Friedlander's Eigenvalue Inequalities and the Dirichlet-to-neumann Semigroup

If Ω is any compact Lipschitz domain, possibly in a Riemannian manifold, with boundary Γ = ∂Ω, the Dirichlet-to-Neumann operator D λ is defined on L 2 (Γ) for any real λ. We prove a close relationship between the eigenvalues of D λ and those of the Robin Laplacian ∆µ, i.e. the Laplacian with Robin boundary conditions ∂ν u = µu. This is used to give another proof of the Friedlander inequalities between Neumann and Dirichlet eigenvalues, λ N k+1 ≤ λ D k , k ∈ N, and to sharpen the inequality to be strict, whenever Ω is a Lipschitz domain in R d. We give new counterexamples to these inequalities in the general Riemannian setting. Finally, we prove that the semigroup generated by −D λ , for λ sufficiently small or negative, is irreducible.


Introduction.
Let Ω ⊂ R d be a bounded domain with ∂Ω = Γ. Let λ D 1 < λ D 2 ≤ λ D 3 ≤ · · · and λ N 1 < λ N 2 ≤ λ N 3 ≤ · · · be the eigenvalues of the Dirichlet and Neumann Laplacians on Ω, respectively. There is a beautiful set of inequalities discovered by Friedlander [9] which compares the elements of these two lists, namely λ N k+1 ≤ λ D k for all k. (1.1) The fundamental tool in his proof is the Dirichlet-to-Neumann operator associated to ∆ − λ; his methods require that ∂Ω be at least C 1 . Friedlander's inequalities have attracted substantial attention since then, starting from a geometric recasting of his argument by the second author [19]. More recently, Filonov [8] discovered a substantially simpler proof of (1.1) based on the minimax characterization of eigenvalues, assuming only that Ω has finite measure and that the inclusion H 1 (Ω) ⊂ L 2 (Ω) be compact. An extension of Filonov's ideas by Gesztesy and Mitrea [10] provides a comparison between generalized Robin and Dirichlet eigenvalues, while Safarov [24] showed how to describe all of this in a purely abstract setting involving only quadratic forms on Hilbert spaces. The present paper is a substantially shortened version of the preprint [3], which apparently provided some motivation for [10], and hence should be placed before that paper in the chronology. We have decided to revise it for publication since we believe that the point of view espoused here is still of interest and should lead to further progress on some of the questions we consider. We return to the use of the Dirichlet-to-Neumann operator, formulated weakly so that our argument applies on Lipschitz domains. (This is still less general than the domains considered by Filonov.) Our starting point is the folklore observation that if λ and µ are real numbers, then µ is an eigenvalue of the Dirichlet-to-Neumann operator D λ associated to ∆ − λ if and only if λ is an eigenvalue of the Robin Laplacian ∆ µ , i.e. the operator ∆ on Ω with boundary condition ∂ ν u = µu. We prove that λ depends strictly monotonically on µ, and vice versa. This has been rediscovered several times before our proof of it in [3]; it is equivalent to the monotonicity for D λ used by Friedlander [9], see also [19], but traces back at least as far as the paper of Grégoire, Nédélec and Planchard [11] in the mid '70's, though they in turn attribute the idea to earlier unpublished work of Caseau. This relationship and monotonicity was known to S.T. Yau in the '70's as well. In any case, this is a lovely set of ideas which deserves to be more widely appreciated and utilized. We show here that it leads directly to yet another proof of (1.1). We also show that (1.1) need not be true for general manifolds with boundary. This was already discussed in [19], and the counterexample given there is any spherical cap larger than a hemisphere. We prove here that (1.1) also fails if Ω is the complement of a sufficiently small set in any closed manifold M .
Our second goal in this paper is to present some facts about the semigroup associated to the Dirichlet-to-Neumann operator D λ (for any λ < λ D 1 ). Specifically, we prove that it is positive and irreducible. While this is somewhat disjoint from the question of eigenvalue inequalities, the proof is yet another illustration of the close link between the Robin Laplacian and D λ . A consequence of this is that the first eigenvalue of D λ is simple and has a strictly positive eigenfunction. Note that this irreducibility of T requires only that Ω be connected, though its boundary may have several components. This reflects the non-local nature of D λ .
We mention also the recent paper [4] which considers a number of issues related to the ones here. For general information about eigenvalue problems we refer to [14] and [15].
