On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains Текст научной статьи по специальности «Математика»

ON THE ASYMPTOTICS OF THE PRINCIPAL EIGENVALUE FOR A ROBIN PROBLEM WITH A LARGE PARAMETER IN PLANAR DOMAINS

Konstantin Pankrashkin

Laboratoire de mathématiques - UMR 8628, Université Paris-Sud, Bâtiment 425, 91405 Orsay Cedex, France konstantin.pankrashkin@math.u-psud.fr http://www.math.u-psud.fr/~pankrash/

PACS 41.20.Cv, 02.30.Jr, 02.30.Tb

Let Q c R2 be a domain having a compact boundary S which is Lipschitz and piecewise C4 smooth, and let v denote the inward unit normal vector on S. We study the principal eigenvalue E(3) of the Laplacian in Q with the Robin boundary conditions df/dv + ¡3f = 0 on S, where 3 is a positive number. Assuming that S has no convex corners, we show the estimate E(¡3) = —¡2 - Ymax3 + O(322 ) as 3 m where Ymax is the maximal curvature of the boundary.

Keywords: eigenvalue, Laplacian, Robin boundary condition, curvature, asymptotics. 1. Introduction

Let Q c R2 be an open connected set having a compact Lipschitz piecewise smooth boundary E. For ft > 0 consider the operator Hp which is the Laplacian / m —Af with the Robin boundary conditions,

/ + ft/ = 0 on E,

where v is the inward unit normal vector. More precisely, Hp is the self-adjoint operator in L2(Q) associated with the sesquilinear form

hp (f,g) = // V/Vg dx — ft f/g da, dom hp = H^Q); (1)

J Jn Js

here a denotes the one-dimensional Hausdorff measure on E. The operator Hp is semibounded from below. If Q is bounded, then Hp has a compact resolvent, and we denote by Ej (ft), j G N, its eigenvalues taken according to their multiplicities and enumerated in the non-decreasing order. If Q is unbounded, then the essential spectrum of Hp coincides with [0, +œ), and the discrete spectrum consists of a finite number of eigenvalues, which we denote again by Ej (ft), j G {1,... ,Np}, and enumerate them in the non-decreasing order taking into account the multiplicities. In the both cases, the principal eigenvalue E(ft) := E1(ft) may be defined through the Rayleigh quotients

cvm • t hp(/,f )

E (ftH^itf , YpÛ-.

o=fedomhp ||f ||L2(n)

It is easy to check that E(ft) < 0: for bounded Q one can test on f =1, and for unbounded Q, one may use f (x) = exp ( — |x|a/2) with small a > 0.

The study of the principal eigenvalue arises in several applications: work [1] discusses the stochastic meaning of the Robin eigenvalues, paper [2] shows that the eigenvalue problem

appears in the study of long-time dynamics related to some reaction-diffusion processes, and a discussion of an interplay between the eigenvalues and the estimate of the critical temperature in a problem of superconductivity may be found in [3].

In the present note, we are interested in the asymptotic behavior of E(ft) for large values of ft. For bounded Q, this question was already addressed in numerous papers. It was conjectured and partially proven in [2] that one has the asymptotics

E(ft) = —Cnft2 + o(ft2) as ft m (2)

for some constant Cn > 0. It seems that the paper [4] contains the first rigorous proof of the above equality for the case of a C1 smooth £, and in that case one has Cn = 1, as predicted in [2]. Under the same assumption, it was shown in [5] that the asymptotics Ej (ft) = —ft2 + o(ft2), ft m hold for any fixed j e N. The paper [6] proved the

asymptotics (2) for domains whose boundary is smooth with the possible exception of a finite number of corners. If the corner opening angles are aj e (0,n) U(n, 2n), j = 1,... , m, and 0 := minaj/2, then Cn = (sin0)-2 if 0 < n/2, otherwise Cn = 1. We remark that the paper [6] formally deals with bounded domains, but the proofs can be easily adapted to unbounded domains with compact boundaries. It should pointed out that domains with cusps need a specific consideration, and the results are different [6,7]. Various generalizations of the above results and some related questions concerning the spectral theory of the Robin Laplacians were discussed in [7-12]. The aim of the present note is to refine the asymptotics (2) for a class of two-dimensional domains. More precisely, we calculate the next term in the asymptotic expansion for piecewise C4 smooth domains whose boundary has no convex corners, i.e. we assume that either the boundary is smooth or that all corner opening angles are larger than n; due to the above cited result of [6] we have Cn = 1 in the both cases.

