ON THE ROBIN EIGENVALUES OF THE LAPLACIAN IN THE EXTERIOR OF A CONVEX POLYGON

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1 NANOSYSTES: PHYSICS, CHEISTRY, ATHEATICS, 15, 6 (1, P ON THE ROBIN EIGENVALUES OF THE LAPLACIAN IN THE EXTERIOR OF A CONVEX POLYGON Konstantin Pankrashkin Laboratoire de mathématiques, Université Paris-Sud Bâtiment 45, 9145 Orsay Cedex, France konstantin.pankrashkin@math.u-psud.fr PACS 41..Cv,.3.Jr,.3.Tb DOI / Let Ω R be the exterior of a convex polygon whose side lengths are l 1,..., l. For a real constant, let H Ω denote the Laplacian in Ω, u u, with the Robin boundary conditions u/ ν = u at Ω, where ν is the outer unit normal. We show that, for any fixed m N, the mth eigenvalue Em( Ω of H Ω behaves as Em( Ω = + µ D m + O( 1/ as +, where µ D m stands for the mth eigenvalue of the operator D 1 D and D n denotes the one-dimensional Laplacian f f on (, l n with the Dirichlet boundary conditions. Keywords: eigenvalue asymptotics, Laplacian, Robin boundary condition, Dirichlet boundary condition. Received: 5 November Introduction 1.1. Laplacian with Robin boundary conditions Let Ω R d, d, be a connected domain with a compact Lipschitz boundary Ω. For >, let H Ω denote the Laplacian u u in Ω with the Robin boundary conditions u/ ν = u at Ω, where ν stands for the outer unit normal. ore precisely, H Ω is the self-adjoint operator in L (Ω generated by the sesquilinear form: h Ω (u, u = u dx u dσ, D(h Ω = W 1, (Ω. Ω Ω Here and below, σ denotes the (d 1-dimensional Hausdorff measure. One checks in the standard way that the operator H Ω is semibounded from below. If Ω is bounded (i.e. Ω is an interior domain, then it has a compact resolvent, and we denote by Em(β, Ω m N, its eigenvalues taken according to their multiplicities and enumerated in the non-decreasing order. If Ω is unbounded (i.e. Ω is an exterior domain, then the essential spectrum of H Ω coincides with [, +, and the discrete spectrum consists of finitely many eigenvalues which will be denoted again by Em(, Ω m {1,..., K }, and enumerated in the non-decreasing order taking into account the multiplicities. We are interested in the behavior of the eigenvalues Em( Ω for large. It seems that the problem was introduced by Lacey, Ockedon, Sabina [11] when studying a reactiondiffusion system. Giorgi and Smits [6] studied a link to the theory of enhanced surface superconductivity. Recently, Freitas and Krejčiřík [1] and then Pankrashkin and Popoff [15] studied the eigenvalue asymptotics in the context of the spectral optimization.