We shall be brief since various of the papers cited above contain good expositions of all the background material needed here, as well as the history of eigenvalue inequalities preceding (1.1). The next section contains a short review of the correspondence between coercive symmetric forms and self-adjoint operators and the weak formulation of normal derivatives on Lipschitz domains, and then records the quadratic forms underlying the various operators we study in this paper. §3 describes the eigenvalue monotonicity and its application to the proof of the eigenvalue inequalities. The Dirichlet-to-Neumann semigroup is the subject of §4.

2.
The Robin Laplacian and Dirichlet-to-Neumann operator. Let H be an infinite dimensional separable Hilbert space and V another Hilbert space which is embedded as a dense subspace in H, so that V ⊂ H ⊂ V * . Suppose that a is a closed, symmetric, real-valued, coercive quadratic form, i.e.
for some ω ∈ R and α > 0. Associated to a is a bounded operator A 1 : V → V * . Also associated to a is an unbounded self-adjoint operator A 2 on H with domain D(A 2 ) ⊂ V ⊂ H. Thus x ∈ D(A 1 ) and A 1 x = y ∈ V * if and only if a(x, v) = y, v for all v ∈ V . The operator A 2 is the part of A 1 in D(A 2 ), and hence we simply write either operator as A and drop the subscript. The form a is accretive (i.e. a(u) ≥ 0 for all u ∈ V ) if and only if A is nonnegative (i.e. Au, u H ≥ 0 for all u ∈ D(A)). Furthermore, A has compact resolvent, and hence discrete spectrum, if and only if the inclusion D(A) → H is compact, which is certainly the case if V → H is compact. Assuming that this is so, then we denote by {e n , λ n } the eigendata for A, so the e n are an orthonormal basis for H, Ae n = λ n e n for all n, and λ 1 ≤ λ 2 ≤ · · · ∞. The standard max-min characterization of the eigenvalues is λ n = sup where G n−1 (V ) denotes the set of all subspaces of V of codimension n − 1. Let (Ω, g) be a compact Riemannian manifold with Lipschitz boundary. In other words, we assume that Ω is a connected, compact subset in a larger smooth manifold M , that the metric g on Ω is the restriction of a smooth metric on M , and that Γ = ∂Ω is locally a Lipschitz graph such that Ω lies locally on one side of Γ. (The results below extend in a straightforward manner if we only assume that M has a C 1,1 structure and that the metric g is Lipschitz.) We refer to [12], [13], [17], [18] for more about the (straightforward) generalizations of the analytic facts used in this paper from the setting of Lipschitz domains in R d to domains in manifolds.
The volume form and gradient for g lead naturally to the Hilbert spaces L 2 (Ω) and H 1 (Ω), as well as the space L 2 (Γ). As usual, H 1 0 (Ω) is the closure of C ∞ 0 (Ω) in H 1 (Ω). The boundary restriction map u → u| Γ := Tr u is well-defined for any u ∈ H 1 (Ω) ∩ C 0 (Ω), and this map extends to a bounded operator Tr : H 1 (Ω) → L 2 (Γ), with nullspace H 1 0 (Ω). We write u |Γ or Tr u interchangeably. We next recall the weak formulations of well-known operators and identities. a) If u ∈ H 1 (Ω), we say that ∆u ∈ L 2 (Ω) if there exists f ∈ L 2 (Ω) such that and we then write ∂ ν u = b. To be explicit, our conventions are that ∆ = −div ∇ and ν is the outer unit normal; also, dV g and dσ g are the volume forms on Ω and Γ associated to g. Here and later we often omit the trace signs under the integral, e.g. simply write Γ bv = Γ bv |Γ . These definitions are set so that Green's formula still holds: Consider the form, for any µ ∈ R, for u, v ∈ H 1 (Ω). It is not hard to show that b µ is coercive, and hence determines an operator ∆ µ . Letting v ∈ H 1 0 (Ω) shows that ∆ µ is just the standard Laplacian in the interior, and we then deduce that u ∈ D(∆ µ ) implies ∂ ν u = µu, at least in the weak sense. Thus, altogether, The special case µ = 0 corresponds to the Neumann Laplacian ∆ N .
We next consider the form The discussion in the next section motivates why the moniker b −∞ is reasonable. The coercivity of this form is obvious, and its corresponding operator is the Dirichlet Laplacian ∆ D .
