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On Courant’s nodal domain property for linear combinations of eigenfunctions (after P. B´ erard and B. Helffer). March 2019 Conference in honour of B.W. Schulze for his 75-th birthday.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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

• f 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 Rd, S2

• r T2. We also discuss the existence of a counterexample in a

C ∞, convex domain Ω in R2 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. B´ erard.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Introduction

Let Ω ⊂ Rd 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

• n ∂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.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Call H(Ω, B) the associated self-adjoint extension of −∆, and list its eigenvalues in nondecreasing order, counting multiplicities, 0 ≤ λ1(Ω, B) < λ2(Ω, B) ≤ λ3(Ω, B) ≤ · · · (2) For any integer n ≥ 1, define the index τ(Ω, B, λn) = min{k | λk(Ω, B) = λn(Ω, B)}. (3) E(λn) will denote the eigenspace associated with λn.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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 β0(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(λn(Ω, B)) , β0(u) ≤ τ(Ω, B, λn) ≤ n . (5) Courant’s nodal domain theorem can be found in Courant-Hilbert [10].

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

The extended Courant nodal property

Given r > 0, denote by L(Ω, B, r) the space L(Ω, B, r) =   

• λj(Ω,B)≤r

cj uj | cj ∈ R, uj ∈ Eλj(Ω,B)    . (6)

Extended Courant Property:= (ECP)

We say that v ∈ L (Ω, B, λn(Ω, B)) satisfies (ECP) if β0(v) ≤ τ(Ω, B, λn) . (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 (G¨

• ttingen, 1932) [16].

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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. ˚
• A. 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.”

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Historical remarks: V. Arnold (1973-1979)

3. As pointed out by V. Arnold [1], when Ω = Sd, (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 S3, and any degree larger than

• r equal to 6, [32].

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

And V. Arnold continues: The point is that for the sphere S2 (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).

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

... 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.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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 R2, 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.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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 Rd, S2 or R3, including convex domains. No algebraic topology will be involved.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Some geometrical statements

There are two statements of V. Arnold for which no proof is available.

◮ ECP is true for the sphere S2. ◮ ECP does not hold for other metrics on the sphere.

Outside the (1D)-case, the only known result is for RP2 and was

• btained by J. Leydold.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Rectangle with a crack

Let R0 be the rectangle ]0, 4π[×]0, 2π[. For 0 < a ≤ 1, let Ca :=]0, a] × {π} and Ra := R0 \ Ca and consider the Neumann

• condition. The setting is described in Dauge-Helffer [11].

We call

• 0 = ν1(0) < ν2(0) < ν3(0) = ν4(0) ≤ · · ·

(8) the Neumann eigenvalues of −∆ in R0 . They are given by the m2

16 + n2 4 for pairs (m, n) of non-negative

integers. Corresponding eigenfunctions are products of cosines. Similarly, the Neumann eigenvalues of −∆ in Ra are denoted by 0 = ν1(a) < ν2(a) ≤ ν3(a) ≤ · · · . (9)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

The first three Neumann eigenvalues for the rectangle R0 are as follows. ν1(0) (0, 0) ψ1(x, y) = 1 ν2(0)

1 16

(1, 0) ψ2(x, y) = cos( x

4)

ν3(0) (0, 1) ψ3(x, y) = cos( y

2)

ν4(0)

1 4

(2, 0) ψ4(x, y) = cos( x

2)

(10)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Dauge-Helffer (1993) prove:

Theorem

For i ≥ 1,

• 1. [0, 1] ∋ a → νi(a) is non-increasing.
• 2. ]0, 1[∋ a → νi(a), is continuous.
• 3. lima→0+ νi(a) = νi(0).

It follows that for 0 < a, small enough, we have 0 = ν1(a) = ν1(0) < ν2(a) ≤ ν2(0) < ν3(a) ≤ ν4(a) ≤ ν3(0) = ν4(0) . (11)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Observe that for i = 1 and 2, ∂ψi

∂y (x, y) = 0. Hence for a small

enough, ψ1 and ψ2 are the first two eigenfunctions for Ra with the Neumann condition with associated eigenvalues ν1(a) = 0 and ν2(a) = 1 4 < ν3(a) . We have αψ1(x, y) + ψ2(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

• r three nodal domains in Ra.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Figure: Rectangle with a crack (Neumann condition)

This proves that (ECP) is false in Ra with Neumann condition. Notice that we can introduce several cracks {(x, bj) | 0 < x < aj}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.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Sphere S2 with cracks

On the round sphere S2, we consider the geodesic lines (x, y, z) → ( √ 1 − z2 cos θi, √ 1 − z2 sin θi, z) through the north pole (0, 0, 1), with distinct θi ∈ [0, π[. Removing the geodesic segments θ0 = 0 and θ2 = π

2 with

1 − z ≤ a ≤ 1, we obtain a sphere S2

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.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

The function z is also an eigenfunction of S2

a with eigenvalue 2.

For a small enough, λ4(a) = 2, with eigenfunction z. For 0 < b < a, the linear combination z − b has five nodal domains in S2

a, see Figure below in spherical coordinates.

