On Courant s nodal domain property for linear combinations of eigenfunctions (after P. Bérard and B. Helffer).

Transcription

1 On Courant s nodal domain property for linear combinations of eigenfunctions (after P. Bérard and B. Helffer). March 2019 Conference in honour of B.W. Schulze for his 75-th birthday. Bernard Helffer, Laboratoire de Mathématiques Jean Leray, Université de Nantes.

2 Abstract We revisit Courant s nodal domain property for linear combinations of eigenfunctions. This property was proven by Sturm (1836) in the case of dimension 1. Although stated as true for the Dirichlet Laplacian in dimension > 1 in a footnote of the celebrated book of Courant-Hilbert (and wrongly attributed to H. Herrmann, a PHD student of R. Courant), it appears to be wrong. This was first observed by V. Arnold in the seventies. In this talk, we present simple and explicit counterexamples to this so-called Herrmann s statement for domains in R d, S 2 or T 2. We also discuss the existence of a counterexample in a C, convex domain Ω in R 2 in relation with the analysis of the number of domains delimited by the level sets of a second eigenfunction for the Neumann problem. We finally discuss the question to have positive statements. This work has been done in collaboration with P. Bérard.

3 Introduction Let Ω R d be a bounded open domain or, more generally, a compact Riemannian manifold with boundary. Consider the eigenvalue problem { u = λ u in Ω, B(u) = 0 on Ω, (1) where B(u) is some boundary condition on Ω, so that we have a self-adjoint boundary value problem (including the empty condition if Ω is a closed manifold). For example, D(u) = u Ω for the Dirichlet boundary condition, or N(u) = u ν Ω for the Neumann boundary condition.

4 Call H(Ω, B) the associated self-adjoint extension of, and list its eigenvalues in nondecreasing order, counting multiplicities, 0 λ 1 (Ω, B) < λ 2 (Ω, B) λ 3 (Ω, B) (2) For any integer n 1, define the index τ(ω, B, λ n ) = min{k λ k (Ω, B) = λ n (Ω, B)}. (3) E(λ n ) will denote the eigenspace associated with λ n.

5 The Courant nodal theorem For a real continuous function v on Ω, we define its nodal set Z(v) = {x Ω v(x) = 0}, (4) and call β 0 (v) the number of connected components of Ω \ Z(v) i.e., the number of nodal domains of v. Courant s nodal Theorem (1923) For any nonzero u E(λ n (Ω, B)), β 0 (u) τ(ω, B, λ n ) n. (5) Courant s nodal domain theorem can be found in Courant-Hilbert [10].

6 The extended Courant nodal property Given r > 0, denote by L(Ω, B, r) the space L(Ω, B, r) = c j u j c j R, u j E λj (Ω,B). (6) λ j (Ω,B) r Extended Courant Property:= (ECP) We say that v L (Ω, B, λ n (Ω, B)) satisfies (ECP) if β 0 (v) τ(ω, B, λ n ). (7) A footnote in Courant-Hilbert [10] indicates that this property also holds for any linear combination of the n first eigenfunctions, and refers to the PhD thesis of Horst Herrmann (Göttingen, 1932) [16].

7 Historical remarks : Sturm (1836), Pleijel (1956). 1. (ECP) is true for Sturm-Liouville equations. This was first announced by C. Sturm in 1833, [30] and proved in [31]. Other proofs were later on given by J. Liouville and Lord Rayleigh who both cite Sturm explicitly. 2. Å. Pleijel mentions (ECP) in his well-known paper [26] on the asymptotic behaviour of the number of nodal domains of a Dirichlet eigenfunction associated with the n-th eigenvalue in a plane domain. At the end of the paper, he writes: In order to treat, for instance the case of the free three-dimensional membrane ]0, π[ 3, it would be necessary to use, in a special case, the theorem quoted in [9]... However, as far as I have been able to find there is no proof of this assertion in the literature.

