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ON THE ROBIN EIGENVALUES OF THE LAPLACIAN IN THE EXTERIOR OF A CONVEX POLYGON
Konstantin Pankrashkin
Laboratoire de mathématiques, Université Paris-Sud Bâtiment 425, 91405 Orsay Cedex, France
konstantin.pankrashkin@math.u-psud.fr
PACS 41.20.Cv, 02.30.Jr, 02.30.Tb DOI 10.17586/2220-8054-2015-6-1-46-56
Let Q c R2 be the exterior of a convex polygon whose side lengths are i\,... ,iM• For a real constant a, let H^ denote the Laplacian in Q, u m —Au, with the Robin boundary conditions du/dv = au at dQ, where v is the outer unit normal. We show that, for any fixed m g N, the mth eigenvalue E^(a) of H behaves as E^m(a ) = —a2 + + O(a-1/2) as a m where ^ stands for the mth eigenvalue of the operator D\©• • •©DM and Dn denotes the one-dimensional Laplacian f m —f" on (0,£n) with the Dirichlet boundary conditions.
Keywords: eigenvalue asymptotics, Laplacian, Robin boundary condition, Dirichlet boundary condition. Received: 5 November 2014
1. Introduction
1.1. Laplacian with Robin boundary conditions
Let Q c Rd, d > 2, be a connected domain with a compact Lipschitz boundary dQ. For a > 0, let H^ denote the Laplacian u m —An in Q with the Robin boundary conditions du/dv = an at dQ, where v stands for the outer unit normal. More precisely, H^ is the self-adjoint operator in L2(Q) generated by the sesquilinear form:
Here and below, a denotes the (d — 1)-dimensional Hausdorff measure.
One checks in the standard way that the operator H^ is semibounded from below. If Q is bounded (i.e. Q is an interior domain), then it has a compact resolvent, and we denote by E^m(P), m e N, its eigenvalues taken according to their multiplicities and enumerated in the non-decreasing order. If Q is unbounded (i.e. Q is an exterior domain), then the essential spectrum of H^ coincides with [0, +rc>), and the discrete spectrum consists of finitely many eigenvalues which will be denoted again by E^(a), m e {1,...,Ka}, and enumerated in the non-decreasing order taking into account the multiplicities.
We are interested in the behavior of the eigenvalues E^(a) for large a. It seems that the problem was introduced by Lacey, Ockedon, Sabina [11] when studying a reaction-diffusion system. Giorgi and Smits [6] studied a link to the theory of enhanced surface superconductivity. Recently, Freitas and Krejcirik [10] and then Pankrashkin and Popoff [15] studied the eigenvalue asymptotics in the context of the spectral optimization.
Let us list some available results. Under various assumptions one showed the asymptotics of the form:
Emm (a) = —CQa2 + o(a2) as a tends to (1)
where CQ > 1 is a constant depending on the geometric properties of Q. Lacey, Ockedon, Sabina [11] showed (1) with m = 1 for C4 compact domains, for which CQ = 1, and for triangles, for which CQ = 2/(1 — cos 9), where 9 is the smallest corner. Lu and Zhu [13] showed (1) with m =1 and CQ = 1 for compact C1 smooth domains, and Daners and Kennedy [2] extended the result to any fixed m e N. Levitin and Parnovski [12] showed (1) with m =1 for domains with piecewise smooth compact Lipschitz boundaries. They proved, in particular, that if Q is a curvilinear polygon whose smallest corner is 9, then for 9 < n there holds CQ = 2/(1 — cos9), otherwise CQ = 1. Pankrashkin [14] considered two-dimensional domains with a piecewise C4 smooth compact boundary and without convex corners, and it was shown that Eq (a) = —a2 — 7a + O(a2/3), where 7 is the maximum of the signed curvature at the boundary. Exner, Minakov, Parnovski [4] showed that for compact C4 smooth domains the same asymptotics E^q(a) = —a2 — 7a + O(a2/3) holds for any fixed m e N. Similar results were obtained by Exner and Minakov [3] for a class of two-dimensional domains with non-compact boundaries and by Pankrashkin and Popoff [15] for C3 compact domains in arbitrary dimensions. Cakoni, Chaulet, Haddar [1] studied the asymptotic behavior of higher eigenvalues.