It follows that (ECP) does not hold on the sphere with cracks.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

The hypercube with Dirichlet boundary condition

We can adapt the method of Gladwell-Zhu in any dimension. Let Cn(π) :=]0, π[ n be the hypercube of dimension n, with either the Dirichlet or Neumann boundary condition on ∂Cn(π). A point in Cn(π) is denoted by x = (x1, . . . , xn). A complete set of eigenfunctions of −∆ for (Cn(π), d) is given by the functions

n

• j=1

sin(kj xj) with eigenvalue

n

• j=1

k2

j ,

for kj ∈ N\{0} , (12) for x = (x1, . . . , xn) ∈]0, π[n. A complete set of eigenfunctions of −∆ for (Cn(π), n) is given by the functions

n

• j=1

cos(kj xj) with eigenvalue

n

• j=1

k2

j ,

for kj ∈ N . (13)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

We make use of the classical Chebyshev polynomials Uk(t), k ∈ N, defined by the relation, sin ((k + 1)t) = sin(t) Uk (cos(t)) , and such that U0(t) = 1, U1(t) = 2t, U2(t) = 4t2 − 1 . The first Dirichlet eigenvalues (as points in the spectrum) are listed in the following table, together with their multiplicities, and eigenfunctions.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Table: First Dirichlet eigenvalues of Cn(π)

Eigenv. Mult. Eigenfunctions n 1 φ1(x) := n

j=1 sin(xj)

n + 3 n φ1(x) U1 (cos(xi)) for 1 ≤ i ≤ n n + 6

n(n−1) 2

φ1(x) U1 (cos(xi)) U1 (cos(xj)) for 1 ≤ i < j ≤ n n + 8 n φ1(x) U2 (cos(xi)) 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) 2 + 2 . (14)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

In order to study the nodal set of the above eigenfunctions or linear combinations thereof, we use the diffeomorphism (x1, . . . , xn) → (ξ1 = cos(x1), . . . , ξn = cos(xn)) , (15) from ]0, π[ onto ] − 1, 1[, and factor out the function φ1 which does not vanish in the open hypercube. We consider the function Ψa(ξ1, . . . , ξn) = ξ2

1 + · · · + ξ2 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 2n + 1 nodal domains, see Figure 3 in dimension 3. For n ≥ 3, we have 2n + 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.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Proposition

For n ≥ 3, the hypercube of dimension n, with Dirichlet boundary condition, provides a counterexample to ECP.

Figure: 3-dimensional cube

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

The equilateral triangle (Dirichlet or Neumann)

Let Te denote the equilateral triangle with sides 1, see Figure 4. The eigenvalues and eigenfunctions of Te, with either Dirichlet or Neumann condition on the boundary ∂Te, 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 Te = [OAB]

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Neumann boundary condition

The sequence of Neumann eigenvalues of the equilateral triangle Te begins as follows, 0 = λ1(Te, N) < 16π2 9 = λ2(Te, N) = λ3(Te, N) < λ4(Te, N) . (16) The second eigenspace has dimension 2, and contains one invariant eigenfunction ϕN

2 under the mirror symmetry w.r.t OM, and

another anti-invariant eigenfunction ϕN

3 .

ϕN

2 is given by

ϕN

2 (x, y) = 2 cos

2πx 3 cos 2πx 3

• + cos

2πy √ 3

• − 1 . (17)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

The set {ϕN

2 + 1 = 0} consists of the two line segments

{x = 3

4} ∩ Te and {x +

√ 3y = 3

2} ∩ Te, which meet at the point

( 3

4, √ 3 4 ) on ∂Te. The sets {ϕ2 + 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 ϕ2 + a in Te jumps from 2 to 3, with the jump occurring for a = 1.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Figure: Levels sets {ϕN

2 + a = 0} for a ∈ {0 ; 0.9 ; 1 ; 1.1}

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

If we take the coordinates X = cos 2π

3 x and Y = cos 2π √ 3y we are

reduced to the level sets of (X, Y ) → X(X + Y ):

Figure: Levels sets in the X, Y variables

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

It follows that ϕN

2 + a = 0, for 1 ≤ a ≤ 1.2, provides a

counterexample to the Extended Courant Property for the equilateral triangle with Neumann boundary condition.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Dirichlet boundary condition

The sequence of Dirichlet eigenvalues of the equilateral triangle Te begins as follows, δ1(Te) = 16π2 3 < δ2(Te) = δ3(Te) = 112π2 9 < δ4(Te). (18) Up to scaling, the first eigenfunction ϕD

1 is given by

ϕD

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 Te .

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

A surprising formula.

The second eigenvalue has multiplicity 2, with one eigenfunction ϕD

2 symmetric with respect to the median OM, and the other ϕD 3

anti-symmetric. Up to scaling, ϕD

2 is given by

ϕD

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)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

First astonished by some numerics, we arrive to the conclusion that the following surprising result could be true:

Lemma

ϕD

2 = −ϕD 1 ϕN 2 .

Proof Express everything in terms of X = cos 2π

3 x and

Y = cos 2π

√

• 3y. 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 ϕN

2 and ϕD 2 /ϕD 1 are the same.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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.

Figure: Level lines of u1,D, u6,D and u6,D

u1,D for the Dirichlet problem in the

regular hexagon (communicated by V. Bonnaillie-No¨ el)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Figure: Level lines of w6,D

w1,D for the Dirichlet problem in the regular

heptagon

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Our theoretical result is the following

Theorem

There exists a one parameter family of C ∞ domains {Ωt, 0 < t < t0} in R2, with the symmetry of the equilateral triangle Te, such that:

• 1. The family is strictly increasing, and Ωt tends to Te, in the

sense of the Hausdorff distance, as t tends to 0.

• 2. For any t ∈]0, t0[, ECP(Ωt) is false.

More precisely,or each t, there exists a linear combination of a symmetric 2nd Neumann eigenfunction and a 1st Neumann eigenfunction of Ωt, with exactly three nodal domains.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

As mentioned before, for the equilateral triangle Te, ECP(Te, 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 Te 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” conjecture1. 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).

1The ”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.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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