2 Robin-Laplacian eigenvalues 47 Let us list some available results. asymptotics of the form: Under various assumptions one showed the E Ω m( = C Ω + o( as tends to +, (1 where C Ω 1 is a constant depending on the geometric properties of Ω. Lacey, Ockedon, Sabina [11] showed (1 with m = 1 for C 4 compact domains, for which C Ω = 1, and for triangles, for which C Ω = /(1 cos θ, where θ is the smallest corner. Lu and Zhu [13] showed (1 with m = 1 and C Ω = 1 for compact C 1 smooth domains, and Daners and Kennedy [] extended the result to any fixed m N. Levitin and Parnovski [1] showed (1 with m = 1 for domains with piecewise smooth compact Lipschitz boundaries. They proved, in particular, that if Ω is a curvilinear polygon whose smallest corner is θ, then for θ < π there holds C Ω = /(1 cos θ, otherwise C Ω = 1. Pankrashkin [14] considered twodimensional domains with a piecewise C 4 smooth compact boundary and without convex corners, and it was shown that E1 Ω ( = γ + O( /3, where γ is the maximum of the signed curvature at the boundary. Exner, inakov, Parnovski [4] showed that for compact C 4 smooth domains the same asymptotics Em( Ω = γ + O( /3 holds for any fixed m N. Similar results were obtained by Exner and inakov [3] for a class of two-dimensional domains with non-compact boundaries and by Pankrashkin and Popoff [15] for C 3 compact domains in arbitrary dimensions. Cakoni, Chaulet, Haddar [1] studied the asymptotic behavior of higher eigenvalues. 1.. Problem setting and the main result The computation of further terms in the eigenvalue asymptotics needs more precise geometric assumptions. To our knowledge, such results are available for the twodimensional case only. Helffer and Pankrashkin [9] studied the tunneling effect for the eigenvalues of a specific domain with two equal corners, and Helffer and Kachmar [8] considered the domains whose boundary curvature has a unique non-degenerate maximum. The machinery of both papers is based on the asymptotic properties of the eigenfunctions: it was shown that the eigenfunctions corresponding to the lowest eigenvalues concentrate near the smallest convex corner at the boundary or, if no convex corners are present, near the point of the maximum curvature, and this is used to obtain the corresponding eigenvalue asymptotics. The aim of the present note is to consider a new class of two-dimensional domains Ω. Namely, our assumption is as follows: The domain R \ Ω is a convex polygon (with straight edges. Such domains are not covered by the above cited works: all the corners are non-convex, and the curvature is constant on the smooth part of the boundary, and it is not clear how the eigenfunctions are concentrated along the boundary. We hope that our result will be of use for the understanding of the role of non-convex corners. In order to formulate the main result, we first need some notation. We denote the vertices of the polygon R \ Ω by A 1,..., A R, 3, and assume that they are enumerated is such a way that the boundary Ω is the union of the line segments := [A n, A n+1 ], n {1,..., }, where we denote A +1 := A 1, A := A. It is also assumed that there are no artificial vertices, i.e. that A n / [A n 1, A n+1 ] for any n {1,..., }. Furthermore, we denote by l n the length of the side, and by D n the Dirichlet Laplacian f f on (, l n viewed as a self-adjoint operator in L (, l n. The main result of the present note is as follows:

3 48 K. Pankrashkin Theorem 1. For any fixed m N there holds: ( 1 Em( Ω = + µ D m + O as tends to +, where µ D m is the mth eigenvalue of the operator D 1 D. The proof is based on the machinery proposed by Exner and Post [5] to study the convergence on graph-like manifolds. In reality, our construction appears to be quite similar to that of Post [16], which was used to study decoupled waveguides. We remark that due to the presence of non-convex corners the domain of the operator H Ω contains singular functions and is not included in W, (Ω, see e.g. Grisvard [7]. This does not produce any difficulties, as our approach is purely variational and is entirely based on analysis of the sesqulinear form.. Preliminaries.1. Auxiliary operators For >, we denote by T the following self-adjoint operator in L (R + : It is well known that: T v = v, D(T = { v W, (R + : v ( + v( = }. spec T = { } [, +, ker(t + = Cϕ, ϕ (s := e s. ( The sesqulinear form t for the operator T looks as follows: t (v, v = v (s ds v(, D(t = W 1, (R +. Lemma. For any v W 1, (R +, there holds: v(s ds ϕ (sv(sds 1 ( v (s ds v( + v(s ds. Proof. We denote by P the orthogonal projector on ker(t + in L (R +, then by the spectral theorem, we have: t (v, v + P v = t (v P v, v P v, for any v D(t. As ϕ is normalized, there holds ϕ (xv(xdx = P v, and we arrive at the conclusion. Another important estimate is as follows, see Lemmas.6 and.8 in [1]: Lemma 3. Let Λ R be an infinite sector of opening θ (, π, then for any ε > and any function v W 1, (Λ there holds: v ds ε v dx + C θ v, θ (, π, dx with C θ = 1 cos θ (3 ε 1, θ [π, π. Λ Λ Λ

