Minimal k-partition for the p-norm of the eigenvalues

Transcription

1 Minimal k-partition for the p-norm of the eigenvalues V. Bonnaillie-Noël DMA, CNRS, ENS Paris joint work with B. Bogosel, B. Helffer, C. Léna, G. Vial Calculus of variations, optimal transportation, and geometric measure theory: from theory to applications Lyon July, 8th 2016

2 Notation Ω R 2 : bounded and connected domain λ 1 (D) < λ 2 (D) eigenvalues of the Dirichlet-Laplacian on D D = (D i ) i=1,...,k : k-partition of Ω (i.e. D i open, D i D j =, and D i Ω) strong if IntD i \ Ω = D i and ( D i )\ Ω = Ω D 1 D 2 D 3 O k (Ω) = {strong k-partitions of Ω}

3 Definition 2-partition Properties 3-partition k-partitions Conclusion k-partitions Examples [Cybulski-Babin-Holyst 05]

4 p-energy D = (D i ) i=1,...,k : k-partition of Ω p-minimal k-partition Definitions 1 p < + p = + ( ) 1/p 1 k Λ k,p (D) = λ 1 (D i ) p Λ k, (D) = max k λ 1(D i ) 1 i k i=1 Optimization problem: let 1 p, L k,p (Ω) = inf Λ k,p(d) D O k (Ω) Comparison k 2, 1 p q < 1 k Λ k, (D) Λ 1/p k,p (D) Λ k,q (D) Λ k, (D) 1 k 1/p L k, (Ω) L k,p (Ω) L k,q (Ω) L k, (Ω) D is called a p-minimal k-partition if Λ k,p (D ) = L k,p (Ω)

5 p-minimal k-partition Existence of minimal partition Theorem For any k 2 and p [1, + ], there exists a regular strong p-minimal k-partition [Bucur Buttazzo-Henrot, Caffarelli Lin, Conti Terracini Verzini, Helffer Hoffmann-Ostenhof Terracini] N(D) N(D) = ( D i Ω) Regular : N(D) is smooth curve except at finitely many points and N(D) Ω is finite (boundary singular points) N(D) satisfies the Equal Angle Property

6 Nodal partition Let u be an eigenfunction of on Ω The nodal domains of u are the connected components of Ω \ N(u) with N(u) = {x Ω u(x) = 0} nodal partition = {nodal domains} Regularity N(u) is a C curve except on some critical points {x} If x Ω, N(u) is locally the union of an even number of half-curves ending at x with equal angle If x Ω, N(u) is locally the union of half-curves ending at x with equal angle Theorem Any eigenfunction u associated with λ k has at most k nodal domains [Courant] u is said Courant-sharp if it has exactly k nodal domains For k 1, L k (Ω) denotes the smallest eigenvalue (if any) for which there exists an eigenfunction with k nodal domains We set L k (Ω) = + if there is no eigenfunction with k nodal domains λ k (Ω) L k (Ω)

7 Proposition Properties Equipartition If D = (D i ) 1 i k is a -minimal k-partition, then D is an equipartition λ 1 (D i ) = λ 1 (D j ), for any 1 i, j k Let p 1 and D a p-minimal k-partition If D is an equipartition, then L k,q (Ω) = L k,p (Ω), for any q p We set p (Ω, k) = inf{p 1, L k,p (Ω) = L k, (Ω)}

8 Proposition 2-partition p = + L 2, (Ω) = λ 2 (Ω) = L 2 (Ω) The nodal partition of any eigenfunction associated with λ 2 (Ω) gives a -minimal 2-partition Examples

9 2-partition p = 1 - p = Proposition Let D = (D 1, D 2 ) be a -minimal 2-partition Suppose that there exists a second eigenfunction ϕ 2 of on Ω having D 1 and D 2 as nodal domains and such that ϕ 2 2 ϕ 2 2 D 1 D 2 Then L 2,1 (Ω) < L 2, (Ω) [Helffer Hoffman-Ostenhof]

10 2-partition p = 1 - p = Applications Let D = (D i ) 1 i k be a -minimal k-partition Let D i D j be a pair of neighbors. We denote D ij = Int D i D i λ 2 (D ij ) = L 2, (Ω) Suppose that there exists a second eigenfunction ϕ ij of on D ij having D i and D j as nodal domains and such that ϕ ij 2 ϕ ij 2 D i D j Then L k,1 (Ω) < Λ k, (D) [Helffer Hoffman-Ostenhof]