8 Historical remarks: V. Arnold ( ) 3. As pointed out by V. Arnold [1], when Ω = S d, (ECP) is related to Hilbert s 16 th problem. Arnold [2] mentions that he actually discussed the footnote with R. Courant, that (ECP) cannot be true, and that O. Viro produced in 1979 counter-examples for the 3-sphere S 3, and any degree larger than or equal to 6, [32].

9 More precisely V. Arnold wrote: Having read all this, I wrote a letter to Courant: Where can I find this proof now, 40 years after Courant announced the theorem?. Courant answered that one can never trust one s students: to any question they answer either that the problem is too easy to waste time on, or that it is beyond their weak powers.

10 And V. Arnold continues: The point is that for the sphere S 2 (with the standard Riemannian metric) the eigenfunctions (spherical functions) are polynomials. Therefore, their linear combinations are also polynomials, and the zeros of these polynomials are algebraic curves (whose degree is bounded by the number n of the eigenvalue). Therefore, from the generalized Courant theorem one can, in particular, derive estimates for topological invariants of the complements of projective real algebraic curves (in terms of the degrees of these curves).

11 ... And then it turned out that the results of the topology of algebraic curves that I had derived from the generalized Courant theorem contradict the results of quantum field theory. Nevertheless, I knew that both my results and the results of quantum field theory were true. Hence, the statement of the generalized Courant theorem is not true.

12 Historical remarks: Gladwell-Zhu (2003) 4. In [12], Gladwell and Zhu refer to (ECP) as the Courant-Herrmann conjecture. They claim that this extension of Courant s theorem is not stated, let alone proved, in Herrmann s thesis or subsequent publications. They consider the case in which Ω is a rectangle in R 2, stating that they were not able to find a counter-example to (ECP) in this case. They also provide numerical evidence that there are counter-examples for more complicated (non convex) domains. They suggest that may be the conjecture could be true in the convex case.

13 Historical remarks: looking for the PHD thesis of H. Herrmann 5. Herrmann s thesis has three parts. Only the second part was accepted by the evaluating committee for publication. This part does not contain any mention of (ECP). The first part, was published later in [17] in Math. Z. in 1936 in a different form. Finally, the third part was never published. The title of this chapter indicates that this part was devoted to the analysis of the Fourier-Robin problem and to analyze how the eigenvalues tend to the eigenvalues of the Dirichlet problem as the Robin parameter tends to +. Nothing to do with (ECP). The purpose in this talk is to provide simple counter-examples to the Extended Courant Property for domains in R d, S 2 or R 3, including convex domains. No algebraic topology will be involved.

14 Some geometrical statements There are two statements of V. Arnold for which no proof is available. ECP is true for the sphere S 2. ECP does not hold for other metrics on the sphere. Outside the (1D)-case, the only known result is for RP 2 and was obtained by J. Leydold.

15 Rectangle with a crack Let R 0 be the rectangle ]0, 4π[ ]0, 2π[. For 0 < a 1, let C a :=]0, a] {π} and R a := R 0 \ C a and consider the Neumann condition. The setting is described in Dauge-Helffer [11]. We call { 0 = ν1 (0) < ν 2 (0) < ν 3 (0) = ν 4 (0) (8) the Neumann eigenvalues of in R 0. They are given by the m n2 4 for pairs (m, n) of non-negative integers. Corresponding eigenfunctions are products of cosines. Similarly, the Neumann eigenvalues of in R a are denoted by 0 = ν 1 (a) < ν 2 (a) ν 3 (a). (9)

16 The first three Neumann eigenvalues for the rectangle R 0 are as follows. ν 1 (0) 0 (0, 0) ψ 1 (x, y) = 1 1 ν 2 (0) 16 (1, 0) ψ 2 (x, y) = cos( x 4 ) ν 3 (0) (0, 1) ψ 3 (x, y) = cos( y 2 ) ν 4 (0) 1 4 (2, 0) ψ 4 (x, y) = cos( x 2 ) (10)