4 Robin-Laplacian eigenvalues 49.. Decomposition of Ω Let us proceed with a decomposition of the domain Ω which will be used through the proof. Let n {1,..., }. We denote by Sn 1 and Sn the half-lines originating respectively at A n and A n+1, orthogonal to and contained in Ω. By, we denote the half-strip bounded by the half-lines Sn 1 and Sn and the line segment, and by we denote the infinite sector bounded by the lines Sn 1 and Sn 1 and contained in Ω. The constructions are illustrated in Fig. 1. We note that the sets and, n {1,..., }, are non-intersecting and that Ω =. We deduce from Lemma 3: Lemma 4. There exists a constant C > such that for any ε >, any n {1,..., } and any v W 1, ( there holds ( v dσ Cε v 1 dx + v dx. ε Furthermore, for each n {1,..., } we denote by Θ n the uniquely defined isometry R R such that: A n = Θ n (, and = Θ n ( (, ln R + : We remark that due to the spectral properties of the above operator T, see (, we have, for any u W 1, (, l n s u( Θ n (t, s ln ds dt u ( Θ n (t, s dt + = l n ( l n u ( Θ n (t, s ds dt s u( Θ n (t, s u ( ds Θn (t, + u ( Θ n (t, s ds dt, which implies, in particular, the following: l n t u( Θ n (t, s ds dt l n l n t u( Θ n (t, s ln ds dt + u ( Θ n (t, dt + =.3. Eigenvalues and identification maps l n s u( Θ n (t, s ds dt u ( Θ n (t, s ds dt u dx u dσ + u dx. (4 We will use an eigenvalue estimate which is based on the max-min principle and is just a suitable reformulation of Lemma.1 in [5] or of Lemma. in [16]:

5 K. Pankrashkin 5 FIG. 1. Decomposition of the domain Proposition 5. Let B and B be non-negative self-adjoint operators acting respectively in Hilbert space H and H and generated by sesqulinear form b and b. Choose m N and assume that the operator B has at least m eigenvalues λ1 λm < inf specess B and that the operator B has a compact resolvent. If there exists a linear map J : D(b D(b (identification map and two constants δ1, δ > such that δ1 (1 + λm 1, and that for any u D(b there holds: kuk kjuk δ1 b(u, u + kuk, b (Ju, Ju b(u, u δ b(u, u + kuk, then λm λm + (λm δ1 + δ (1 + λm, 1 (1 + λm δ1 where λm is the mth eigenvalue of the operator B Proof of Theorem 1 Dirichlet-Neumann bracketing Consider the following sesqulinear form: hω,d (u, u = ZZ X D(hΩ,D = u dx + ZZ X Λn W1, (Λn Z u dx Πn u dσ, Ln f1, (Πn, W f1, (Πn := f W 1, (Πn : f = at S 1 S, W n n and denote by HΩ,D the associated self-adjoint operator in L (Ω. Clearly, the form hω,d is a restriction of the initial form hω, and due to the max-min principle we have: Ω Ω,D Em ( Em (,

6 Robin-Laplacian eigenvalues 51 where Em Ω,D ( is the mth eigenvalue of H Ω,D have the decomposition: H Ω,D = (as soon at it exists. On the other hand, we ( D n G D n,, where ( D n is the Dirichlet Laplacian in L ( and G D n, is the self-adjoint operator in L ( generated by the sesquilinear form: gn,(u, D u = u dx u dσ, D(g D 1, n, = W (. Consider the following unitary maps: U n : L ( L ( (, l n R +, Un f := f Θ n, n {1,..., }, then it is straightforward to check that U n G D n,un = D n 1+T 1. As the operators ( D n are non-negative, it follows that spec ess H Ω,D = [, + and that Em Ω,D ( = + µ D m, which gives the majoration: Em( Ω + µ D m, (5 for all m with µ D m <. In particular, the inequality (5 holds for any fixed m as tends to +. Similarly, we introduce the following sesquilinear form: ( h Ω,N (u, u = u dx + u dx D(h Ω,N = W 1, ( W 1, (, u dσ, and denote by H Ω,N the associated self-adjoint operator in L (Ω. Clearly, the initial form h Ω is a restriction of the form h N,Ω, and due to the max-min principle we have: Em Ω,N ( Em(, Ω where Em Ω,N ( is the mth eigenvalue of H Ω,N, and the inequality holds for those m for which Em( Ω exists. On the other hand, we have the decomposition: H Ω,N = ( N n G N n,, where ( N n denotes the Neumann Laplacian in L ( and G N n, is the self-adjoint operator in L ( generated by the sesquilinear form gn,(u, N u = u dx u dσ, D(gn, N = W 1, (. There holds U n G N n,un = N n 1 + T 1, where N n is the operator f f on (, l n with the Neumann boundary condition viewed as a self-adjoint operator in the Hilbert space L (, l n, n {1,..., }. The operators ( N n are non-negative, and we have spec ess H Ω,N = [, + and Em Ω,N ( = + µ N m, where µ N m is the mth eigenvalue of the operator N 1 N. Thus, we obtain the minorations: H Ω and E Ω m( + µ N m, (6