11 Ω =,? 2-partition p = 1 Ω = ϕ 2 : symmetric eigenfunction associated with λ 2 (Ω) ϕ 2 2 < D 1 ϕ D 2 L 2,1 (Ω) < L 2, (Ω)

12 2-partition p = 1 Ω =,? Ω = ϕ 2 : symmetric eigenfunction associated with λ 2 (Ω) ϕ 2 2 < D 1 ϕ D 2 L 2,1 (Ω) < L 2, (Ω) is a -minimal 2-partition but not a 1-minimal 2-partition

13 Ω =,? 2-partition p = 1 is a -minimal 2-partition but not a 1-minimal 2-partition Angular sector with opening π/4 ϕ 2 : symmetric eigenfunction associated with λ 2 (Ω) 0.37 ϕ 2 2 < D 1 ϕ D 2 L 2,1 (Ω) < L 2, (Ω) is a -minimal 2-partition but not a 1-minimal 2-partition The inequality L 2,1 (Ω) < L 2, (Ω) is generically satisfied [Helffer Hoffmann-Ostenhof]

14 Lower bounds Square, equilateral triangle, disk ( 1 k ) 1/p k λ i (Ω) p L k,p (Ω) i=1 Explicit eigenvalues for,, Ω λ m,n (Ω) m, n π 2 (m 2 + n 2 ) m, n π2 (m 2 + mn + n 2 ) m, n 1 jm,n 2 m 0, n 1 (multiplicity)

15 Theorem Bounds for p = λ k (Ω) L k, (Ω) L k (Ω) If L k, = L k or L k, = λ k, then λ k (Ω) = L k, (Ω) = L k (Ω) with a Courant sharp eigenfunction associated with λ k (Ω) [Helffer Hoffmann-Ostenhof Terracini] Theorem There exists k 0 such that λ k < L k for k k 0 [Pleijel] Explicit upper-bound for k 0 [Bérard-Helffer 16, van den Berg-Gittins 16]

16 Upper bounds Square, disk L k,p (Ω) Λ k, (D ) Explicit upper bound for L k,p ( ) λ 1 (Σ 2π/k ) with Σ 2π/k : angular sector of opening 2π/k Explicit upper bound for L k,p ( ) inf {λ m,n( ) mn = k} λ k,1 ( ) m,n 1

17 Examples for p = Minimal nodal partitions Let Ω =, or, λ k (Ω) = L k, (Ω) = L k (Ω) iff k = 1, 2, 4 -minimal nodal partitions

18 Properties Dichotomy for the case p = Let k > 2 To determine a -minimal k-partition, we consider the eigenspace E k associated with λ k Two cases: If there exists u E k with k nodal domains, then u produces a minimal k-partition and any minimal k-partition is nodal L k, (Ω) = λ k (Ω) = L k (Ω) [Bipartite case] If µ(u) < k for any u E k we have to find another strategy [Non bipartite case]

19 Known results in the non bipartite case, p = Sphere and fine flat torus Theorem The minimal 3-partition for the sphere is [Helffer Hoffmann-Ostenhof Terracini] Theorem Let 0 < b a and T (a, b) = (R/aZ) (R/bZ) the flat torus D k (a, b) = {] i 1 k a, i k a[ ]0, b[, 1 i k } k even and b a 2 k D k(a, b) is minimal k odd and b a < 1 k D k(a, b) is minimal [Helffer Hoffmann-Ostenhof] k odd and 1 k b a l D k (a, b) is minimal [BN-Léna 16] The question is open for any other domain (in the non bipartite case)

20 Topological configurations Euler formula 3 types of configurations Xα O M X0 X1 O M X0 X1 O M Xα O M X0 O M X1 X0 O M X1 Question If Ω is symmetric, does it exist a symmetric minimal 3-partition?