17 Dauge-Helffer (1993) prove: Theorem For i 1, 1. [0, 1] a ν i (a) is non-increasing. 2. ]0, 1[ a ν i (a), is continuous. 3. lim a 0+ ν i (a) = ν i (0). It follows that for 0 < a, small enough, we have 0 = ν 1 (a) = ν 1 (0) < ν 2 (a) ν 2 (0) < ν 3 (a) ν 4 (a) ν 3 (0) = ν 4 (0). (11)

18 Observe that for i = 1 and 2, ψ i y (x, y) = 0. Hence for a small enough, ψ 1 and ψ 2 are the first two eigenfunctions for R a with the Neumann condition with associated eigenvalues ν 1 (a) = 0 and We have ν 2 (a) = 1 4 < ν 3(a). αψ 1 (x, y) + ψ 2 (x, y) = α + cos( x 4 ). We can choose the coefficient α ] 1, +1[ in such a way that these linear combinations of the first two eigenfunctions have two or three nodal domains in R a.

19 Figure: Rectangle with a crack (Neumann condition) This proves that (ECP) is false in R a with Neumann condition. Notice that we can introduce several cracks {(x, b j ) 0 < x < a j } k j=1 so that for any d {2, 3,... k + 2} there exists a linear combination of 1 and cos( x 4 ) with d nodal domains.

20 Sphere S 2 with cracks On the round sphere S 2, we consider the geodesic lines (x, y, z) ( 1 z 2 cos θ i, 1 z 2 sin θ i, z) through the north pole (0, 0, 1), with distinct θ i [0, π[. Removing the geodesic segments θ 0 = 0 and θ 2 = π 2 with 1 z a 1, we obtain a sphere S 2 a with a crack in the form of a cross. We consider the Neumann condition on the crack. We then easily produce a function in the space generated by the two first eigenspaces of the sphere with a crack having five nodal domains.

21 The function z is also an eigenfunction of S 2 a with eigenvalue 2. For a small enough, λ 4 (a) = 2, with eigenfunction z. For 0 < b < a, the linear combination z b has five nodal domains in S 2 a, see Figure below in spherical coordinates. It follows that (ECP) does not hold on the sphere with cracks.

22 Figure: Sphere with crack, five nodal domains Remark. Removing more geodesic segments around the north pole, we can obtain a linear combination z b with as many nodal domains as we want.

23 The hypercube with Dirichlet boundary condition We can adapt the method of Gladwell-Zhu in any dimension. Let C n (π) :=]0, π[ n be the hypercube of dimension n, with either the Dirichlet or Neumann boundary condition on C n (π). A point in C n (π) is denoted by x = (x 1,..., x n ). A complete set of eigenfunctions of for (C n (π), d) is given by the functions n n sin(k j x j ) with eigenvalue kj 2, for k j N\{0}, (12) j=1 for x = (x 1,..., x n ) ]0, π[ n. A complete set of eigenfunctions of for (C n (π), n) is given by the functions n n cos(k j x j ) with eigenvalue kj 2, for k j N. (13) j=1 j=1 j=1

24 We make use of the classical Chebyshev polynomials U k (t), k N, defined by the relation, and such that sin ((k + 1)t) = sin(t) U k (cos(t)), U 0 (t) = 1, U 1 (t) = 2t, U 2 (t) = 4t 2 1. The first Dirichlet eigenvalues (as points in the spectrum) are listed in the following table, together with their multiplicities, and eigenfunctions.

