On Pleijel's nodal domain theorem for the Robin problem

We prove an improved Pleijel nodal domain theorem for the Robin eigenvalue problem. In particular, we remove the restriction, imposed in previous work, that the Robin parameter be non‐negative. We also improve the upper bound in the statement of the Pleijel theorem. In the particular example of a Euclidean ball, we calculate the explicit value of the Pleijel constant for a generic constant Robin parameter, and we show that it is equal to the Pleijel constant for the Dirichlet Laplacian on a Euclidean ball.


Introduction
Let Ω ⊆ R d be an open, bounded, and connected domain with C 1,1 boundary, and let h ∈ L ∞ (Ω).We consider the eigenvalue problem for the Robin Laplacian where ∆ = − div grad is the positive Laplacian and n is the unit outward normal along ∂Ω.Its spectrum consists of a discrete sequence of real numbers bounded from below, each with finite multiplicity.We denote its eigenvalues by {µ k } ∞ k=1 .Let {u k } ∞ k=1 be a corresponding sequence of eigenfunctions which form an orthonormal basis of L 2 (Ω).Finally, let N k be the number of nodal domains of u k , which is the number of connected components of Ω \ u −1 k (0).We use the notation ∆ R,h Ω to refer to the Robin Laplacian in (1).By the Courant nodal domain theorem we know that N k ≤ k for any k ≥ 1.
In 1956, Pleijel [Ple56] studied the asymptotic behaviour of the number of the nodal domains of Dirichlet eigenfunctions in a planar domain.Let Ω ⊆ R 2 be open, bounded, connected, and Jordan measurable.Let N D k denote the number of nodal domains of the eigenfunction of the Laplacian on Ω corresponding to the kth Dirichlet eigenvalue (with multiplicity).The Pleijel nodal domain theorem states that lim sup This theorem was extended to the manifold setting in dimension two by Peetre [Pee57] and to any dimension by Bérard and Meyer [BM82], with the constant γ(2) replaced by , where j (d−2)/2 is the smallest positive zero of the Bessel function J (d−2)/2 .Note that γ(d) < 1 for d ≥ 2 [BM82].More recently Bourgain [Bou15], Donnelly [Don14], and Steinerberger [Ste14] showed that γ(d) is not an optimal upper bound.Their results give an improved version of the Pleijel theorem, i.e. lim sup where ǫ(d) is a positive constant depending only on d.These results hold for the Dirichlet Laplacian as well as for the Laplacian on a closed manifold.The first result in extending the Pleijel theorem to other boundary conditions is due to I. Polterovich [Pol09], who extended the Pleijel theorem to the Neumann Laplacian on domains Ω ⊂ R 2 with analytic boundary.Léna [Lén19] extended the Pleijel theorem to the Robin problem with non-negative parameter (which includes the Neumann problem) on a bounded and C 1,1 domain Ω in any dimension, using a different approach.
The goal of this paper is twofold.First, we prove an improved Pleijel theorem for the Robin problem (1) without any assumption on the sign of h.Our result extends Léna's result in two ways.It removes the assumption that h is non-negative.In addition, it shows that the constant in the Pleijel nodal domain theorem for the Robin problem can be improved.
Second, we calculate the exact Pleijel constant for the Robin problem on the Euclidean unit ball, with h being a constant (of either sign), and show that for generic h it is the same as the Pleijel constant for the Dirichlet problem on the same Euclidean unit ball.
Our first main result is the following.
Remark 1.2.The statement of Theorem 1.1 holds if Ω is an open, bounded, connected domain with C 1,1 boundary in a complete Riemannian manifold, see Theorem 2.5 below.
The proof of Theorem 1.1 closely follows the proof of the Pleijel nodal domain theorem for the Robin problem with positive parameter by Léna [Lén19], incorporating the proof of the improved Pleijel theorem by Bourgain [Bou15], Donnelly [Don14] and Steinerberger [Ste14].In particular, the methods of the latter three authors are applied to a subset of nodal domains which Léna calls the bulk domains.These are the nodal domains for which the L 2 -norm of the corresponding eigenfunction is concentrated away from the boundary.
