded variation on . Some of the standard references for this subject are the books of Evans, Gariepy [1] and E. Giusti [2]. We will state below some of the results we will be using in the sequel, whose proofs can be found in the given references.

## Intro to Shape Optimization

**Shape Optimization Introductory Course**

This course was given by Dorin Bucur in November 2010 in Timisoara, and was my first contact with the field of shape optimization. As an outsider, at that time, I was really impressed by this course and by the problems presented here. The course does not go into deep details, and it is aimed to be a presentation of the field.

**1. The Dido Problem.** This is an ancient problem. The hystorical sources say that queen Dido was allowed to have as much land as can be encompassed by an oxhide. Being very smart, she cut the oxhide into a very thin and long strip and encompassed a semicircle having the sea as the straight boundary.

Therefore, the problem was to maximize the area enclosed by a fixed perimeter, (a similar problem is to minimize the perimeter which encloses a fixed area).

## Shortest Fence which blocks the vision

The owner of a square shaped land was an unpleasant person. He didn’t mind people seeing inside his property, but he didn’t want that his neighbors would be able to see across his property. To put his plan into action he built fences on three edges of his square shaped property. His goal was achieved, but of course the solution is not optimal, since the man is not the smartest person around. What is the optimal shape of the fence which blocks the vision in the sense that it costs the least (i.e. it has the shortest length)? Does such as optimal fence exist?

A simpler question? What is the optimal fence in case the property has a triangular shape?

One variant would be to build the fence on the two diagonals, which yields a length of . This is smaller than which is the length of the fence the owner built, but it is not optimal. A better configuration is depicted in the picture. Still there is a better configuration.

## Minkowski content and the Isoperimetric Inequality

The proof of the isoperimetric inequality given below relies on the Brunn-Minkowski inequality and on the concept of Minkowski content.

In what follows we say that a curve parametrized by is simple if is injective. It is a closed simple curve if and is injective on . We say that a curve is quasi-simple if it the mapping is injective with perhaps finitely many exceptions.

## Existence of Maximal Area Polygon with given sides

Suppose are given positive real numbers such that there exists a polygon with vertices having as lengths of its size. Then among all polygons having these given side lengths there exists one which has maximal area.

Since I have used in the linked post the fact that the maximal area polygon with given sides exists, I will give a proof in the following.

## Proof of the Isoperimetric Inequality 3

I will present here a third proof for the planar Isoperimetric Inequality, using some simple notions of differential curves. For this suppose that the simple closed plane curve has length and encloses area . Then

and equality holds if and only if is a circle.