1.2. 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 two-dimensional 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 Q. Namely, our assumption is as follows:
The domain R2 \ Q 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 R2 \ Q by Ab...,AM e R2, M > 3, and assume that they are enumerated is such a way that the boundary dQ is the union of the M line segments Ln := [An,An+1 ], n e {1,...,M}, where we denote AM+1 := A1, A0 := AM. It is also assumed that there are no artificial vertices, i.e. that An e [An-1,An+1] for any n e {1,...,M}.
Furthermore, we denote by tn the length of the side Ln, and by Dn the Dirichlet Laplacian f m —f" on (0,£n) viewed as a self-adjoint operator in L2(0,£n). The main result of the present note is as follows:
Theorem 1. For any fixed m e N there holds:
Emm (a) = —a2 + + of—U) as a tends to
Wa/
where ^^ is the mth eigenvalue of the operator D1 ® ■ ■ ■ ® DM.
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 Hq contains singular functions and is not included in W2'2(Q), 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.
2. Preliminaries
2.1. Auxiliary operators
For a > 0, we denote by Ta the following self-adjoint operator in L2(R+): T«v = —v", D(T„) = {v e W2,2(R+) : v'(0) + av(0) = 0}. It is well known that:
g—as
spec Ta = {—a2}U [0, ker(T + a2) = (s) := —=. (2)
2a
The sesqulinear form ta for the operator Ta looks as follows:
ta(v,v) |v'(s)|2ds — a|v(0)|2, D(ta) = W1,2(R+).
0
Lemma 2. For any v e W1,2 (R+) there holds:
Iv(s)I2ds —
2 oo oo
< ^J |v'(s)|2ds — a|v(0)|2 + a2 J |v(s)|2ds^.
0 0 0
22
Proof. We denote by P the orthogonal projector on ker(Ta + a2) in L2(R+), then by the spectral theorem, we have:
ta(v,v) + a2|Pv|2 = ta(v - Pv,v - Pv) > 0,
for any v g D(ta). As is normalized, there holds
0
and we arrive at the conclusion. □
Another important estimate is as follows, see Lemmas 2.6 and 2.8 in [12]:
Lemma 3. Let A c R2 be an infinite sector of opening 9 g (0,2n), then for any e > 0 and any function v g W1,2 (A) there holds:
i |v|2ds < e // |Vv|2dx + C /7 |v|2dx with C = ( 1 -cos 9, 9 G (0,n) (3)
di V e V I 1, 9 G [n 2n).
2.2. Decomposition of Q
Let us proceed with a decomposition of the domain Q which will be used through the proof. Let n e {1,..., M}. We denote by Sn and S2 the half-lines originating respectively at An and An+i, orthogonal to Ln and contained in Q. By nn, we denote the half-strip bounded by the half-lines Sn and Sn and the line segment Ln, and by An we denote the infinite sector bounded by the lines S2n-l and Sln and contained in Q. The constructions are illustrated in Fig. 1. We note that the 2M sets An and nn, n e {1,...,M}, are non-intersecting and that Q = (JM1 An U UM1 H«. We deduce from Lemma 3:
Lemma 4. There exists a constant C > 0 such that for any e > 0, any n e {1,..., M} and any v e W1,2 (An) there holds
1
|v|2da < CeU |Vv|2dx + -2 |v|2dx).