The proof uses C 1,1 regularity of the boundary, and it is open whether the Pleijel theorem holds under weaker regularity assumptions.For example, the square is not a C 1,1 domain, and the Pleijel theorem for the Robin Laplacian on the square remains open.The only available result in this setting is the study of bounds on the number of Courant-sharp Robin eigenvalues.A Robin eigenvalue is called Courant-sharp if it has a corresponding eigenfunction with exactly k nodal domains; note that an immediate consequence of the Pleijel theorem for C 1,1 domains is that the number of Courant-sharp Robin eigenvlaues is finite.Gittins and Helffer [GH19,GH21] studied upper bounds on the number of Courant-sharp eigenvalues of the Robin problem on a square when the Robin parameter h is constant.In particular, they show that the Robin Laplacian on a square with constant parameter h has finitely many Courant-sharp eigenvalues.
We also note that a bound on the number of Courant-sharp eigenvalues of the Robin problem with non-negative parameter on domains with C 2 boundary was obtained by Gittins and Léna [GL20].They made use of monotonicity results between Robin, Dirichlet and Neumann eigenvalues which only hold for non-negative Robin parameter.
The second part of this paper deals with calculating the Pleijel constant for the Robin eigenvalue problem ∆ where the supremum is taken over all planar domains with regular boundary (e.g.Lipschitz).If the conjecture is true, the inequality is sharp, as Polterovich has shown that the upper bound is attained for rectangles [Pol09].
The value of the Pleijel constant for a given domain is generally unknown.Bobkov [Bob18] calculated Pl(∆ D Ω ) for some simple domains, including the Euclidean disk in R 2 .When Ω = B is a Euclidean disk in R 2 , he showed that Pl(∆ D B ) = 0.461 • • • We show that in the example of the unit disk, with constant Robin parameter h = σ, the value of the Pleijel constant is generically independent of the boundary condition.More precisely, we introduce the following condition on σ: Assumption 1.3.The positive roots of the equation zJ ′ m+d/2−1 (z) where J ν (z) is the Bessel function, do not coincide for different values of m ∈ N 0 .
This condition should be thought of as a variant of Bourget's hypothesis [Wat44, p. 485].It is generic in the sense that it may fail for at most a countable set of σ ∈ R, as we show in Proposition 3.2.Under this assumption we are able to prove that the Pleijel constant for the Robin problem is the same as for the Dirichlet problem.
Theorem 1.4.Let B be a Euclidean ball in R d .Assume that h = σ and that Assumption 1.3 holds for σ.Then Pl(∆ R,σ B ) = Pl(∆ D B ).In particular for d = 2 we have Pl(∆ R,σ B ) = Pl(∆ D B ) = 0.461 • • • .The proof begins by using separation of variables to write down the Robin spectrum.Assumption 1.3 shows that there is no accidental coincidence resulting in mixing of spherical harmonics from different eigenspaces, and thus each Robin eigenfunction is a spherical harmonic in the angular variables times a Bessel function in the radial variable.We show using Bessel function asymptotics that for each Robin eigenfunction, there is a corresponding Dirichlet eigenfunction with the same nodal count, and with an eigenvalue sufficiently close to the Robin eigenvalue.This is enough to prove the theorem.
It is an intriguing question to investigate the independence of the Pleijel constant from the boundary condition.More precisely Open Question 1.5.Can we show that Pl(∆ R,h Ω ) = Pl(∆ D Ω ) for any bounded connected domain Ω with regular enough boundary (e.g.Lipschitz or C 1,1 boundary)?

Proof of Theorem 1.1
The proof of Theorem 1.1 consists of two parts: the first part relies on the proof of the Pleijel theorem for the Robin problem with non-negative parameter by Léna [Lén19], and the second part on the proof of the improved Pleijel theorem by Bourgain [Bou15], Donnelly [Don14] and Steinerberger [Ste14].Let us first review the main ideas of the previous work.
The classical proof of the Pleijel theorem for the Dirichlet Laplacian uses only the Faber-Krahn inequality (for all nodal domains) and the Weyl law.However, for the Neumann and Robin eigenvalue problems, the Faber-Krahn inequality does not hold for the nodal domains which are adjacent to the boundary.For bounded planar domains with analytic boundary, I. Polterovich obtained the Pleijel nodal domain theorem using an estimate by Toth and Zelditch [TZ09] that the growth rate of the number of nodal domains adjacent to the boundary is smaller than the index k.Léna extended the Pleijel theorem to the Robin problem with non-negative parameter on a bounded domain Ω ⊂ R n with C 1,1 boundary using a completely different approach.