Let us formulate the assumptions and the result more carefully. Let £k, k = 1,..., n, be non-intersecting C4 smooth connected components of the boundary £ such that £ = (J£k. Denote by the length of £k and consider a parametrization of the closure £k by the arc length, i.e. let [0,-k] 3 s m rk(s) = (Tk,1(s), rk,2(s)) e £k be a bijection with |r'k| = 1, such that rk e C4([0,-4], R2), and we assume that the orientation of each rk is chosen in such a way that vk(s) := (— r'k 2(s), r'k ^s)) is the inward unit normal vector at the point rk(s) of the boundary. If two components £j, £k meet at some point P := r(-j) = rk(0), then two options are allowed: either £j U £k is C4 smooth near P or the corner opening angle at P measured inside Q belongs to (n, 2n).

We denote by Yk(s) the signed curvature of the boundary at the point rk(s) and let Ymax denote its global maximum:

Yk(s) := rk,i(s)rfc,2(s) — rk,i(s)rk,2(s), Ymax := ^x max Yk(s);

ke{i,...,n> se[o,4]

note that the decomposition of the boundary £ into the pieces £k is non-unique, but the value Ymax is uniquely determined. Our result is as follows:

Theorem 1. Under the preceding assumptions there holds

E(ft) = —ft2 — Ymaxft + Otft2) as ft m

We believe that it is hard to improve the asymptotics without any additional information on the set at which the curvature attains its maximal value. For example, one may expect that the case of a curvature having isolated maxima and the case of a piecewise constant curvature should give different resolutions of the remainder, and we hope to progress in this direction in subsequent works.

At first sight, the Robin eigenvalue problem may look rather similar to the eigenvalue problem for ^-potentials supported by curves, see e.g. [13-15]. This first impression is wrong, and the result of Theorem 1 concerning the secondary asymptotic term is very different from the one obtained in the papers [13,14] for strong ^-potentials; nevertheless, a part of the machinery of [13] plays an important role in our considerations. On the other hand, the asymptotic behavior of the principal Robin eigenvalue shows some analogy with the lowest eigenvalue of the Neumann magnetic Laplacian studied in the theory of superconductivity [16-18].

2. Dirichlet-Neumann bracketing on thin strips

In this section we introduce and study an auxiliary eigenvalue problem, and the results obtained will be used in the next section to prove theortem 1. )

Let - > 0 and let r : [0,-] m R2, s m r(s) = ^i(s), ^(s)) e R2, be an injective C4 map such that |r'(s)| = 1 for all s e (0,-). We denote

S := rt(0,-)), k(s) := ^(s^s) — ^(s^s), Kmax := max k(s),

K := max U(s)! + max U^s) + max ^"(s)!. se[0/] 1 1 se[o,^] 1 1 se[o,^] 1 1

Due to k e C2([0,-]), the above quantity K is finite. For a > 0, consider the map

x (C.a) m r2, *.(.,„)= —

As shown in [13, Lemma 2.1], for any a e (0,a0), a0 := (2K)-1, the map $a defines a diffeomorphism between the domains Da := (0,-) x (0,a) and Qa := $«(□«). In what follows, we always assume that a e (0, a0) and we will work with the usual Sobolev space Hi(Qa) and its part HH0i(Qa) := {/ e Hi (Qa) : / |~dQa= 0}; here the symbol means the trace of the function on the indicated part of the boundary.

Here, we introduce two sesquilinear forms in L2(Qa). The first one, hN,a, is defined on dom hN,a := Hi(Qa) by the expression

h?'a(/,g)= // V/Vgdx - ß /gda, JJna Js

and the second one, h^0", is its restriction to dom := H^Q^. Both forms are densely defined, symmetric, closed and semibounded from below, and we denote

En/d (ft, a) = inf ^ J. (3)

0=fGdomllf HL2(n0)

We show the following results:

Lemma 2. There exists ai > 0 such that for any a e (0,ai) one has the estimate

En/d (ft, a) = —ft2 — Kmaxft + O(ft2) as ft m

The rest of this section is devoted to the proof of lemma 2. We first introduce a suitable decomposition of Qa and then provide two-side eigenvalue estimates using operators with separated variables.

