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Asymptotic behaviour ofextremal domains

for Laplace eigenvalues

Neuch

âtel,

June

201

7

Pedro FreitasInstituto Superior Técnico andGroup of Mathematical PhysicsUniversidade de Lisboa

Grupo deFísica Matemáticada Universidade de Lisboa

and are optimised by one and(~130 years: Dirichlet, Neumann andRobin with positive parameter at the boundary

Extremals for the Dirichletproblem exist within the class of

quasi-open sets (Bucur and Mazzoleni & Pratelli (2012))

two (equal) balls, respectively

There is no nice structurefor intermediate frequencies

(numerical results within the last 15 years)

Extremal domains are not described by known functions.

There is, in general, no symmetry.

Brief summary of what is known

Grupo deFísica Matemáticada Universidade de Lisboa

2D

3D

4D

Dirichlet

Theorem Numerical

Brief (pictorial) summary of what is known

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2D

Grupo deFísica Matemáticada Universidade de Lisboa

2D

Theorem [A. Berger 2015] In dimension 2 and for k larger than 4, theDirichlet eigenvalue is never minimised by a ball or unions of balls.

This is, however, no longer true in higher dimensions.

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3D

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

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Can we say something about what happensat the other end of the spectrum?

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two ways oftackling

the problem

considerspecific

examples

studyproperties of thegeneral problem

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Specific examples: rectangles

Eigenvalues are given explicitly:

They satisfy Pólya's inequality:

Problem:

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Example of the type of function we aretalking about minimising

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Grupo deFísica Matemáticada Universidade de Lisboa

Theorem [Antunes & F. (2013)]

Idea of the 2-step proof:

apply to optimal rectangles to obtain uniform boundedness of

use estimates from the Gauss circle problem to obtain convergenceof the perimeter to that of the square

1.

2.

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 pointcounting problems [Laugesen & Liu (2016), Ariturk & Laugesen(2017),Marshall & Steinerberger (2017)].

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The general problem

What are the properties of the sequence ?

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Theorem [Colbois & El Soufi (2014)] The sequence of minimal values

is sub-additive and it satisfies

In particular, the following two statements are equivalent:

(1)

(2) (Pólya's conjecture)

is such that

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Combining the two approaches: averages

Define

and consider the problem of determining

where X could be one of

or

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Theorem [F. (2017)] The sequence is sub-additive and

Fixed volume

where

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Theorem [F. (2017)] The sequence satisfies

Fixed surface area

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Prove that minimisers of the Dirichlet problem with fixedarea among tiling domains converge to the regular hexagon.

Prove minimisers of the kth Dirichlet eigenvalue with fixedvolume approach the ball as k goes to infinity (and hencePólya's conjecture holds!)

Prove the ball minimises the (n+1)th Dirichlet eigenvaluein Rn among domains with fixed volume

Prove that k equal balls minimise the kth Robin eigenvalue forsufficiently small (positive) boundary parameter and fixed volume

P1:

P5:

P2:

P3:

P4:

Top 5 open problems

Prove that minimisers of the Dirichlet problem with fixedsurface area converge to the ball.