Minimal k -partition for the p -norm of the eigenvalues V. - PowerPoint PPT Presentation

Definition 2-partition Properties 3-partition k -partitions Conclusion Minimal k -partition for the p -norm of the eigenvalues V. Bonnaillie-No¨ el DMA, CNRS, ENS Paris joint work with B. Bogosel, B. Helffer, C. L´ ena, G. Vial Calculus of variations, optimal transportation, and geometric measure theory: from theory to applications Lyon July, 8th 2016

Definition 2-partition Properties 3-partition k -partitions Conclusion 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 D 1 (i.e. D i open, D i ∩ D j = ∅ , and D i ⊂ Ω) D 3 strong if Int D i \ ∂ Ω = D i and ( ∪ D i ) \ ∂ Ω = Ω D 2 ◮ O k (Ω) = { strong k -partitions of Ω }

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

Definition 2-partition Properties 3-partition k -partitions Conclusion p -minimal k -partition Definitions ◮ p -energy D = ( D i ) i =1 ,..., k : k -partition of Ω 1 ≤ p < + ∞ p = + ∞ � 1 / p � k 1 � λ 1 ( D i ) p Λ k , p ( D ) = Λ k , ∞ ( D ) = max 1 � i � k λ 1 ( D i ) k i =1 ◮ Optimization problem: let 1 ≤ p ≤ ∞ , L k , p (Ω) = D∈ O k (Ω) Λ k , p ( D ) inf ◮ Comparison ∀ k ≥ 2 , ∀ 1 ≤ p ≤ q < ∞ 1 k 1 / p Λ k , ∞ ( D ) ≤ Λ 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)

Definition 2-partition Properties 3-partition k -partitions Conclusion 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 ) = ∪ ( ∂ D i ∩ Ω) N ( D ) • 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

Definition 2-partition Properties 3-partition k -partitions Conclusion 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)

Definition 2-partition Properties 3-partition k -partitions Conclusion Properties Equipartition Proposition 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) }

Definition 2-partition Properties 3-partition k -partitions Conclusion 2-partition p = + ∞ Proposition 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

Definition 2-partition Properties 3-partition k -partitions Conclusion 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 ]

Definition 2-partition Properties 3-partition k -partitions Conclusion 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 ]

Definition 2-partition Properties 3-partition k -partitions Conclusion 2-partition p = 1 ◮ Ω = � , � ? ◮ Ω = △ ϕ 2 : symmetric eigenfunction associated with λ 2 (Ω) � � | ϕ 2 | 2 < | ϕ 2 | 2 ≃ 0 . 505 0 . 495 ≃ D 1 D 2 L 2 , 1 (Ω) < L 2 , ∞ (Ω)

Definition 2-partition Properties 3-partition k -partitions Conclusion 2-partition p = 1 ◮ Ω = � , � ? ◮ Ω = △ ϕ 2 : symmetric eigenfunction associated with λ 2 (Ω) � � | ϕ 2 | 2 < | ϕ 2 | 2 ≃ 0 . 505 0 . 495 ≃ D 1 D 2 L 2 , 1 (Ω) < L 2 , ∞ (Ω) is a ∞ -minimal 2-partition but not a 1-minimal 2-partition

Definition 2-partition Properties 3-partition k -partitions Conclusion 2-partition p = 1 ◮ Ω = � , � ? 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) � � | ϕ 2 | 2 < | ϕ 2 | 2 ≃ 0 . 63 0 . 37 ≃ D 1 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 ]

Definition 2-partition Properties 3-partition k -partitions Conclusion Lower bounds Square, equilateral triangle, disk � 1 / p � k 1 � λ i (Ω) p ≤ L k , p (Ω) k i =1 Explicit eigenvalues for � , △ , � Ω λ m , n (Ω) m , n π 2 ( m 2 + n 2 ) m , n ≥ 1 � 9 π 2 ( m 2 + mn + n 2 ) 16 △ m , n ≥ 1 j 2 m ≥ 0, n ≥ 1 (multiplicity) � m , n

Definition 2-partition Properties 3-partition k -partitions Conclusion Bounds for p = ∞ Theorem 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 [ B´ erard-Helffer 16, van den Berg-Gittins 16 ]

Definition 2-partition Properties 3-partition k -partitions Conclusion Upper bounds Square, disk L k , p (Ω) ≤ Λ k , ∞ ( D ⋆ ) Explicit upper bound for � L k , p ( � ) ≤ λ 1 (Σ 2 π/ k ) with Σ 2 π/ k : angular sector of opening 2 π/ k Explicit upper bound for � L k , p ( � ) ≤ m , n ≥ 1 { λ m , n ( � ) | mn = k } ≤ λ k , 1 ( � ) inf

Definition 2-partition Properties 3-partition k -partitions Conclusion 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

Definition 2-partition Properties 3-partition k -partitions Conclusion 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]

Definition 2-partition Properties 3-partition k -partitions Conclusion 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 / a Z ) × ( R / b Z ) the flat torus �� i − 1 k a , i � � D k ( a , b ) = 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-L´ ena 16 ] The question is open for any other domain (in the non bipartite case)

Definition 2-partition Properties 3-partition k -partitions Conclusion Topological configurations Euler formula ⇒ 3 types of configurations X α X 0 X 1 X 0 X 1 • • • • • • • • • • • O M O M O M X 0 X 0 • • X α • • • • • • • O M O M O M • • X 1 X 1 Question If Ω is symmetric, does it exist a symmetric minimal 3-partition ?

Definition 2-partition Properties 3-partition k -partitions Conclusion Non bipartite symmetric ∞ -minimal 3-partition First configuration: One critical point on the symmetry axis D = ( D 1 , D 2 , D 3 ) minimal 3-partition D 1 ⇒ ( D 1 , D 3 ) minimal 2-partition for Int ( D 1 ∪ D 3 ) D 3 • a x 0 • b • ⇒ nodal partition on Int ( D 1 ∪ D 3 ) D 2 [ BN–Helffer–Vial 10 ]