21 Non bipartite symmetric -minimal 3-partition First configuration: One critical point on the symmetry axis D 1 D = (D 1, D 2, D 3 ) minimal 3-partition a x 0 D 3 b (D 1, D 3 ) minimal 2-partition for Int(D 1 D 3 ) D 2 nodal partition on Int(D 1 D 3 ) [BN Helffer Vial 10]

22 Non bipartite symmetric -minimal 3-partition First configuration: One critical point on the symmetry axis Introduce a mixed Dirichlet-Neumann problem Ω + ϕ = λϕ in Ω + a x 0 b n ϕ = 0 on [x 0, b] ϕ = 0 elsewhere (λ 2 (x 0 ), ϕ x0 ) second eigenmode x 0 λ 2 (x 0 ) is increasing the nodal line starts from (a, b) and reaches the boundary [BN Helffer Vial 10]

23 Non bipartite symmetric -minimal 3-partition First configuration: One critical point on the symmetry axis Introduce a mixed Dirichlet-Neumann problem Ω + ϕ = λϕ in Ω + a x 0 b n ϕ = 0 on [x 0, b] ϕ = 0 elsewhere (λ 2 (x 0 ), ϕ x0 ) second eigenmode x 0 λ 2 (x 0 ) is increasing the nodal line starts from (a, b) and reaches the boundary [BN Helffer Vial 10]

24 Non bipartite symmetric -minimal 3-partition First configuration: One critical point on the symmetry axis Introduce a mixed Dirichlet-Neumann problem Ω + ϕ = λϕ in Ω + a x 0 b n ϕ = 0 on [x 0, b] ϕ = 0 elsewhere (λ 2 (x 0 ), ϕ x0 ) second eigenmode x 0 λ 2 (x 0 ) is increasing the nodal line starts from (a, b) and reaches the boundary [BN Helffer Vial 10]

25 Non bipartite symmetric -minimal 3-partition First configuration: One critical point on the symmetry axis Introduce a mixed Dirichlet-Neumann problem Ω + ϕ = λϕ in Ω + a x 0 b n ϕ = 0 on [x 0, b] ϕ = 0 elsewhere (λ 2 (x 0 ), ϕ x0 ) second eigenmode x 0 λ 2 (x 0 ) is increasing the nodal line starts from (a, b) and reaches the boundary [BN Helffer Vial 10]

26 Non bipartite symmetric -minimal 3-partition First configuration: examples Λ 3, (D 0) Λ 3, (D 1 ) Λ 3, (D 0) Λ 3, (D 1)

27 Non bipartite symmetric -minimal 3-partition Second and third configurations: Two critical points on the symmetry axis a D3 D2 x0 x1 D1 b a Ω + x0 x1 b Mixed Neumann-Dirichlet-Neumann problem ϕ = λϕ in Ω + n ϕ = 0 on [a, x 0 ] [x 1, b] ϕ = 0 elsewhere D1 D3 a x0 x1 b a D2 Ω + x0 x1 b Mixed Dirichlet-Neumann-Dirichlet problem ϕ = λϕ in Ω + n ϕ = 0 on [x 0, x 1 ] ϕ = 0 elsewhere No candidate for the square, disk, angular sectors with two critical points!

28 -minimal 3-partition Candidates Λ 3, (D 0) Λ 3, (D 1 ) Λ 3, (D 0) Λ 3, (D 0) Applications 0.75 D 1 ϕ 2 2 > 2 D 2 ϕ L 3,1 ( ) < Λ 3, (D)

29 Numerical simulations p-minimal 3-partition for the square Since Λ DN and L 3 = 10π π 2 = λ 3 < L 3, Λ DN ( ) 1/p π 2 2 p +5 p +5 p 3 L3,p Λ DN π 2 L 3, p = 1, 2, 5, Λ 3, Λ 3,p Λ DN [Bogosel-BN16]

30 Numerical simulations p-minimal 3-partition Conjecture For the square : p L 3,p ( ) is increasing p (, 3) = + For the disk: p (, 3) = 1 For the equilateral triangle: p (, 3) = 1 is a p-minimal 3-partition for any p 1

31 Definition 2-partition Properties 3-partition k-partitions Conclusion k-partitions Examples [Cybulski-Babin-Holyst 05]

32 Iterative methods Penalization 1. Instead of looking for k domains (D 1,..., D K ), we look for a k-upple of functions (ϕ 1,..., ϕ k ) M with { } k M = (ϕ 1,..., ϕ k ), ϕ i : Ω [0, 1] measurable, ϕ i = 1 a.e. Ω. 2. Penalized eigenvalue problem on Ω i=1 v i + 1 ε (1 ϕ i)v i = λ(ε, ϕ i )v i in Ω Note that lim ε 0 λ(ε, ϕ i ) = λ 1(D i ) 3. Penalized optimization problem ( 1/p 1 k M(ε, k) = inf λ p 1 i)) k (ε, ϕ, (ϕ 1,..., ϕ k ) M i=1 In some sense, lim ε 0 M(ε, k) = L k,p (Ω) 4. Projected-gradient descent with adaptive step [Bourdin-Bucur-Oudet 09]