Define Ua : L2(Q„) ^ L2(D„) by (Uf) (s,u) = ^/1 — u«(s)f ($a(s,«)). Clearly, is a unitary operator, and one has Ua(H^Q^) = H 1(Da) and

Ua(^}(na)) = Hl(na) := {f e H 1(Da) : f (0, ■) = f (€, ■) = 0 and f (-,a) = 0},

where the restrictions should be again understood as the traces. Using integration by parts, one may easily check that for any f, g e H1 (Qa), one has hN'a(f, g) = qN'a(Uaf, Uag), where the form qN'a is defined on the domain dom qN'a := H 1(Da) by the expression

/f 1 d/ög f f d f dg

La (1 - ,*(.))» && dS dU dUdU d"

// V(s,u)/g ds du — ß / /(s, 0)g(s, 0) ds

•/./□a ./0

1 _ 1 K(s) _

- / K(s) /(s, 0)g(s, 0) ds + - / ( ' /(s, a)g(s, a) ds (4)

2 Jo 2 Jo 1 — aK(s)

1 r u

+ -k'(£) / -3 /(€,u)g(€,u)du

2

1—

0 1 — UK

1 r U

— 2k'(0) / (-7"7)3 /(0, u)g(0, u) du

2 Jo (1 — uk(0)J

with

uk"(s) 5u2k'(s)2 k(s)2

V(s,u) := -—-3 +--——4 +--—-2.

2(1 — uk(s)) 3 4(1 — uk(s)) 4 4(1 — uk(s)) 2

Similarly, for any /,g G H1^), one has hD'a(/,g) = qD'a(Ua/, U0g), where is the

restriction of to the domain dom qß5'0 := ii01(^0); note that for /, g G dom q^'0 the three last terms on the right-hand side of (4) vanish. Using the unitarity of U0 we may rewrite the equalities (3) in the form:

qN/D'0(/,/)

en/d (ß, a) = inf -. (5)

7 0=f Gdom qN ll/

We would like to reduce the estimation of these quantities to the study of the eigenvalues of certain one-dimensional operators.

Using the one-dimensional Sobolev inequality on (0,£) we see that one can find a constant C > 0 independent of a, such that for all / G H 1(^0), one has

jf /(0,u)|2 du + ^ /(£,u)|2 du < C(JJ d/2dsdu + JJ |/|2dsdu).

One can also find a constant v > 0, such that |V(s,u)| < v for all (s,u) G □ and all a G (0,a0). Furthermore, again for (s,u) G □ and any a G (0,a0), we have

K(s)

21 < 2K, - < -— < 2.

1 — aK(s) '3 1 — uk(s)

For any M G N, we denote

S := M, Ij :=j - j □m := Ij X (0,a),

:= inf k(s), kMj := sup k(s), j = 1,..., M,

se/M " se/j

M

and introduce functions kM : (0,£) ^ R as follows: kM(s) := kM, if s G Ij, and kM(j£) := 0

M'

\-i

for j = 1,..., M — 1. In addition, we assume that 0 < a < (10KC) i. Now, we introduce two new sesquilinear forms which will be used to obtain a two-side estimate for EN/D(ft, a). The first one, t-,M,a, is defined by

MM

domt-'M'" = H^ (J DÏm) * © H\DÏm) , j=i j=i

t-'M'"(/,g) = - 4aKC ) JLa fs dsds du+/La Hds du

- (v + 4aKC) // /g ds du -i (ß + km(s)) /M)g(s, 0) ds

— K / /(s, a)g(s, a) ds.

J0

The second one, t+,M,a, is defined on the domain dom t+,M,a = 0M=i _H0i(Da,M),

Hij) := { / e Hi(aa,M) : /j — ¿, ■) = /j ■) = 0 and /(-,a) = 0},

through

'+"</•') = V! did! ds du + || ds du

pi

+ v II /g ds du - f (ß + ^^ / (s, 0)g(s, 0) ds.

'□a ./0

N,a ^ j j.-,M,a

, , , - i / i / » » i z"- ' ' — ' '

equalities

.-M,at e e\ ^ N,af £ c\ £ - 1 No.