Furthermore, for each n e {1,..., M} we denote by 6n the uniquely defined isome-try R2 ^ R2 such that:
An = 6n(0,0) and nn = 6n((0,€n) x R+) :
We remark that due to the spectral properties of the above operator Ta, see (2), we have, for any u e W 1,2(nn),
in OO
d
—u{ On(t,s))
d s dt — a
u(On(t, s))
dt + a2
0 0
in OO
d
—u(0n(t, s ))
d a
u(On(t, 0))
in OO
00
+ a
u(8n(t, s ))
d dt
u(©n(t, s ))
00
which implies, in particular, the following:
in <x
d s j dt > 0,
0
d
~u( 6n(t,s))
2
ds dt
00
in OO
<
d
in OO
— u( @n(t,s))
ds dt +
00
00 in OO
a
u(On(t, 0))
d
—u{ On(t,s))
ds dt
dt + a J J uyQn(t,s)j ds dt 0 0 0
=U1 vu i 2dx—a J1 u i 2da+a2 U1.1 2 dx (4)
nn Ln nn
2.3. Eigenvalues and identification maps
We will use an eigenvalue estimate which is based on the max-min principle and is just a suitable reformulation of Lemma 2.1 in [5] or of Lemma 2.2 in [16]:
n
2
2
2
2
2
2
2
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 G N and assume that the operator B has at least m eigenvalues Ai < • • • < Am < 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 Si, S2 > 0 such that Si < (1 + Am)-i, and that for any u G D(b) there holds:
IMI2 — ||Ju||2 < Si(b(u,u) + ||u||2), b'( Ju, Ju) — b(u, u) < S2 ^b(u, u) + ||u||2^ ,
then
\ ' ^ A + (AmSi + S2)(1 + Am)
Am < Am + , x x r- ,
1 — (1 + Am )Si where A'm is the mth eigenvalue of the operator B'.
3. Proof of Theorem 1
3.1. Dirichlet-Neumann bracketing
Consider the following sesqulinear form:
M ^ M
(u,u) = V / / |Vu|2dx + V / / |Vu|2dx — a |u|2da ,
n=i an n=i * n
) = 0 W1,2(An) © 0 W1,2(En),
n=1 n=1
Wo1'2(n„) := {/ G W1,2(n„) : / = 0 at ^USn},
and denote by the associated self-adjoint operator in L2(fi). Clearly, the form is a restriction of the initial form h^, and due to the max-min principle we have:
Em (a) < (a),
where Ed,d(a) is the mth eigenvalue of H^,D (as soon at it exists). On the other hand, we have the decomposition:
M ( ) M Hr = ©( — AD) © © GDa,
n=1 n=1
where (—AD) is the Dirichlet Laplacian in L2(An) and GD,a is the self-adjoint operator in L2(nn) generated by the sesquilinear form:
gDa(u,u) = jj |Vu|2dx — a J |u|2da, D^J = W01,2(nn).
nn Ln
Consider the following unitary maps:
Un : L2(nn) ^ L2 ((0, in) x R+), Unf := f ◦ 8n, n e{1,...,M},
then it is straightforward to check that UnGDaU* = Dn®1+Ta01. As the operators (—AD) are non-negative, it follows that specess H^,D = [0, +rc>) and that Ed,d(a) = —a2 + ^D, which gives the majoration:
ED(a) <—a2 + ^D, (5)
for all m with ^D < a2. In particular, the inequality (5) holds for any fixed m as a tends to
Similarly, we introduce the following sesquilinear form:
(u,u) = IÎ |Vu|2dx + ¿f IÎ |Vu|2dx — a I |u|2da ra=1a ™=1 n L
An nn Ln
MM
D(h^N) = © W 1,2(An) © © W 1,2(nn),
n=1 n=1
and denote by the associated self-adjoint operator in L2(Q). Clearly, the initial form h^ is a restriction of the form hN,n, and due to the max-min principle we have:
ED,N(a) < ED(a),
where Ed,n (a) is the mth eigenvalue of H^,N, and the inequality holds for those m for which ED(a) exists. On the other hand, we have the decomposition:
M M
en = ©(—an ) ® © cNa,
n=1 n=1
where (—AN) denotes the Neumann Laplacian in L2(An) and G^« is the self-adjoint operator in L2(nn) generated by the sesquilinear form
C(u,u) = JJ |Vu|2dx — a J |u|2da, D««) = W 1,2(nn).
nn Ln
There holds UnGN^UJ = Nn 0 1 + Ta 0 1, where Nn is the operator f ^ — f" on (0,4) with the Neumann boundary condition viewed as a self-adjoint operator in the Hilbert space L2(0,in), n e {1,...,M}. The operators (—AN) are non-negative, and we have specess H^,N = [0, +rc>) and Ed,n(a) = —a2 + ^D, where ^D is the mth eigenvalue of the operator N1 © ■ ■ ■ © NM. Thus, we obtain the minorations:
H^ > —a2 and E^(a) > —a2 + ^D, (6)
which holds for any fixed m as a tends to By combining the inequalities (5) and (6) we also obtain the rough estimate:
ED(a) = —a2 + O(1) for any fixed m and for a tending to (7)
3.2. Construction of an identification map
In order to conclude the proof of Theorem 1, we will apply Proposition 5 to the operators:
B = H2 + a2, B' = D1 ©•••© Dn, which will allow us to obtain another inequality between the quantities:
Ad ED(a) + a , AD ^D.