7 5 K. Pankrashkin which holds for any fixed m as tends to +. By combining the inequalities (5 and (6 we also obtain the rough estimate: E Ω m( = + O(1 for any fixed m and for tending to +. (7 3.. Construction of an identification map In order to conclude the proof of Theorem 1, we will apply Proposition 5 to the operators: B = H Ω +, B = D 1 D n, which will allow us to obtain another inequality between the quantities: λ m = E Ω m( +, λ m = µ D m. Note that for any fixed m N, one has λ m = O(1 for large, see (7. Therefore, it is sufficient to construct an identification map J = J as in Proposition 5 with δ 1 + δ = O( 1/. Recall that the respective forms b and b in our case are given by: b(u, u = h Ω (u, u + u, b (f, f = l n D(b = D(h Ω = W 1, (Ω, f n (t dt, D(b = { f = (f 1,..., f : f n W 1, (, l n }. Here and below, by u we mean the usual norm in L (Ω. The positivity of b is obvious, and the positivity of b follows from (6. Consider the maps: P n, : W 1, ( L (, l n, (P n, u(t = ϕ (su ( Θ n (t, s ds, n {1,..., }. If u W 1, (Ω, then u W 1, ( for any n {1,..., }, and one can estimate, using the Cauchy-Schwarz inequality: (Pn, u( + (Pn, u(l n ( u Θn (, s ds + ( u Θn (l n, s ds = u dσ + u dσ. As S n 1 S 1 n =, we can use Lemma 4 with ε = 1, which gives: ( (Pn, u( + (Pn, u(l n = S 1 n ( u dσ For each n {1,..., }, we introduce a map: S 1 n C S n u dσ + S n u dσ ( u dx + u dx. (8 π n : (, l n {, l n }, π n (t = for t < l n, π n(t = l n otherwise,

8 Robin-Laplacian eigenvalues 53 and choose a function: ρ n C ( [, l n ] ( ln with ρ n ( = ρ n (l n = 1 and ρ n Finally, we define: J : W 1, (Ω L (, l n, =. (J u n (t = (P n, u(t (P n, u ( π n (t ρ n (t. We remark that (J u n W 1, (, l n for any u W 1, (Ω and n {1,..., }, i.e. J maps D(b into D(b and will be used as an identification map Estimates for the identification map we estimate: Take any δ >. Using the following inequality: (a 1 + a (1 δa 1 1 δ a, a 1, a, ( u J u = u dx + u dx ( P n, u (t ( P n, u ( π(t ρ(t dt Λ n ( l n u dx + u dx (1 δ ( P n, u (t dt + 1 l n ( P n, u ( π(t ρ(t dt δ Λ n ( l n = u dx + u dx ( P n, u (t dt Λ n +δ l n We have the trivial inequality: I 1 1 ( P n, u (t dt + 1 δ l n l n =: I 1 + I + I 3 + I 4. ( u dx + To estimate the term I, we use Lemma and then (4: I = 1 l n ( l n 1 ( ( u Θn (t, s ds ( ( P n, u ( π(t ρ n (t dt u dx. ϕ (su ( Θ n (t, s ds dt s u( Θ n (t, s ( ds u Θn (t, + u dx u dσ + u dx, ( u Θn (t, s ds dt

9 54 K. Pankrashkin which gives: I 1 + I 1 ( h Ω (u, u + u. Furthermore, with the help of the Cauchy-Schwarz inequality, we have: I 3 δ l n u ( Θ n (t, s ds dt = δ u dx δ u, To estimate the last term, I 4, we introduce the following constant: R := max { ln ρn (t } dt : n {1,..., }, then, using first the estimate (8, and then the inequality (4, I 4 R δ sup t (,l n ( P n, u ( π n (t R ( (Pn, u( + (Pn, u(l n δ RC ( u dx + u dx δ RC [ ( u dx + u dx δ ( + u dx u dσ + = RC ( h Ω δ (u, u + u. ] u dx Choosing δ = 1/ and summing up the four terms, we see that: u J u c ( 1 h Ω (u, u + u + u c ( 1 b(u, u + u, with a suitable constant c 1 >. Now, we need to compare b (J u, J u and b(u, u. inequality: Take δ (, 1 and use the (a 1 + a (1 + δa 1 + δ a, a 1, a,

