Description

One of the most classic examples is the isoperimetric problem which asserts that among all planar Euclidean domains of equal perimeter, the disc of the same perimeter has maximal area. Legend has it that Queen Dido solved a version of this problem when she founded the ancient city of Carthage.

A first step towards resolving the aforementioned question is to show that an optimal shape exists within the chosen collection. Determining the geometric properties of such an optimal shape can help to narrow down the potential optimal candidates. For example, do they have symmetries? Are they connected? If not, how many connected components do they have?

Consider a bounded set in the Euclidean plane and fix its boundary. We can think of this set as a drumhead so that when we strike it, it vibrates. The displacement of this drumhead is governed by a partial differential equation. In fact, the frequencies at which the drumhead vibrates correspond to the eigenvalues of the Laplace operator with a Dirichlet (zero) boundary condition.

A fascinating question is which set(s) minimise the Dirichlet eigenvalues among all bounded sets of prescribed area in the Euclidean plane? In other words, which shapes minimise the frequencies of the drumhead?

Among planar domains of given area, it is well known that the first Dirichlet eigenvalue is minimised by the disc and the second Dirichlet eigenvalue is minimised by the disjoint union of two discs each of equal area. But for the higher Dirichlet eigenvalues, the minimising sets of prescribed area in the Euclidean plane are not known to date!

Among all triangles, respectively quadrilaterals, of prescribed area the equilateral triangle, respectively square, minimises the first Dirichlet eigenvalue. However, for \(n\)-gons of prescribed area with \(n > 4\), it is a challenging open problem to determine whether the regular \(n\)-gon minimises the first Dirichlet eigenvalue.

A first goal of this project is to study some of the fundamental concepts in the field of shape optimisation. Students will then have the opportunity to study other topics in this field including the shape optimisation of the eigenvalues of the Laplacian.

Prerequisites and Corequisites

PDE III/IV and Differential Geometry III could be useful but are not essential.

Some Resources

Antoine Henrot Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics (2006) Birkhäuser Basel.

Antoine Henrot Shape optimization and spectral theory. De Gruyter (2017).

Antoine Henrot, Michel Pierre. Shape Variation and Optimization: A Geometrical Analysis. EMS Tracts in Mathematics, Volume: 28 (2018).

Paul J. Nahin When Least Is Best : How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible. Princeton, NJ : Princeton University Press (2011).