Note that for any fixed m e N, one has AD = O(1) for large a, see (7). Therefore, it is sufficient to construct an identification map J = Ja as in Proposition 5 with + i2 = O(a-1/2). Recall that the respective forms b and b' in our case are given by:
b(u,u) = h^(u, u) + a2||u||2, D(b) = D(h^) = W M(Q),
m in
b '(f,f) = W f(t)|2dt, D(b ') = {f =(f1,...,fM): fn e W01,2(0,in)}. n=1 0
Here and below, by ||u|| we mean the usual norm in L2(Q). The positivity of b' is obvious, and the positivity of b follows from (6). Consider the maps:
OO
Pn,a : W 1,2(nn) ^ L2(0,in), (Pn,«u)(t) = J ^a(s)u(8n(t,s))ds, n e {1,... , M}.
0
If u e W 1,2(Q), then u e W 1,2(nn) for any n e {1,...,M}, and one can estimate, using the Cauchy-Schwarz inequality:
|(Pn,«u)(0)|2 +|(Pn,«u)(£n)|2 < J |u(6n(0,s))|2ds + J |u(6n(in,s))|2ds
0 0
= J |u|2da ^y |u|2da.
As Sn-1 U Sn = dAn, we can use Lemma 4 with e = a-1, which gives:
m m „ „
^(|(Pn,au)(0)|2 + |(Pn,«u)(£n)r) < / |u|2da + |u|2da)
^ A, ,2J CMf rr
n^/|u|2da < a UJJ
n=1a a n=1 A
u|2da 11 |Vu|2dx + a2// |u|2dx). (8)
dan n=1 an
For each n e {1, . . . , M}, we introduce a map:
in
nn : (0,in) ^ {0,in}, nn(t) = 0 for t < y, nn(t) = in otherwise,
and choose a function: pn e CC([0,£n]) with pn(0) = pn(€n) = 1 and Pny-^j = 0. Finally, we define:
M
Ja : W 1,2(H) ^ 0 L2(0,€„), (J«u)„(t) = (Pra>au)(t) - (Pn,au)(nra(i))pra(i).
n=1
We remark that (Jau)n e W01,2(0,€ra) for any u e W 1,2(H) and n e {1,..., M}, i.e. Ja maps D(b) into D(b') and will be used as an identification map.
3.3. Estimates for the identification map
Take any 5 > 0. Using the following inequality:
(«1 + «2)2 > (1 - 5)«2 - 1
we estimate:
(a1 + a2)2 > (1 — 5)a1 —- a2, a1, a2 > 0,
5
m„„ m„„ y 2
u||2 -||J«u||2 = j / / |u|2dx + j / / |u|2dx - |(Pra>au)(i) - (Pra>au)(n(i))p(i)| dt)
n=1 an n=1 nn 0
M „ „ M ¿n ¿n
jrJJ |u|2dx + ^^I' |u|2dx - (1 - 5) j |(P„,a u) (t) |2dt + |(P„,aU)(n(i))p(i)|2di
n=1 an n=1 nn 0 0
j U |u|2dx + j [JJ |u|2dx - yn|(Praau)(i)|2di
2
dt
|u|
n=1an n=1 nn 0
M ¿n M
+5 j i |(Pra>au)(t)|2dt + 1 j i |(Pra>au)(n(i))pra(t)
n=1 n n=1 £
0
: /1 + /2 + Is + /4
We have the trivial inequality:
/1 - ( // |yu|2dx + a2 J J |u|2dx).