Since our proof relies on Léna's proof, we briefly explain the main ideas.Léna considered two families of nodal domains, which he called the bulk and boundary nodal domains.A nodal domain D for which the L 2 -norm of the eigenfunction on D concentrates away from the boundary of Ω is called a bulk nodal domain, while one for which the L 2 -norm of the eigenfunction on D has significant mass near the boundary of Ω is referred to as a boundary nodal domain.He then showed that the growth rate of the number of boundary nodal domains is lower than k.Thus the boundary nodal domains do not contribute to the limit of the fraction N k /k as k → ∞.We use the same decomposition of nodal domains into bulk and boundary domains and extend Léna's result without imposing any constraint on the Robin parameter h.Roughly speaking, we show that the contribution from h − = max{−h, 0} contributes to the lower order terms but disappears when divided by k in the limit.
To get an improved version of the Pleijel theorem for the Laplacian, the main idea (as in [Bou15,Don14,Ste14]) is to follow the original proof of the Pleijel theorem but use a quantitative version of the Faber-Krahn inequality and show that it leads to an improvement of the upper bound in the Pleijel theorem.We adapt this idea for the Robin eigenvalue problem by applying a quantitative version of the Faber-Krahn inequality for the bulk domains only.
We begin with the same general approach as outlined in [Lén19].We first give two key propositions concerning the eigenfunctions.The first involves boundary regularity, generalizing [Lén19, Proposition 1.6].Throughout this section, we will mainly use the same notations as in [Lén19].
Ω is an element of C 1 (Ω).Proof.The statement follows from some elliptic regularity arguments.The same line of argument is used to obtain the same statement for Steklov eigenfuctions by Decio [Dec22], and for Robin eigenfunctions with positive parameter by Léna [Lén19].
Specifically, let u be an eigenfunction of ∆ R,h Ω with eigenvalue µ.Then u ∈ H 1 (Ω) (see e.g.[Dan09]), and its trace is in , and g = −hu, we conclude that u ∈ W 2,2 (Ω).We get u ∈ W 2,p (Ω) for all p < ∞ by bootstrapping the argument, and conclude that u ∈ C 1 ( Ω) by the Sobolev embedding theorem.We refer the reader to the proof of [Dec22, Proposition 2] for details.
The second is a replacement for Green's formula, generalizing [Lén19, Propositions 1.7 and 4.3].
Proposition 2.2.Let H = h − ∞ where h − (x) = max{−h(x), 0}.There exists a constant C depending only on Ω such that if u is any eigenfunction of ∆ R,h Ω with eigenvalue µ, and D is any nodal domain of u, then Let α > 0 be such that α is a regular value of both ũ and ũ| ∂Ω .This ensures that [Lén19] for details).Note that by Sard's theorem there exists an infinite sequence of such regular values approaching zero.We can now apply Green's theorem to the function Plugging in the information we know, Since u and u α are both positive on Γ α and on D α , we conclude that Step 2. Let us bound the second term in the right-hand side of inequality (4) in terms of the L 2 -norms of u and ∇u.Let ρ(x) be a function on Ω which agrees with dist(x, ∂Ω) in a neighborhood of the boundary ∂Ω.By Lemma 2.3, below, we have that dist(x, ∂Ω) is C 1,1 on a neighbourhood of ∂Ω.Thus, we can assume that ρ ∈ C 1,1 (Ω).Then by Green's theorem, By the Cauchy-Schwarz inequality, Plugging this into (4) and dividing through by Dα u 2 dx, we get Let us denote Here, The result of Proposition 2.2 now follows by taking the limit as α tends to zero along the sequence of regular values guaranteed by Sard's theorem.Note that both u and |∇u| are continuous on Ω by Proposition 2.1.
We now prove the regularity of the distance function on a neighborhood of ∂Ω which is used in the second step of the proof of Proposition 2.2 above.
We say ∂Ω has a positive reach if there exists δ > 0 such that for any x ∈ ∂Ω δ := {y ∈ R n : dist(y, ∂Ω) < δ}, there exists a unique nearest point in ∂Ω.We denote the supremum of such δ by reach(∂Ω).We refer to [Fed59, page 432] for more details on sets with positive reach.
Lemma 2.3.Let Ω ⊂ R n be an open bounded connected domain with C 1,1 boundary.Then ∂Ω has a positive reach and dist(x, ∂Ω) is C 1,1 on a neighbourhood of ∂Ω.
