## Numerical method – minimizing eigenvalues on polygons

I will present here an algorithm to find numerically the polygon with sides which minimizes the -th eigenvalue of the Dirichlet Laplacian with a volume constraint.

The first question is: how do we calculate the eigenvalues of a polygon? I adapted a variant of the method of fundamental solutions (see the general 2D case here) for the polygonal case. The method of fundamental solutions consists in finding a function which already satisfies the equation on the whole plane, and see that it is zero on the boundary of the desired shape. We choose the fundamental solutions as being the radial functions where are some well chosen source points and . We search our solution as a linear combination of the functions , so we will have to solve a system of the form

in order to find the desired eigenfunction. Since we cannot solve numerically this system for every we choose a discretization on the boundary of and we arrive at a system of equations like:

and this system has a nontrivial solution if and only if the matrix is singular. The values for which is singular are exactly the square roots of the eigenvalues of our domain .

## Eigenvalues – from finite dimension to infinite dimension

We can look at a square matrix and see it as a table of numbers. In this case, matrices and below are completely different:

If instead we look at a square matrix as at a linear transformation things change a lot. Since the transformation is arbitrary, it seems normal that does not act in every direction in the same way, and some directions are privileged in the sense that the transformation is a simple dilatation in those special directions, i.e. there exists and a non-zero vector (the direction) such that . The values and the corresponding vectors are so important for the matrix that they almost characterize it; hence their names are *eigenvalue* and *eigenvector* which means own value and own vector (eigen = own in German). It turns out that and above both have the same eigenvalues , and because they are distinct, both the matrices are similar to the diagonal matrix ( and are similar if there exists invertible such that ).

## Agreg 2012 Analysis Part 2

**Part 2. Some elements of Spectral Analysis**

In this part we prove that the spectrum of a bounded linear operator is non-empty, and we look at the characteristics of the spectrum of a compact operator.

Let be a complex Banach space which is not reduced to . (it is known that ) For we define as the set of those such that is bijective, and denote .

We define the spectrum by . In particular, if is an eigenvalue for we have and therefore . (but note that may contain elements which are not eigenvalues)

1. Suppose that . Prove that and .

2. Prove that if then and

3. Prove that is an open set in and for every the application is analytic in a neighborhood of any point .

4. Deduce that for every , is a non-void and compact.

## Agreg 2012 Analysis Part 1

**Part 1. Finite dimension**

The goal is to prove the following theorem:

**Theorem 1.** Let be a square matrix with non-negative coefficients. Suppose that for every with non-negative coordinates, the vector has strictly positive components. Then

- (i) the spectral radius is a simple eigenvalue for ;
- (ii) there exists an eigenvector of associated to with strictly positive coordinates.
- (iii) any other eigenvalue of verifies ;
- (iv) there exists an eigenvector of associated to with strictly positive components.

1. Consider such that . Prove that for distinct we have . Deduce that there exists such that .

2. Prove that the coefficients of are strictly positive.

3. For we denote . Prove that if and only if there exists such that .

4. Denote . Consider and denote . Prove that

5. Denote . Prove that is an interval which does not reduces to , it is bounded and closed.

6. Denote . Prove that if verifies then we have . Deduce that is an eigenvalue of and that for this eigenvalue there exists an eigenvector with coordinates strictly positive.

7. Consider . Prove that and implies . Deduce that and every other eigenvalue of verifies .

8. Prove that every eigenvector of which has positive coordinates is proportional to .

## IMC 2013 Problem 1

**Problem 1.** Let and be real symmetric matrices with all eigenvalues strictly greater than . Let be a real eigenvalue of matrix . Prove that .

## Eigenvalues via Fundamental Solutions

Eigenvalue problems like

can be solved numerically in a variety of ways. Probably the best known one is the finite element method. I will present below the sketch of an algorithm which does not need meshes, and when implemented correctly, can decrease computational costs.

The idea of Fundamental Solution first appeared in the 60s and was initially used to find solutions of the Laplace equation in a domain. It later was extended to more general equations and eigenvalue problems. The method uses (as the title says) some particular fundamental solutions of the studied equation to create an approximation of the solution as a linear combination of them. The advantage is that the fundamental solutions are sometimes known in analytic form, and the only thing that remains to do is to find the optimal coefficients in the linear combination. A detailed exposure of the method can be found in the following article of Alvez and Antunes.

## First Dirichlet eigenvalue is simple for connected domains

Suppose is a connected open set and consider the first two eigenvalues of the Laplace operator with Dirichlet boundary conditions . Then .