n=1 an an
To estimate the term /2, we use Lemma 2 and then (4):
M ¿n C CO
/2 = ¿/(/ |u(e„(M))|2ds -| I ^a(s)u(era(t,s))ds|^ dt
n=1
1
< —
0 0 0
d ^ | .„2
M ¿n CO CO
a2
j/ (/ |dd:u(en(i,s))| ds - a|u(eB (t, 0))|2 + a2 J |u(e„(t,s))|2d^ dt n=10 0 0
- ¿2 j ( // |Vu|2dx - i |u|2da + a2 if |u|2dx) ,
which gives:
I1 + I2 < (h«(u,u) + a2||u||2). Furthermore, with the help of the Cauchy-Schwarz inequality, we have:
h <
5//M
5 |u(0n(t,s))
n=1 0 0
To estimate the last term, 14, we introduce the following constant:
ds dt = 5 V 11 |u|2dx < 5||u|
nn
R := maxj / |pn(t)|2dt : n e {1,... ,M}},
then, using first the estimate (8), and then the inequality (4),
M
R
< —
- 5
I4 < Y y^ sup (Pn,«u) (nn(t))
5 n=1 ^(0,in)
M
^(|(Pn,«u)(0)|2 + |(Pn,«u)(£n)|2)
n=1
< RC {U |Vu|2dx + a2 U |u|2dx)
n=1 an an
RC r 5 ( // |Vu|2dx + a2 [f |u|2dx)
n=1
M
+ 5 ( yy |Vu|2dx — J |u|2 da + a2 J J |u|2dx)
nn Ln nn
h^ (u, u) + a2||u||).
— (hn 5a ^
Choosing 5 = a 1/2 and summing up the four terms, we see that:
||u||2 — || Jau||2 < —^ ^h^(u, u) + a2||u||2 + ||u||2) = (b(u, u) + ||u||2)
with a suitable constant c1 > 0.
Now, we need to compare b'(Jau, Jau) and b(u,u). Take 5 e (0,1) and use the inequality:
2
(«1 + «2)2 < (1 + 5) a2 + — «2, «1, «2 > 0,
5
in
2
Robin-Laplacian eigenvalues Then:
b'(J«u, J«u) — b(u, u) = E (Pn,au)' — pn[(Pn,«u) o fin] dt — (h«(u,u) + a2||u||2)
n=i
M in M in
M r 2 9 _ _ r
<(1+ S)£ (Pn,au)' dt P n[(Pn,au) o fi„]
n=i0 n=i0
E (JJ |Vu|2dx + a2 JJ |u|2dx)
dt
M
— ^ ( yy |Vu|2dx — j |u|2da + a jJ |u|2dx)
n=i rr r rr
(9)
n„
< (1+S) E/
n=i0
(Pn,au)'
2 2
dt + -0
M
n=i
P n [(Pn,«u) o fin]
dt
— E (JJ |Vu|2dx — y |u|2da + a2 JJ |u|2 dx).
n=i rr r rr
nn
Using first the Cauchy-Schwarz inequality and then inequality (4), we have:
¿n ¿n C
|(Pn,«u)'|2dt <
00
^u(6„(t,s)) 2ds dt < JJ |Vu|2dx — J |u|2da + a2 j I |u|2dx.
nn Ln nn
Substituting the last inequality into (9), we arrive at:
b'( J«u, Jau) — b(u,u) < S f JJ |Vu|2dx — J |u|2da + a2 JJ |u|2dxj
n=i n L n
nn Ln nn
2 + 2
M
(10)
n=i
Pi [(Pn,«u) O fin]
dt.
Furthermore, using the constant:
R ' := max j f |pn(t)|2dt : n G {1,... ,M}},
and the inequality (8), we have:
m 2 M
y, I pn [(Pn,«u) O fin] dt < R 'V sup |(Pn,«u^ n n=^ n=i
M
< R ' EE (|(Pn,«u)(0)|2 + |(Pn,«u)(4)f) < — E( // |Vu|2 dx + a2 [f |u|2dxV
n=i a n=iM^ A
2
in
in
2
in
2
2
The substitution of this inequality into (10) and the choice 5 = a 1/2 then leads to:
b'( Ju, Ju) — b(u, u) < —^ ( h^(u, u) + a2||u||2 ) < —^ ( b(u, u) + ||u||2 ) ,
ya V / va V '
with a suitable constant c2 > 0. By Proposition 5, for any fixed m g N and for large a, we have the estimate ^ < ED(a) + a2 + O(a-1/2). The combination with (5) gives the result.
Acknowledgments
The work was partially supported by ANR NOSEVOL (ANR 2011 BS01019 01) and GDR Dynamique quantique (GDR CNRS 2279 DYNQUA).
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