In [KP81], Krantz and Parks showed that if ∂Ω is C 1 and has a positive reach then dist(x, ∂Ω) is also C 1 .Note that assuming ∂Ω is C 1 is not enough to conclude that it has a positive reach, see [KP81,Example 4].The proof of Lemma 2.3 follows the same idea of the proof of [KP81, Theorem 2] together with the results of [Fed59, Section 4].
Proof of Lemma 2.3.The fact that reach(∂Ω) > 0 is a consequence of [Fed59, Theorem 4.12].For the regularity of the distance function, we repeat the same argument as in the proof of [KP81, Theorem 2].Namely, for any p ∈ ∂Ω, we choose a coordinate system putting p at the origin such that in a neighbourhood U of p = 0 we have with f (0) = 0 and ∇f (0) = 0. Let δ ∈ (0, reach(∂Ω)) be sufficiently small so that for every (x, y) ∈ B n δ (0), there exists a unique nearest point (t, (5) Using (5), we can write . By plugging this into (6) and using (5), we get The right-hand side of (7) is C 0,1 .Therefore by [Fed59, Theorem 4.7] we conclude that the above identity holds on B n δ (0), and thus that d is C 1,1 on B n δ (0).Since ∂Ω is compact, the result follows by taking a finite open cover.
Remark 2.4.Although Propositions 3.2 and 2.2 are stated in the Euclidean setting, they continue to hold for a bounded C 1,1 domain in a complete Riemannian manifold.Indeed the argument showing the regularity of the eigenfunctions in Proposition 2.1 and the Green's formula used in the proof of Proposition 2.2 clearly extend to the Riemannian setting.Moreover, the proof of the regularity of ρ is verbatim as in the proof of Lemma 2.3 if ∂Ω has a positive reach, which itself follows from [Fed59, Theorem 4.12] by considering a finite cover of ∂Ω.
where A(D) is the Fraenkel asymmetry and | • | denotes the n−volume of the set.We refer to the proof of Claim 2 below where the choices of ǫ 1 and ǫ 2 are discussed.Crucially, we emphasise that their values are independent of ǫ.Recall that Let N ⋆ be the number of ǫ-bulk nodal domains satisfying condition ⋆ = I, II or II.Note that N B (k, ǫ) = N I + N II + N III .
For any ǫ-bulk nodal domain D, we have the following upper bound for λ D 1 (D) calculated in [Lén19, page 291]: where C 0 is a constant independent of µ k .Applying Proposition 2.2 we get where We first estimate the number ǫ-bulk nodal domains in each category I, II, III.
Case I.For any ǫ-bulk nodal domain D j of type I, we trivially have After summing over all ǫ-bulk nodal domains of type I and rearranging inequality (10), we get where Ω I is the union of all ǫ-bulk nodal domains of type I.
Case II.For any ǫ-bulk nodal domain D j of type II, we use the quantitative Faber-Krahn inequality [BDPV15] to get ) where ǫ 3 = c(n)ǫ 2 2 .We use inequality (9) and sum over all type II ǫ-bulk nodal domains to get where Ω II is the union ǫ-bulk nodal domains of type II.
Case III.For any ǫ-bulk nodal domain D j of type III, the Faber-Krahn inequality implies that We use inequality (9) and sum over all ǫ-bulk nodal domains of type III to get where Ω III is the union ǫ-bulk nodal domains of type III.Now let us recall the Weyl asymptotic for the Robin problem (see e.g.[BS80, FG12]): and the definition of γ(n), where α ⋆ = |Ω⋆| |Ω| , ⋆ = I, II, III.
To conclude the proof, we need to prove the following statements.
Claim 1.For the ǫ-boundary nodal domains we have lim sup Proof of Claim 1.The proof follows from [Lén19, Section 2.4] and we do not repeat the details of the calculation.The main idea is to extend u 1 by reflection to a function v on a neighbourhood of each ǫ-boundary nodal domain and apply the Faber-Krahn inequality on each connected component of the interior of the support of v. Here, the regularity of the eigenfunction discussed in Proposition 2.1 is needed.
Then we can get an estimate for N b (k, ǫ): where For any fixed ǫ > 0, the leading term in the right-hand side of (16) goes to zero as k → ∞ because of the Weyl law (15), completing the proof of Claim 1.
Proof of Claim 2. The proof of the claim follows by adapting the argument given in [Don14, Page 59].Recall that for any nodal domain D of type III, we have The second inequality means that there exists an approximate ball B such that |D| = |B| and |D∆B| < ǫ 2 |B|.We now show that for any pair of type III nodal domains D and D, their approximate balls B and B almost have the same radius and their overlap can be made small if we choose ǫ, ǫ 1 , and ǫ 2 small enough.
