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Friedlander's Eigenvalue Inequalities and The COMMUNICATIONS ON doi:10.3934/cpaa.2012.11.2201 PURE AND APPLIED ANALYSIS Volume 11, Number 6, November 2012 pp. 2201{2212 FRIEDLANDER'S EIGENVALUE INEQUALITIES AND THE DIRICHLET-TO-NEUMANN SEMIGROUP Wolfgang Arendt Institute of Applied Analysis University of Ulm, D - 89069 Ulm, Germany Rafe Mazzeo Department of Mathematics Stanford University, Stanford, CA 94305, USA Abstract. If Ω is any compact Lipschitz domain, possibly in a Riemannian manifold, with boundary Γ = @Ω, the Dirichlet-to-Neumann operator Dλ is defined on L2(Γ) for any real λ. We prove a close relationship between the eigenvalues of Dλ 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 N Friedlander inequalities between Neumann and Dirichlet eigenvalues, λk+1 ≤ D λk , k 2 N, and to sharpen the inequality to be strict, whenever Ω is a Lipschitz d domain in R . We give new counterexamples to these inequalities in the general Riemannian setting. Finally, we prove that the semigroup generated by −Dλ, for λ sufficiently small or negative, is irreducible. d D 1. Introduction. Let Ω ⊂ R be a bounded domain with @Ω = Γ. Let λ1 < D D N N N λ2 ≤ λ3 ≤ · · · and λ1 < λ2 ≤ λ3 ≤ · · · 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 N D λk+1 ≤ λk 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 C1. 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 H1(Ω) ⊂ L2(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 2000 Mathematics Subject Classification. Primary: 35P15; Secondary: 47D06. Key words and phrases. Eigenvalue inequalities, Dirichlet-to-Neumann operator. 2201 2202 WOLFGANT ARENDT AND RAFE MAZZEO 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 Dλ 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 Dλ used by Friedlander [9], see also [19], but traces back at least as far as the paper of Gr´egoire,N´ed´elecand 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 D associated to the Dirichlet-to-Neumann operator Dλ (for any λ < λ1 ). 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 Dλ. A consequence of this is that the first eigenvalue of Dλ 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 Dλ. 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 exposition- s 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 cor- respondence 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. x3 de- scribes the eigenvalue monotonicity and its application to the proof of the eigenvalue inequalities. The Dirichlet-to-Neumann semigroup is the subject of x4. 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. 2 2 a(u) + !kukH ≥ αkukV for all u 2 V ∗ for some ! 2 R and α > 0. Associated to a is a bounded operator A1 : V ! V . Also associated to a is an unbounded self-adjoint operator A2 on H with domain ∗ D(A2) ⊂ V ⊂ H. Thus x 2 D(A1) and A1x = y 2 V if and only if a(x; v) = hy; vi for all v 2 V . The operator A2 is the part of A1 in D(A2), 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 2 V ) if and only if A is nonnegative (i.e. hAu; uiH ≥ 0 for all FRIEDLANDER'S EIGENVALUE INEQUALITIES 2203 u 2 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 fen; λng the eigendata for A, so the en are an orthonormal basis for H, Aen = λnen for all n, and λ1 ≤ λ2 ≤ · · · % 1. The standard max-min characterization of the eigenvalues is λn = sup inffa(u): u 2 Vn−1; jjujj = 1 : (2.1) Vn−12Gn−1(V ) where Gn−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 C1;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 Rd to domains in manifolds. The volume form and gradient for g lead naturally to the Hilbert spaces L2(Ω) 1 2 1 1 and H (Ω), as well as the space L (Γ). As usual, H0 (Ω) is the closure of C0 (Ω) in 1 H (Ω). The boundary restriction map u 7! ujΓ := Tr u is well-defined for any u 2 H1(Ω) \C0(Ω), and this map extends to a bounded operator Tr : H1(Ω) ! L2(Γ), 1 with nullspace H0 (Ω). We write ujΓ or Tr u interchangeably. We next recall the weak formulations of well-known operators and identities. a) If u 2 H1(Ω), we say that ∆u 2 L2(Ω) if there exists f 2 L2(Ω) such that Z Z 1 ru · rv dVg = fv dVg for all v 2 H0 (Ω): Ω Ω 1 2 2 b) Suppose that u 2 H (Ω) and ∆u 2 L (Ω). We say that @ν u 2 L (Γ) if there exists b 2 L2(Γ) such that Z Z 1 ru · rv − ∆u v dVg = bv dσg for all v 2 H (Ω); Ω Γ and we then write @ν u = b. To be explicit, our conventions are that ∆ = −div r and ν is the outer unit normal; also, dVg and dσg are the volume forms on Ω and Γ associated to g.
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