One clearly has the inclusions dom t+' M ' " C dom q^ ' " C dom qN " C dom t- ' M ' " and the in-

t-'M'"(/,/) < qß (/, /), / G domqß ■

qN'a(/ / ) = qD'a(/ / / G dom qDa

«,?'"(/,/) < t,l'M'"(/,/), / G domt+'M",

which justify the estimates

EM CM < En GM < ED (M < E+GM, (6)

where we denote

t±'M'"(/, /)

EM (ß,a):= inf

0=f edom t±,M,a 11/ ll^Oa)

Now, we are going to estimate E±(ft, a) using separation of variables. Note that the forms t±,M,a are densely defined, semibounded from below and closed in L2(Oa), therefore, they define some self-adjoint operators T±,M,a in L2(Oa), and E±(ft, a) = inf specT±,M,a. On the

a

other hand, due to the fact that the domains M are disjoint and isometric to one another , we can identify y±,M,a ~ 0m=i T±jM,a, where T±jM,a are self-adjoint operators acting in L2(D^,a), П<5,а := (0,5) x (0,a), and associated respectively with the sesqulinear forms t±jM,a,

'4 ^N fS f"d/dg^^ , fs f" d/dg

~2

i-M'a(/'g) = (9 - 4aKC4 I dSdSduds + L I dUdUduds

rS pa K+ \ i'S _

- (v + 4aKC) / /g du ds - [ß + )/ /(s, 0)g(s, 0) ds

0 0 0

S

— X/ f (s, a)g(s, a) ds, domj 'a = H (□¿a), ./0

ija(f,g) = 4i L ds dsduds+i I iufududs

J fg du ds — (ft + Jo f (s, 0)g(s, 0) ds,

dom j = {f e H1 (□¿a) : f (0, ■) = f (i, ■) = 0 and f (■, a) = 0}.

It is routine to check that T±jM'a = QM ® 1 + 1 ® L±a, where QM are the operators acting in L2(0, £) as follows:

QMf = — (9 — f" — (v + 4aXC) f,

dom QM = {/ e H2(0,i) : /'(0) = /'(¿) = 0},

QM / = -4/'' + v/,

dom QM = {/ e H2(0,i) : /(0) = /(i) = 0},

and are the self-adjoint operators in L2(0, a) both acting as L±a f = — f" on the domains

T2(n . fl(n\ , A« , KMj

dom= {/ G H2(0,a) : /'(0) + (ft + - j/(0) = 0, /'(a) — Kf (a) = 0

domL+j = {/ G H2(0,a) : /'(0) + (ft + Kf )/(0) = 0, /(a) = 0}.

The spectra of QM can be calculated explicitly; in particular, one has

4n2 4n2 M 2

inf spec QM = — v — 4aKC, inf spec QM = —— + v = —--+ v.

o2 £2

Therefore, denoting E±'j (ft, a) := inf spec LÎ^', we arrive at

ß, a '

EM (в, a) = min (inf spec TßjM'a) = —v — 4aKC + min E"'j (в, a),

22

4^2^ 2

E+(ft, a) = min (inf specT+j 'a) = ——--+ v + minE+J(ft, a).

To study the lowest eigenvalues of L^, we prove two auxiliary estimates.

(7)

p,a'

Lemma 3. For a, ft, 7 > 0, let Aap7 denote the self-adjoint operator in L2(0,a) acting as / m — /'' on the functions / G H2(0,a) satisfying the boundary conditions

/'(0) + ft/(0) = /'(a) — 7/(a) = 0, and let E(a, ft, 7) be its lowest eigenvalue. Let ft > 27

and fta > 1, then ft2 < —E(a, ft, 7) < ft2 + 123ft2e-2^a

Proof. Let k > 0. Clearly, E = —k2 is an eigenvalue of Aa,^,7 if one can find (Ci,C2) e C2 \ {(0,0)} such that the function / : x m Ciekx + C2e-kx belongs to the domain of Aa,^,7. The boundary conditions give

0 = / '(0) + ft/(0) = (ft + k)Ci + (ft — k)C2,

0 = / '(a) — 7/(a) = (k — 7)ekaCi — (k + 7 )e-kaC2,

and one has a non-zero solution if the determinant of this system vanishes, i.e. if k satisfies the equation (k + ft)(k + 7)e-ka = (k — ft)(k — 7)eka. Let us look for solutions k e (ft, One

may rewrite the preceding equation as

g(k) = h(k), g(k) = k—ft, h(k) = e2ka. (8)

Both Unctions g and h are continuous. It is readily seen that the function g is strictly decreasing on (ft, with g(ft+) = and g(+rc>)=1. Conversely, for ft > 27, the Unction h is strictly increasing in (ft, being the product of two strictly increasing positive functions,

and we have h(ft+) = e2^a(ft — 7)/(ft + 7) < and h(+rc>) = These properties of g and h show that there exists a unique solution k = k(a,ft,7) e (ft, of (8) and that E (a, ft, 7) = —k(a,ft,7)2.