By the Faber-Krahn inequality we have For any fixed ǫ, we use the estimate in (9).It implies that for k ≥ k(ǫ) we can have Putting ( 17) and ( 18) together, we get .
Hence, if we choose ǫ, ǫ 1 > 0 small enough, it guarantees that the approximate balls almost have the same radius.Moreover, we have , and by a choice of ǫ 2 small enough, their overlap can be made small.Note that the ǫ 1 and ǫ 2 can be chosen independent of ǫ.We can complete the argument by showing that the proportion of Ω covered by type III nodal domains is uniformly bounded away from 1.By the result of Rogers [Rog58], the packing density ρ(n) of balls of the same radius in R n is strictly less than one.Roughly speaking, we can obtain a packing for the domain Ω by slightly reducing the size of the balls.The ratio α III of the volume of type III nodal domains to the total volume is then approximately equal to (1 + o(1))ρ(n).See [Don14, Page 59] for more details.Therefore, lim sup By Claim 2, we have α III ≤ c(n) < 1.Note that ǫ 3 does not depend on ǫ.The statement of Theorem 1.1 follows by taking ǫ → 0.
Theorem 2.5.The statement of Theorem 1.1 holds for Ω a bounded open domain with C 1,1 boundary in a complete Riemannian manifold.
Proof.It is enough to show that the key ingredients of the proof in the Euclidean setting can be adapted to the Riemannian setting.From Remark 2.4, we know that the Propositions 2.1 and 2.2 hold in the Riemannian setting.We consider the decomposition of nodal domains into ǫ-bulk and ǫ-boundary nodal domains as defined in (8) and adapt the argument on pages 8-9 as in [Don14, Section 3].
The main idea is to use a weaker version of the Faber-Krahn inequality and its quantitative version, which hold only for ǫ-bulk nodal domains with volume smaller than a given threshold number.The number of ǫ-bulk nodal domains with volume larger than the threshold number is bounded independent of k and so does not contribute to the limit.Hence we can repeat the same argument as in the proof of Theorem 1.1.We refer the reader to [Don14, Section 3] for more details.

Pleijel constant for the Robin problem on a Euclidean ball
Throughout this section we will consider the Robin problem on a unit ball in R d , d ≥ 2. We assume throughout that h = σ is a constant, so our boundary condition is ∂ n u + σu = 0. Note that σ ≥ 0 yields non-negative spectrum, but we need not assume this.
We begin by writing down the spectrum of this problem for arbitrary σ ∈ R. To start, consider the Dirichlet spectrum of the ball in R d .As it is well known, it consists of the squares of zeroes of Bessel functions, specifically 1 and j ν,k is the kth positive zero of J ν (z).These eigenvalues have multiplicity κ m,d , where Note that when d = 2, κ m,d = 2 for all m ∈ N. The multiplicity κ m,d is simply the multiplicity of the space of spherical harmonics with eigenvalue m(m + d − 2).We now consider the Robin spectrum when σ ≥ 0. In this case, the eigenvalues are all non-negative.They are again with multiplicity κ m,d .However, the values x ν,k are not just Bessel function zeroes but are the kth non-negative roots of the equation There is exactly one such x ν,k in between each pair of roots of J ν (k) [Wat44,.Furthermore, x ν,1 < j ν,1 .For σ < 0, there are finitely many negative Robin eigenvalues.All of the values x 2 ν,k are still eigenvalues, regardless of the sign of σ.And for m ≥ −σ, we still have 0 ≤ x ν,1 < j ν,1 .However, when m < −σ, this no longer holds and instead x ν,1 > j ν,1 .The reason is that there is now a negative eigenvalue, specifically −(x ν,0 ) 2 , where x ν,0 is the unique positive root of the equation with I ν (z) the modified Bessel function.This eigenvalue has multiplicity κ m,d as before.Thus the number of negative Robin eigenvalues, with multiplicity, is To summarize, for m ≥ −σ, each x 2 ν,k is an eigenvalue, and 0 However, for m < −σ, each x 2 ν,k is an eigenvalue, but there is also a corresponding negative eigenvalue −x 2 ν,0 , and we have instead that 0 < j ν,1 < x ν,1 < • • • < x ν,k < j ν,k+1 < x ν,k+1 < . . .