Since H 1 (Ω) is compactly included in L 2 (Ω), each of these operators has discrete spectrum. We write . Hence the Robin eigenvalues interpolate between the Dirichlet and Neumann eigenvalues.
We now define, for each λ ∈ R, the Dirichlet-to-Neumann operator D λ . If λ ∈ R\σ(∆ D ), then the classical definition is that if g is a (sufficiently smooth) function on Γ and u is the unique function on Ω such that (∆ − λ)u = 0, Tr u = g, then D λ g = ∂ ν u |Γ . Note that u is indeed uniquely defined if and only if λ / ∈ σ(∆ D ). There are several equivalent ways to circumvent this apparent need to avoid the Dirichlet eigenvalues. The first and most classical is simply to consider the Cauchy data subspace, sometimes also called the Calderon subspace, which is defined for any λ ∈ R by It follows from Proposition 1 below that C λ is a closed subspace of L 2 (Γ) × L 2 (Γ). If λ / ∈ σ(∆ D ), then C λ intersects {0} × L 2 (Γ) only at the origin, and hence there is a densely defined closed operator D λ on L 2 (Γ) for which C λ is the graph.
We may also consider C λ as a multi-valued selfadjoint operator when λ ∈ σ(∆ D ). In order to avoid this, we define D λ as follows. Let λ ∈ σ(∆ D ) and define K(λ) := {h ∈ L 2 (Γ) : (0, h) ∈ C λ }; clearly ). In this way, the operator D λ is defined for all λ ∈ R. It follows from our definition of the normal derivative that D λ is symmetric. In order to show that D λ is self-adjoint (i.e., that (is − D λ ) is invertible for s ∈ R \ {0}), we use the following result by Grégoire, Nédélec and Planchard [11,Proposition 1].
This solution u is uniquely determined by the condition that . The relationship with D λ is as follows. Proposition 2. The operator D λ is selfadjoint for every λ ∈ R. In fact, for s ∈ R \ {0} the resolvent is given by ). In particular, D λ has compact resolvent.
We will see later, in Theorem 3.1, that the operator D λ is bounded below. Thus its spectrum consists of eigenvalues α k (λ), k = 1, 2, . . . , which we arrange in increasing order repeated according to multiplicity. Remark 1. Using Proposition 1 Grégoire et al. [11] define the unitary operator B(s) on L 2 (Γ): for λ = s 2 (where C λ is considered as a multi-valued operator, which is such that its resolvent is single-valued). Hence as λ increases, the poles of D λ as λ crosses a Dirichlet eigenvalue transform to a more innocuous spectral flow across the value 1.
We conclude this discussion by an alternative form definition of D λ in the case where λ ∈ σ(∆ D ).
Since Tr : Then a λ is a closed, symmetric form on L 2 (Γ) and D λ is the associated self-adjoint operator. We refer to [3] for more details.
We now describe how the Robin eigenvalues λ k (µ) vary with µ.
Proposition 3. For each k, the function λ k (µ) is strictly decreasing and satisfies Proof. It follows from the definition of b µ and the max-min definition of eigenvalues that λ k is at least nonincreasing. To see that it decreases strictly, suppose that λ k (µ 1 ) = λ k (µ 2 ) for some µ 1 < µ 2 . Then setting λ := λ k (µ 1 ), it follows from Theorem 3.1 that µ ∈ σ(D λ ) for all µ ∈ [µ 1 , µ 2 ]. But this is impossible since σ(D λ ) is discrete. Standard eigenvalue perturbation theory shows that each λ k is continuous in µ, and is even analytic if one follows the eigenvalue branches correctly across their crossings, see [16]). We refer to [3,Theorem 2.4] for the proof that lim On the other hand, if there exist k ∈ N and λ ∈ R such that λ k (µ) > λ > −∞ for all µ ∈ R, then by Theorem 3.1, for that value of λ, σ(D λ ) ⊂ {µ ∈ R, λ j (µ) = λ, j = 1, . . . k − 1}, which is a finite set. This is impossible.
We are now almost in a position to reprove the Friedlander eigenvalue inequalities.
This same set of test functions was used in [9] and later in [8] for the same purpose.
Using this Lemma and Theorem 3.1, we now obtain the strict Friedlander inequalities.
A quantitative version of this inequality which appears in [9] for λ ∈ σ(∆ D ) can be proved by similar considerations.