10 Robin-Laplacian eigenvalues 55 Then: b (J u, J u b(u, u = l n (1 + δ [ ] (P n, u ρ n (Pn, u π n dt (h Ω (u, u + u l n (P n, u dt + δ l n ( u dx + u dx ( u dx u dσ + (1 + δ l n ( (P n, u dt + δ ] ρ n[ (Pn, u π n dt l n u dx u dσ + u dx ] ρ n[ (Pn, u π n dt u dx. Using first the Cauchy-Schwarz inequality and then inequality (4, we have: l n (Pn, u dt l n t u( Θ n (t, s ds dt l n u dx u dσ + u dσ + Substituting the last inequality into (9, we arrive at: ( b (J u, J u b(u, u δ u dx Furthermore, using the constant: R := max + δ { ln ρ n u dx u dx. (9 [ (Pn, u π n ] dt. (1 ρ n (t } dt : n {1,..., }, and the inequality (8, we have: R l n ρ n [ (Pn, u π n ] dt R ( (P n, u( + (P n, u(l n R C sup t (,l n (Pn, u ( π n (t ( u dx + u dx.

11 56 K. Pankrashkin The substitution of this inequality into (1 and the choice δ = 1/ then leads to: b (Ju, Ju b(u, u c ( h Ω (u, u + u c ( b(u, u + u, with a suitable constant c >. By Proposition 5, for any fixed m N and for large, we have the estimate µ D m E Ω m( + + O( 1/. The combination with (5 gives the result. Acknowledgments The work was partially supported by ANR NOSEVOL (ANR 11 BS119 1 and GDR Dynamique quantique (GDR CNRS 79 DYNQUA. References [1] F. Cakoni, N. Chaulet, H. Haddar. On the asymptotics of a Robin eigenvalue problem. Comptes Rendus ath., 351 (13 14, P (13. [] D. Daners, J.B. Kennedy. On the asymptotic behaviour of the eigenvalues of a Robin problem. Differential Integr. Equ., 3 (7 8, P (1. [3] P. Exner, A. inakov. Curvature-induced bound states in Robin waveguides and their asymptotical properties. J. ath. Phys., 55 (1, P. 111 (14. [4] P. Exner, A. inakov, L. Parnovski. Asymptotic eigenvalue estimates for a Robin problem with a large parameter. Portugal. ath., 71 (, P (14. [5] P. Exner, O. Post. Convergence of spectra of graph-like thin manifolds. J. Geom. Phys., 54 (1, P (5. [6] T. Giorgi, R. Smits. Eigenvalue estimates and critical temperature in zero fields for enhanced surface superconductivity. Z. Angew. ath. Phys., 58 (, P (7. [7] P. Grisvard. Singularities in boundary value problems. Recherches en athématiques Appliques, No. (199. [8] B. Helffer, A. Kachmar. Eigenvalues for the Robin Laplacian in domains with variable curvature. arxiv: (14. [9] B. Helffer, K. Pankrashkin. Tunneling between corners for Robin Laplacians. J. London ath. Soc., 91 (1, P. 548 (15. [1] P. Freitas, D. Krejčiřík. The first Robin eigenvalue with negative boundary parameter. arxiv: , (14. [11] A.A. Lacey, J.R. Ockendon, J. Sabina. ultidimensional reaction diffusion equations with nonlinear boundary conditions. SIA J. Appl. ath., 58 (5, P (1998. [1]. Levitin, L. Parnovski. On the principal eigenvalue of a Robin problem with a large parameter. ath. Nachr., 81 (, P (8. [13] Y. Lou,. Zhu. A singularly perturbed linear eigenvalue problem in C 1 domains. Pacific J. ath., 14 (, P (4. [14] K. Pankrashkin. On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains. Nanosystems: Phys. Chem. ath., 4 (4, P (13. [15] K. Pankrashkin, N. Popoff. ean curvature bounds and eigenvalues of Robin Laplacians. arxiv: , (14. [16] O. Post. Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case. J. Phys. A, 38 (, P (5.

Eigenvalues of Robin Laplacians on infinite sectors and application to polygons

Eigenvalues of Robin Laplacians on infinite sectors and application to polygons Magda Khalile (joint work with Konstantin Pankrashkin) Université Paris Sud /25 Robin Laplacians on infinite sectors 1 /