When enumerating both Dirichlet and Robin spectra we have to put the eigenvalues in order.So let us now do this.Define the index function ι(m, k) to be the smallest possible index of the Robin eigenvalue µ m,k = x 2 m,k , with a similar definition for ι D (m, k).The words "smallest possible" are required because these eigenvalues have multiplicity, but nothing in the subsequent work would change even if different choices were made for each pair (m, k).
The following technical lemma is key to the proof of Theorem 1.4.
We defer the proof for now and instead use Lemma 3.1 to prove Theorem 1.4.
Proof of Theorem 1.4.By Assumption 1.3, all Robin eigenfunctions have the form for some pair (m, k), with f m (θ) being a spherical harmonic (an element of Y m (S d−1 )).The same is true for Dirichlet eigenfunctions with x ν,k replaced by j ν,k .Consider a sequence of Robin eigenfunctions u i , corresponding to eigenvalues µ ℓ i , for which By the previous paragraph, there is a sequence (m i , k i ) and a sequence of elements With this notation, ℓ i = ι(m i , k i ).Now consider the corresponding sequence of Dirichlet eigenfunctions We know that for each i, N (u i ) = N (w i ), as both are equal to the number of nodal domains of f m i (θ) times k i .And the index of u i as a Dirichlet eigenfunction is precisely ι D (m i , k i ).By Lemma 3.1, Thus any limit point of Robin nodal quotients is also a limit point of Dirichlet nodal quotients.Repeating the argument word for word shows that the reverse is true as well.Therefore the sets of limit points, and thus their suprema, are the same, completing the proof.
3.1.Proof of Lemma 3.1.We begin with a few preliminary observations.By Weyl's law for the Dirichlet eigenvalues, there is a nonzero constant c d such that as Additionally, the multiplicity coefficients κ m,d are polynomials of degree d − 2 in m.Thus there is a nonzero constant cd such that as N → ∞, By combining (20) and (21), we know that for each k > 1, When k = 1 we have the bound Finally we need some results on spacing of Bessel function zeroes.These well-known bounds follow immediately from, e.g., [Hor17, Theorem 2].For all ν ≥ 1/2 and all k, and when ν < 1/2, j ν,k+1 − j ν,k ≤ π.
Now we prove Lemma 3.1.
Proof of Lemma 3.1.Consider ι(m, k).We will find upper and lower bounds for it.
Begin with an upper bound.Letting η = ℓ + d/2 − 1 and using (24) we have Considering the j = 1 and j > 1 terms separately, Using (25) for the middle term and (24) for the last term, Rewriting the middle term and relabeling j − 1 as j in the last term gives the upper bound ι(m, k) Now divide by ι D (m, k) and consider the limit as x m,k goes to infinity (which is equivalent to j m,k going to infinity and happens for any infinite subsequence of distinct (m, k)).The first term on the right of (28) goes to zero when divided by ι D (m, k).The second term on the right of ( 28) is, by (23), We claim that j ν,k+1 /j ν,k → 1, which would show that the right-hand side of ( 29) is 1 + o(1).Indeed, it is equivalent to show that the following goes to zero: For any subsequence with ν < 1/2, this goes to zero because the numerator is bounded by π by ( 27) and the denominator goes to infinity.For all other such subsequences, (26) gives But asymptotics of Bessel zeroes together with the fact that j ν,1 > ν [ODL + 20, (10.21.Remark 3.3.Observe that even without making Assumption 1.3, there exists a basis of Robin eigenfunctions on a Euclidean ball which are separated solutions.For that basis, the proofs in this section show that Pleijel's theorem holds with the same constant as for the Dirichlet ball. Proof.quality (4) is the same as the proof of [Lén19, Propositions 1.7 and 4.3].Let ũ : U → R be a C 1 -extension of u to an open neighborhood U of Ω.Since u ∈ C 1 ( Ω) by Proposition 2.1, such an extension exists.Let D be the nodal domain of ũ containing D. For α > 0, we denote D α = D ∩ {u(x) > α}, Dα := D ∩ {ũ(x) > α}, and thus Γα u 2 ds ≤ Dα u 2 |∆ρ| dx + Dα 2|u| |∇u| |∇ρ| dx.Since ρ depends only on Ω and is in C 1,1 (Ω), there exists a constant C such that Γα u 2 ds ≤ C Dα u 2 dx + C Dα |u| |∇u| dx.