25 Table: First Dirichlet eigenvalues of C n (π) Eigenv. Mult. Eigenfunctions n 1 φ 1 (x) := n j=1 sin(x j) n + 3 n φ 1 (x) U 1 (cos(x i )) for 1 i n n(n 1) n φ 1 (x) U 1 (cos(x i )) U 1 (cos(x j )) for 1 i < j n n + 8 n φ 1 (x) U 2 (cos(x i )) for 1 i n For the above eigenvalues, the index is given by, τ(n + 3) = 2, τ(n + 6) = n + 2, τ(n + 8) = n(n + 1) (14)

26 In order to study the nodal set of the above eigenfunctions or linear combinations thereof, we use the diffeomorphism (x 1,..., x n ) (ξ 1 = cos(x 1 ),..., ξ n = cos(x n )), (15) from ]0, π[ onto ] 1, 1[, and factor out the function φ 1 which does not vanish in the open hypercube. We consider the function Ψ a (ξ 1,..., ξ n ) = ξ ξ 2 n a which corresponds to a linear combination Φ in E(n) E(n + 8). Given some a, (n 1) < a < n, this function has 2 n + 1 nodal domains, see Figure 3 in dimension 3. For n 3, we have 2 n + 1 > τ(n + 8). The function Φ therefore provides a counterexample to ECP for the hypercube of dimension at least 3, with Dirichlet boundary condition.

27 Proposition For n 3, the hypercube of dimension n, with Dirichlet boundary condition, provides a counterexample to ECP. Figure: 3-dimensional cube

28 Remark 1. An interesting feature of this example is that we get counterexamples to ECP for linear combinations which involve eigenvalues with high energy while the other examples only involve eigenvalues with low energy. Remark 2. Similar results can be obtained for the hypercube with Neumann boundary condition

29 The equilateral triangle (Dirichlet or Neumann) Let T e denote the equilateral triangle with sides 1, see Figure 4. The eigenvalues and eigenfunctions of T e, with either Dirichlet or Neumann condition on the boundary T e, can be completely described. We show that the equilateral triangle provides a counterexample to the Extended Courant Property for both the Dirichlet and the Neumann boundary condition. Figure: Equilateral triangle T e = [OAB]

30 Neumann boundary condition The sequence of Neumann eigenvalues of the equilateral triangle T e begins as follows, 0 = λ 1 (T e, N) < 16π2 = λ 2 (T e, N) = λ 3 (T e, N) < λ 4 (T e, N). 9 (16) The second eigenspace has dimension 2, and contains one invariant eigenfunction ϕ N 2 under the mirror symmetry w.r.t OM, and another anti-invariant eigenfunction ϕ N 3. ϕ N 2 is given by ( ) ( ( ) ( )) 2πx 2πx 2πy ϕ N 2 (x, y) = 2 cos cos + cos 1. (17) 3 3 3

31 The set {ϕ N = 0} consists of the two line segments {x = 3 4 } T e and {x + 3y = 3 2 } T e, which meet at the point ( 3 4, 3 4 ) on T e. The sets {ϕ 2 + a = 0}, with a {0, 1 ε, 1, 1 + ε}, and small positive ε, are shown in Figure 6. When a varies from 1 ε to 1 + ε, the number of nodal domains of ϕ 2 + a in T e jumps from 2 to 3, with the jump occurring for a = 1.

32 Figure: Levels sets {ϕ N 2 + a = 0} for a {0 ; 0.9 ; 1 ; 1.1}

33 If we take the coordinates X = cos 2π 2π 3 x and Y = cos 3 y we are reduced to the level sets of (X, Y ) X (X + Y ): Figure: Levels sets in the X, Y variables

34 It follows that ϕ N 2 + a = 0, for 1 a 1.2, provides a counterexample to the Extended Courant Property for the equilateral triangle with Neumann boundary condition.

35 Dirichlet boundary condition The sequence of Dirichlet eigenvalues of the equilateral triangle T e begins as follows, δ 1 (T e ) = 16π2 3 < δ 2 (T e ) = δ 3 (T e ) = 112π2 9 < δ 4 (T e ). (18) Up to scaling, the first eigenfunction ϕ D 1 is given by ϕ D 1 (x, y) = 8 sin 2πy 3 sin π( 3x + y) 3 sin π( 3x y) 3, (19) which shows that it does not vanish inside T e.