To obtain the required estimate we use again the monotonicity of h on (ft, and the inequality ft > 27. We have

|±| = g(k) = h(k) >h(ft+)= ft^e- > ^,

which gives (1 — 3e-2^a)k < (1 + 3e-2^a)ft. The assumption fta > 1 gives the inequality 3e-2^a < 1/2, and we arrive at

1 1 3P-2^a

k < 1 + 3 0« ft < (1 + 3e-2^a)(1 + 15e-2^a)ft < (1 + 41e-2^a)ft 1 — 3e 2^a

and k2 < (1 + 41e-2^a)2ft2 < (1 + 123e-2^a)ft2. Together with the inclusion k e (ft, this gives the result. □

Lemma 4. For a, ft > 0, let na,^ denote the self-adjoint operator in L2(0,a) acting as / m — /'' on the functions / e H2(0,a) satisfying the boundary conditions /'(0) + ft/(0) = /(a) = 0, and let E(a,ft) be its lowest eigenvalue. Assume that fta > 4/3, then ft2 — 4ft2e-^a < —E(a, ft) < ft2.

Proof. Let k > 0. Proceeding as in the proof of lemma 3, we see that E = —k2 is an eigenvalue of na,^ if k satisfies the equation (ft + k)e-ka = (ft — k)eka. As the left sideof the equation is strictly positive, the right side must be positive as well, which means that all solutions k belong to (0,ft). Let us rewrite the equation in the form g(k) = 0 with g(k) := log(ft + k) — log(ft — k) — 2ka. One has g(0) = 0, the function g is strictly decreasing in (0, k0) and strictly increasing in (k0,ft), with k0 := \Jft2 — ft/a. Moreover, g(ft—) = Therefore, the equation g(k) = 0 has a unique solution in (k0,ft). It fol-

lows from the assumption fta > 4/3 that k0 > ft/2, and we can represent k = ft — s with some s e (0,ft/2). Using again the condition g(k) = 0, we arrive at the inequality log s = log(2ft — s) — 2fta + 2sa < log(2ft) — fta, which gives s < 2fte-^a and k = ft — s > ft(1 — 2e-^a). Finally, —E(a,ft) = k2 > ft2(1 — 2e-^a)2 > ft2(1 — 4e-^a). Together with the first inequality k < ft this gives the desired estimate . □

Let us complete the proof of lemma 2. Denote a1 := min {a0, (10XC)—-1} and pick any a e (0,a1), and let ft > 3X + 1 + 4/(3a). Applying lemma 3 to each of the operators L—'¿j and lemma 4 to each of the operators L+a, we arrive at the estimates

E- ,j (в, a) > - в + Ф)2 - 123 (в + KMj N 2

2

2

exp

- 2a в +

KM,j

E+ j (в, a) <-( в + 2 + 4 в + KMjN 2

2

2

exp

- a в +

к

M,j 2

To simplify the form of the remainders, we choose fta > 0 sufficiently large such that for ft > fta we have

„ K)2

в + ~2J eXP

- 2a (в - f) +^в + у)2 exp - a(,

К \ 2

-*(в- !)

1

< -,

- в,

then for в > ва + 3K + 1 + 4/(3a) and all j = 1,..., M, we have

K2 1

E- ,з (в, a) > -в2 - kM,7в - -г" - ö, E+,з (в, a) < -в2 - kM,7-в + тг.

4в

в

Using the inequality KMj — Kmax, we obtain

K2 1

min E ,з (в, a) > -в2 - Ктахв----~Б.

з 4 в

(9)

Conversely, let l G {1,..., M} be such that kM such that k(s) = Kmax. Using the Taylor expansion near s, we obtain

M l = ктах. This means that there exists s e I

i

M

- + К

KM , l — KM , l — ктах = ктах ^^ .

