Elliptic PDEs, vibrations of a plate, set discrepancy and packing problems Stefan Steinerberger Mathematisches Institut Universität Bonn Oberwolfach Workshop 1340 Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 1 / 23
Vibrations of a string. Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 2 / 23
u = λu Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 3 / 23
I feel that these informations about the proper oscillations of a membrane, valuable as they are, are still very incomplete. (Hermann Weyl, Gibbs lecture 1950) Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 4 / 23
(Image credit: Douglas Stone, Yale) Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 5 / 23
(Image credit: Aurich, Bäcker, Schubert, Taglieber, Physica D) Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 6 / 23
Given a domain Ω R 2 and φ n (x) = λ n φ n (x) on Ω φ n (x) = 0 on Ω what bounds can be proven on on the number of connected components of Ω \ {x Ω : φ n (x) = 0}? Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 7 / 23
Courant s nodal line theorem (1924) The number N(n) of connected components of Ω \ {x Ω : φ n (x) = 0} satisfies N(n) n. Pleijel s estimate (1956) The number N of connected components of Ω \ {x Ω : φ n (x) = 0} satisfies lim sup n N(n) n ( ) 2 2 0.69... j Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 8 / 23
Polterovich, 2009 Remark 2.5. [...] and nodal domains of an eigenfunction can not be all disks at the same time. [...] we suggest that for any regular bounded planar domain with either Dirichlet or Neumann lim sup k n k k 2 π 0.63 If true, this estimate (which is quite close to Pleijel s bound) is sharp and attained for the basis of separable eigenfunctions on a rectangle. Bourgain (2013) The number N of connected components of Ω \ {x Ω : φ n (x) = 0} satisfies lim sup n N(n) n ( ) 2 2 3 10 9 j Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 9 / 23
Proof relies on Theorem (Blind). A collection of disks with radii r 1, r 2,... such that inf i,j r i r j 3 4 has packing density bounded from above by π/ 12. Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 10 / 23
Philosophy Given Ω R 2 open and bounded. Consider a partition Geometric discrepancy is the Ω = N Ω i. i=1 worst deviation of set of points in a cube from the ideal while we will be interested in the average deviation of a Ω i from a disc. Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 11 / 23
Fraenkel asymmetry Given a domain Ω, we define the Fraenkel asymmetry via A(Ω) = inf B Ω B, Ω where the infimum ranges over all disks B R 2 with B = Ω and is the symmetric difference Ω B = (Ω \ B) (B \ Ω). Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 12 / 23
Fraenkel asymmetry Scale-invariant quantity with A(Ω) = inf B Ω B Ω 0 A(Ω) 2. Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 13 / 23
Apollonian packing Can a set be decomposed into (infinitely) many discs? Certainly....but some are small. Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 14 / 23
Deviation measure Given a decomposition we define D is scale invariant and satisfies Ω = N Ω i, i=1 D(Ω i ) = Ω i min 1 j N Ω j. Ω i 0 D(Ω i ) 1. Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 15 / 23
Geometric uncertainty principle N Ω = Ω i, i=1 A(Ω) = inf B Ω B, D(Ω i ) = Ω i min 1 j N Ω j Ω Ω i Theorem For N sufficiently large (depending on Ω) ( N i=1 ) ( N ) Ω i Ω A(Ω Ω i i) + Ω D(Ω i) 1 60000. i=1 Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 16 / 23
Extensions ( N i=1 ) ( N ) Ω i Ω A(Ω Ω i i) + Ω D(Ω i) 1 60000. i=1 The statement remains true in higher dimensions with some optimal constant c n > 0. Conjecture. We have 0.07 c 2 < c 3 < c 4 < 2, where c 2 0.07 comes from assuming that the hexagonal packing is extremal. Is it? What can be said about extremizers? Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 17 / 23
Extensions ( N i=1 ) ( N ) Ω i Ω A Ω i K (Ω i ) + Ω D(Ω i) c K > 0. i=1 The statement also remains true if we replace the ball B in the definition of Fraenkel asymmetry A by any strictly convex body K. Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 18 / 23
Sketch of the global proof I Replace all partition elements by Fraenkel balls. Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 19 / 23
Sketch of the global proof Remove all the big balls. There are few of them anyway. Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 20 / 23
Sketch of the global proof Remove all the balls intersecting another ball too strongly. There are few of them anyway. Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 21 / 23
Sketch of the global proof The remaining balls are roughly equal sized and keep a certain distance from each other. Shrink them a tiny bit to make them disjoint. Use Blind s result on the packing density of disks. Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 22 / 23
A more subtle question Define for any set S R 2 A S (Ω) = inf S Ω S, Ω where the infimum is taking over all rescaled and rotated translates of S. Define the evil set E containing all sets in the Euclidean plane such that they admit a tiling of the plane by just rotation and translation. Let K R 2 be some set and assume i=1 inf A E (K) > ε. E E Does this imply a geometric uncertainty principle ( N ) ( N ) Ω i Ω A Ω i K (Ω i ) + Ω D(Ω i) i=1 c(ε)? Stefan Steinerberger (U Bonn) A Geometric Uncertainty Principle Oberwolfach Workshop 1340 23 / 23