36 A surprising formula. The second eigenvalue has multiplicity 2, with one eigenfunction ϕ D 2 symmetric with respect to the median OM, and the other ϕd 3 anti-symmetric. Up to scaling, ϕ D 2 is given by ϕ D 2 (x, y) = sin ( 2π 3 (5x + 3y) ) sin ( 2π 3 (5x 3y) ) + sin ( 2π 3 (x 3 3y) ) sin ( 2π 3 (x + 3 3y) ) + sin ( 4π 3 (2x + 3y) ) sin ( 4π 3 (2x 3y) ). (20)

37 First astonished by some numerics, we arrive to the conclusion that the following surprising result could be true: Lemma ϕ D 2 = ϕ D 1 ϕ N 2. Proof Express everything in terms of X = cos 2π 3 x and Y = cos 2π 3 y. We have then to verify an equality between two polynomials of the variables X and Y. We deduce from the lemma that the counterexample for Neumann is identical to the counterexample for Dirichlet! The level sets of ϕ N 2 and ϕd 2 /ϕd 1 are the same.

38 Numerical simulations for Regular polygons. In (2D) Gladwell-Zhu were not successful for the square. One can be successfull for the hexagone for Neumann and for Dirichlet (Numerics or theory). We have also looked at the Rhombus. u Figure: Level lines of u1,d, u6,d and u6,d for the Dirichlet problem in the 1,D regular hexagon (communicated by V. Bonnaillie-Noe l) Bernard Helffer, Laboratoire de Mathe matiques Jean Leray, Universite On de Courant s Nantes. nodal domain property for linear combinations of ei

39 Figure: Level lines of w 6,D w 1,D heptagon for the Dirichlet problem in the regular

40 Regular examples for ECP. All the previous examples are singular (domains or surfaces with cracks), or have a nonsmooth boundary (polygonal domains). A natural question is whether one can construct counterexamples to ECP with a C boundary. Numerical simulation for the equilateral triangle with rounded corners (the corners of the triangle are replaced with circular caps tangent to the sides) suggest that this should be true. Note however that a triangle with rounded corners is C 1, not C 2.

41 The pictures in the first row of the figure below display the level sets and nodal domains of a second Neumann eigenfunction φ of the equilateral triangle with rounded corners, as calculated by matlab. The function is almost symmetric with respect to one of the axes of symmetry of the triangle. The pictures in the second row display the nodal sets of the function a + φ for two values of a. They provide a numerical evidence that ECP is not true for the equilateral triangle with rounded corners, and Neumann boundary condition.

42 Our theoretical result is the following Theorem There exists a one parameter family of C domains {Ω t, 0 < t < t 0 } in R 2, with the symmetry of the equilateral triangle T e, such that: 1. The family is strictly increasing, and Ω t tends to T e, in the sense of the Hausdorff distance, as t tends to For any t ]0, t 0 [, ECP(Ω t ) is false. More precisely,or each t, there exists a linear combination of a symmetric 2nd Neumann eigenfunction and a 1st Neumann eigenfunction of Ω t, with exactly three nodal domains.

43 As mentioned before, for the equilateral triangle T e, ECP(T e, a) is false for both the Dirichlet, and the Neumann boundary conditions. The idea of the proof of our theorem is to show that one can find a deformation of T e such that the symmetric second Neumann eigenfunction deforms nicely. For this purpose, we revisit a deformation argument given by Jerison and Nadirashsvili (2000) in the framework of the hot spots conjecture 1. This argument permits to control with respect to t the nodal deformation of a suitable symmetric eigenfunction. Note that the symmetry considered by Jerison and Nadirashvili is different (two perpendicular axes). 1 The hot spots conjecture says that the eigenfunction corresponding to the second eigenvalue of the Laplacian with Neumann boundary conditions attains its maximum and minimum on the boundary of the domain.

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