(10)

In the previous considerations the number M was arbitrary, and now we pick M e [ft1, 2ft 3 ] n N, then

min E+,j (в, a) — E^(в^) < -в2 - ^в + тг

j M,l

1

в

M'

1

в

il 1

= -в2 - Ктахв + TT в + ä — -в2 - Ктах в + К^в 3 + ^ (11)

Substituting the estimates (9) and (11) into (7) we arrive at

E^, a) — -в2 - Ктахв + К^в3 + "Г +

2 1 4n2M2

+v

-в2 - Ктахв + К +

в

16П2N 2 1

3 + v + 1.

K2 1

EM (в, a) — -в2 - Ктахв - к - v - 4aKC - 1,

and the assertion of lemma 2 follows from the two-side estimates (6)

2

3. Proof of Theorem 1

:(0,4) x (0,a) ^ R2, $fc>0(s,u)

k = 1,..., n.

We continue using the notation introduced just before theorem 1. For a > 0, consider the maps

TM(s) — urk,2(s)N ,rfc,2(s) + urk,i(s),

As in section 2, we can find a0 > 0 such that for any a e (0, a0) these maps are diffeo-morphic between Ok,a := (0,-4) x (0,a) and Qk,a := $fc,a(Ofc,a), that Qk,a C Q, and that Qj,a H Qk,a = 0 for j = k. Note that the last property follows from the fact that the opening

angles of the boundary corners (if any) are reflex. In addition, we set Q0,a := Q \ ^ Ufc=i Qfc,a).

Denote H0(Qfc,a) := {/ e Hi(Qfc,a) : / rdnfca\Efc = 0}, k = 1,..., n, and introduce two new

sesquilinear forms hN/D,a in L2(Q), both defined by the same expression as h^ on the domains

dom h

N,0

ß

and define

n / n

0 H 1(nfc>a), dom hD'a = Ho1(na>0) U ( 0 H^«) fc=0 ^ fc=1

N/D,0(/,/)

En/d (ß, a) :=

inf

a=f edom hN/D'a

h.

2

L2(Q)

(12)

Due to the inclusions dom hD,a C dom h^ C dom hN,a, we have the inequalities

En (ft, a) < E (ft) < Ed (ft, a).

Furthermore, due to the fact that the parts Qk a are disjoint and that the set £ H dQ0 a is finite (this is exactly the set of the corners), we have the equality EN/D (ft, a) = minfce{0,...,ra} Ek,N/D (ft, a), with

Eo,n(ß, a) := Efc)w(ß, a) := Eo,d(ß, a) = Efc,D(ß, a) :=

inf

a=f eH1(Qo,a)

IV^i2(Ho,a)

2

L2(Ho,a)

inf

a=f €H1(Qfc,a)

IIV/1|2

¿2(Hfc,a)

2

L2(£fc)

k = 1,

, n,

¿2(Hfc,a)

inf

a=f eHi(Qo,a) inf

a=f e-H01(nfc,a)

IV/IIi2(no,a)

2

L2(Ho,a)

IIV/II2

L2(Hfc,a)

2

L2(Sfc)

k = 1,

n.

We have clearly E0,N/D(ft, a) > 0. Furthermore, in virtue of lemma 2 we can find a > 0 such that for each k e {1,...,n} for ft m we have

Efc,N/D (ß, a) = -ß2 - 7fc,maxß + O (ß2) ,

7fc,i

max 7fc(s), sepA ]

which gives EN/D(ft, a) = —ft2 — 7maxft + 0(ft2), and the assertion of theorem 1 follows from the two-side estimate (12).

Remark 5. A more detailed asymptotic analysis is beyond the scope of the present note, but we mention one case in which the remainder estimate can be slightly improved with minimal efforts. Namely, assume that one of the following conditions is satisfied:

2

2

• the boundary £ is of class C4 (i.e. there are no corners),

• the curvature does not attain its maximal value Ymax at the corners,

then

E (в) = -в2 - Ymax в + ) as в ^ (13)

Indeed, let us pick any k e {1,..., n} such that Yk ,max = Ymax and revise the proof of lemma 2 with Г := Гк, к := Yfc and t := tk. For any s e [0,t] with k(s) = Kmax we have then k'(s) = 0, and we may replace the inequality (10) with

+ 2 2 Kt2

KM,l — KM,l — Kmax = Kmax 2 ,

and by choosing M e [^в, 2 -^в] n N we arrive at the estimate en/d(в, a) = —в2 — Kma^ + 0(\/в) as в ^ which in turn gives the asymptotics

(13).

Acknowledgments

The research was partially supported by ANR NOSEVOL and GDR Dynamique quantique.

References

[1

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