Asymptotic behaviour of extremal domains for Laplace eigenvalues

Transcription

1 N eu ch ât el,j un e Asymptotic behaviour of extremal domains for Laplace eigenvalues Pedro Freitas Instituto Superior Técnico and Group of Mathematical Physics Universidade de Lisboa

2 Brief summary of what is known and are optimised by one and two (equal) balls, respectively Extremals for the Dirichlet problem exist within the class of quasi-open sets There is no nice structure for intermediate frequencies (~130 years: Dirichlet, Neumann and Robin with positive parameter at the boundary (Bucur and Mazzoleni & Pratelli (2012)) (numerical results within the last 15 years) Extremal domains are not described by known functions. There is, in general, no symmetry.

3 Brief (pictorial) summary of what is known Dirichlet 2D 3D 4D Theorem Numerical

4 2D

5 2D Theorem [A. Berger 2015] In dimension 2 and for k larger than 4, the Dirichlet eigenvalue is never minimised by a ball or unions of balls. This is, however, no longer true in higher dimensions.

6 3D

7 Fixed surface area [Antunes and F. (2016)]

8 Can we say something about what happens at the other end of the spectrum?

9 two ways of tackling the problem consider specific examples study properties of the general problem

10 Specific examples: rectangles Eigenvalues are given explicitly: They satisfy Pólya's inequality: Problem:

11 Example of the type of function we are talking about minimising

12

13 Theorem [Antunes & F. (2013)] Idea of the 2-step proof: 1. apply to optimal rectangles to obtain uniform boundedness of 2. use estimates from the Gauss circle problem to obtain convergence of the perimeter to that of the square This approach was extended to the Neumann problem [van den Berg, Bucur & Gittins (2016)], 3 dimensions [van den Berg & Gittins (2017)], n-dimensions [Gittins & Larson (2017)] and abstract lattice point counting problems [Laugesen & Liu (2016), Ariturk & Laugesen(2017), Marshall & Steinerberger (2017)].

14 The general problem What are the properties of the sequence?

15 Theorem [Colbois & El Soufi (2014)] The sequence of minimal values is such that is sub-additive and it satisfies In particular, the following two statements are equivalent: (1) (2) (Pólya's conjecture)

16 Combining the two approaches: averages Define and consider the problem of determining where X could be one of or

17 Fixed volume Theorem [F. (2017)] The sequence is sub-additive and where

18 Fixed surface area Theorem [F. (2017)] The sequence satisfies

19 Top 5 open problems P1: Prove the ball minimises the (n+1)th Dirichlet eigenvalue in Rn among domains with fixed volume P2: Prove that k equal balls minimise the kth Robin eigenvalue for sufficiently small (positive) boundary parameter and fixed volume P3: Prove that minimisers of the Dirichlet problem with fixed surface area converge to the ball. P4: Prove that minimisers of the Dirichlet problem with fixed area among tiling domains converge to the regular hexagon. P5: Prove minimisers of the kth Dirichlet eigenvalue with fixed volume approach the ball as k goes to infinity (and hence Pólya's conjecture holds!)