33 Algorithm Let ρ > 0, ε > 0 Initialisation k vectors Φ 0 l given randomly Iteration Step p: for any l = 1,..., k: 1. Compute the first eigenmode (λ(φ l ), U(Φ l )) of A(ε, Φ l) 2. Gradient descent : 3. Projection on S : p+1 Φ l = Φ p+1 l ρ Φ p λ(φ l) l Φ p+1 l = Π S Φp+1 l

34 p-minimal 4-partition λ 4 (Ω) = L 4, (Ω) = L 4 (Ω) if Ω =,, -minimal 4-partitions : Conjecture For the disk: p (, 4) = 1 For the square : p (, 4) = 1 is a p-minimal 4-partition for any p 1

35 p-minimal 4-partition Equilateral triangle Conjecture p L 4,p ( ) is increasing p = 1, 2, 5, 50 p = Λ 4, Λ 4,p

36 p-minimal 5-partition Symmetric candidates for the square Use a Dirichlet-Neumann approach to find some symmetric equipartition 0.72 ϕ 2 2 < 4 D 1 ϕ D 2 L 5,1 ( ) < Λ 5, (D)

37 p-minimal 5-partition Square π 2 = λ 5 < L 5, < L 5 26π ( ) 1/p π 2 2 p +5 p +5 p +8 p +10 p 4 L5,p π 2 L 5,1 Mixed Dirichlet-Neumann approach L 5,p Λ DN

38 p-minimal 5-partition Square π 2 = λ 5 < L 5, Λ DN ( ) 1/p π 2 2 p +5 p +5 p +8 p +10 p 5 L5,p π 2 L 5,1 Λ DN p = 1, 2, 5, Λ 5, 120 Λ 5,p Λ DN [Bogosel-BN16]

39 p-minimal 5-partition Equilateral triangle Λ 5, Λ 5,p [Bogosel-BN16]

40 p-minimal 5-partition Conjecture For the square : For the equilateral triangle : p L 5,p ( ) is increasing p L 5,p ( ) is increasing p (, 5) = + p (, 5) = + For the disk: p (, 5) = 1 For any p 1, a p-minimal 5-partition is

41 p-minimal 6-partition Square p = 1, 2, 5, π 2 = λ 6 < L 6, < L 6 = 13π Λ 6, Λ 6,p Conjecture p L 6,p is increasing p (, 6) = +

42 p-minimal 6-partition Disk is not Courant-sharp then not minimal λ 1 (Σ π/3 ) L 6,p < λ 1 (Σ π/3 ) p 1

43 p-minimal 6-partition Disk L 6,p < λ 1 (Σ π/3 ) p 1 p = 1, 2, 5, Λ 6, Λ 6,p [Bogosel-BN16] Conjecture p L 6,p ( ) is increasing p (, 6) = +

44 p-minimal 6-partition Equilateral triangle Conjecture p (, 6) = 1

45 p-minimal 6-partition Equilateral triangle Candidates of 6-partitions for p = Best candidate x opt Eigenvalues : ,

46 p-minimal 7-partition Square π 2 = λ 7 < L 7, < L 7 50π Λ 7, Λ 7,p Λ 7, 165 Λ 7,p

47 p-minimal 7-partition Disk λ 1 (Σ 2π/7 ) 48.86, L 7, < λ 1 (Σ 2π/7 ) Λ 7, Λ 7,p p (, 7) = 1? [Bogosel-BN16]

48 Conclusion k = 2 k = 3 k = 4 k = 5 k = 6

49 Conclusion k = 7 k = 8 k = 9 k = 10

50 Definition 2-partition Properties 3-partition k-partitions Conclusion Asymptotics k Hexagonal conjecture I The limit of Lk, (Ω)/k as k + exists and λ1 (7) Lk (Ω) = k + k Ω lim I The limit of Lk,1 (Ω)/k as k + exists and lim k + k = 15 λ1 (7) Lk,1 (Ω) = k Ω k = 20 k = 25 k = 30 [Bourdin Bucur Oudet, BN Le na]