We now consider the functions µ k : (−∞, λ D k ) → R which are the inverses of the λ k (µ), k = 1, 2, . . .. These are well-defined by the strict monotonicity of the λ k , of course, and each µ k is continuous (see [3]), strictly decreasing and satisfies Theorem 3.1 now gives the following description of the spectrum of D λ , where we use the convention λ D 0 = −∞. Proposition 5. For any λ ∈ R, choose n ∈ N such that λ D n−1 ≤ λ < λ D n . Then σ(D λ ) = {µ j (λ) : j ≥ n} .
We conclude this section with a broader class of counterexamples of (1.1) when Ω is no longer Euclidean than those presented in [19]. We first quote an old result by Rauch and Taylor which is a special case of Theorem 2.3 in [23]: Remark 2. The precise criterion in [23] is that the capacities of the sets K n tend to 0. Proposition 6. Choose any k ∈ N such that λ k (M ) < λ k+1 (M ). Then for n sufficiently large, λ D k (Ω n ) < λ N k+1 (Ω n ). Proof. This is immediate from On the other hand, a straightforward perturbation result using the variational characterization of the eigenvalues also proves the following.
Proposition 7. Let (M d , g) be any compact Riemannian manifold and let k ∈ N. Then for any λ > 0 there exists an r 0 which depends on λ and g such that λ N k+1 (Ω) < λ D k (Ω) for any Lipschitz domain Ω in M which is contained in a geodesic ball B r0 (p), and for all k such that λ D k (Ω) ≤ λ. Note that from this sort of perturbation argument, it is impossible to discern whether these inequalities hold for all k independent of the size of Ω.

Positivity.
We now turn to a study of the semigroup generated by −D λ on L 2 (Γ). Some of the facts established here appear also in the paper [7]. Proof. If w is any function, then we recall the standard notation w + = max{w, 0} and w − = − min{w, 0}. If u ∈ H 1 (Ω), then both u ± ∈ H 1 (Ω) as well. Let ϕ = Tr u, which is an element of the space V consisting of all boundary traces of elements of H 1 (Ω). Then the terms ϕ ± in its decomposition are precisely the boundary traces Tr u ± , and in particular both ϕ ± ∈ V as well.
By the Beurling-Deny criterion (see [6] or [22,Theorem 2.6]), the semigroup T (t) is positive if and only if a λ (ϕ + , ϕ − ) ≤ 0 for all ϕ ∈ V . Now suppose that u ∈ H 1 (λ), and write u ± = u ± 0 + u ± 1 ∈ H 1 0 ⊕ H 1 (λ). We use this short notation here even though u = 0 is not the positive part u 0 (and similarly for by the Poincaré inequality. In the last identity we used that fact that 0. This actually characterizes the first eigenvalue: whenever λ ∈ R is an eigenvalue with positive eigenvector, then λ = λ 1 (B). This set of results is frequently referred to as the Krein-Rutman Theorem, see [20] for more information. The following comparison result will be used below. LetT be another C 0 -semigroup on L p (Y ) whose generatorB has compact resolvent. If (see [1,Theorem 1.3]). An example of this is the semigroup generated by the Robin Laplacian, or slightly more generally, the Laplacian ∆ β with boundary conditions ∂ ν u = βu |Γ for some fixed β ∈ L ∞ (Γ). To make this precise, let Ω be a compact manifold with Lipschitz boundary Γ, as before, and fix β ∈ L ∞ (Γ). Then define the form with domain H 1 (Ω). The associated self-adjoint operator is ∆ β , and has domain Moreover, −∆ β generates a positive irreducible C 0 -semigroup T β on L 2 (Ω) and if β ≤ β then 0 ≤ T β (t) ≤ Tβ(t).
We refer to [5] for this and further information. The Krein-Rutman Theorem shows that ifβ ≤ β, then Let us return to our primary goal, which is to prove the Theorem 4.2. Suppose that Ω is connected, and let λ < λ D 1 . Then the semigroup T generated by −D λ on L 2 (Γ) is irreducible. Remark 3. It is somewhat surprising that this holds only assuming that Ω, but not necessarily Γ, is connected. There is an elegant criterion by Ouhabaz [22] which shows that the semigroup generated by ∆ β is irreducible, and we shall use this last result to deduce the irreducibity of T . It does not seem to be easy to prove the irreducibility more directly using the usual criteria such as the one in [22].