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Naiara Arrizabalaga, Loïc Le Treust, Albert Mas, Nicolas Raymond
To cite this version:
Naiara Arrizabalaga, Loïc Le Treust, Albert Mas, Nicolas Raymond. The MIT Bag Model as an infinite mass limit. Journal de l’École polytechnique - Mathématiques, École polytechnique, 2019, 6, pp.329-365. �10.5802/jep.95�. �hal-01863065v2�
NAIARA ARRIZABALAGA, LO¨IC LE TREUST, ALBERT MAS, AND NICOLAS RAYMOND
Abstract. The Dirac operator, acting in three dimensions, is considered. As- suming that a large massmą0 lies outside a smooth enough and bounded open set ΩĂR3, it is proved that its spectrum approximates the one of the Dirac op- erator on Ω with the MIT bag boundary condition. The approximation, modulo an error of orderop1{?
mq, is carried out by introducing tubular coordinates in a neighborhood ofBΩ and analyzing one dimensional optimization problems in the normal direction.
Contents
1. Introduction 2
1.1. Context 2
1.2. The Dirac operator with large effective mass 2
1.3. Squared operators, heuristics, and main results 3
1.4. Organization of the paper 7
2. About the exterior optimization problem 8
2.1. Organization of the section 8
2.2. Existence, uniqueness and Euler-Lagrange equations 9
2.3. Agmon estimates 10
2.4. Optimization problem in a tubular neighborhood 12 2.5. One dimensional optimization problem with parameters 14
2.6. Asymptotic study of Λm,m´1{2pvq. 16
2.7. End of the proof of Proposition 2.1 22
3. A vectorial Laplacian with Robin-type boundary conditions 22
3.1. Preliminaries: proof of Lemma 1.13 22
3.2. Asymptotics of the eigenvalues 23
3.3. Proof of Proposition 3.1 23
4. Proof of the main theorem 26
4.1. First term in the asymptotic 26
4.2. Second term in the asymptotic 28
Appendix A. Sketch of the proof of Lemma 4.3 34
Fundings 35
References 35
2010 Mathematics Subject Classification. 35J60, 35Q75, 49J45, 49S05, 81Q10, 81V05, 35P15, 58C40.
Key words and phrases. Dirac operator, relativistic particle in a box, MIT bag model, spectral theory.
1
1. Introduction
1.1. Context. This paper is devoted to the spectral analysis of the Dirac operator with high scalar potential barrier in three dimensions. More precisely, we will assume that there is a large mass m outside a smooth and bounded open set Ω. From physical considerations, see [8, 10], it is expected that, when m becomes large, the eigenfunctions of low energy do not visit R3zΩ and tend to satisfy the so-called MIT bag condition onBΩ. This boundary condition, that we will define in the next section, is usually chosen by the physicists [13, 10, 11], in order to get a vanishing normal flux at the bag surface. It was originally introduced by Bogolioubov in the late 601s [8] to describe the confinement of the quarks in the hadrons with the help of an infinite scalar potential barrier outside a fixed set Ω. In the mid 701s, this model has been revisited into a shape optimization problem named MIT bag model [13, 10, 11] in which the optimized energy takes the form
ΩÞÑλ1pΩq `b|Ω|,
whereλ1pΩqis the first nonnegative eigenvalue of the Dirac operator with the bound- ary condition introduced by Bogolioubov, |Ω| is the volume of Ω Ă R3 and b ą 0.
The interest of the bidimensional equivalent of this model has recently been renewed with the study of graphene where this condition is sometimes called “infinite mass condition”, see [1, 7]. The aim of this paper is to provide a mathematical justification of this terminology, and extend to dimension three the work [16]. More precisely, we show the convergence of the eigenvalues for the Dirac operator with high scalar potential barrier to the ones of the MIT bag Dirac operator. In dimension two, this follows by the convergence of the spectral projections shown in [16]. Regarding the first eigenvalue of the MIT bag Dirac operator, we also find the first order term in the asymptotic expansion of the eigenvalues given by the high scalar potential barrier, showing its dependence on geometric quantities related to BΩ. This is a novel result with respect to the ones in [16].
1.2. The Dirac operator with large effective mass. In the whole paper, Ω denotes a fixed bounded domain of R3 with C2,1 boundary.
Let us recall the definition of the Dirac operator associated with the energy of a relativistic particle of mass m0 P R and spin 12, see [17]. The Dirac operator is a first order differential operator pH,DompHqq, acting on L2pR3;C4q in the sense of distributions, defined by
(1.1) H “cα¨D`m0c2β , D“ ´i~∇,
where DompHq “ H1pR3;C4q, c ą 0 is the velocity of light, ~ ą 0 is Planck’s constant, α“ pα1, α2, α3qand β are the 4ˆ4 Hermitian and unitary matrices given by
β “
ˆ 12 0 0 ´12
˙
, αk “
ˆ 0 σk
σk 0
˙
for k “1,2,3. Here, the Pauli matrices σ1, σ2 and σ3 are defined by
σ1 “
ˆ 0 1 1 0
˙
, σ2 “
ˆ 0 ´i i 0
˙
, σ3 “
ˆ 1 0 0 ´1
˙ , and α¨X denotes ř3
j“1αjXj for any X “ pX1, X2, X3q. In the following, we shall always use units with ~“c“1.
The Dirac and Pauli matrices are chosen in such a way that the Dirac operator pH,DompHqqis self-adjoint, and satisfies
H2 “14pm20´∆q,
(see for instance [17, Section 1.1]). Let us also mention that its spectrum is p´8,´|m0|s Y r|m0|,`8q.
In this paper, we consider particles with large effective mass m " m0 outside Ω. Their kinetic energy is associated with the self-adjoint operatorpHm,DompHmqq defined by
Hm “α¨D` pm0`mχΩ1qβ ,
where Ω1 is the complementary set of Ω, χΩ1 is the characteristic function of Ω1 and DompHmq “ H1pR3;C4q. The essential spectrum ofpHm,DompHmqqis
p´8,´|m0`m|s Y r|m0`m|,`8q.
In this paper, the mass m0 is not assumed to be positive since this assumption is not used in the proofs (see also Remark 1.10).
Notation 1.1. In the following, Γ :“ BΩ and for all x P Γ, npxq is the outward- pointing unit normal vector to the boundary, Lpxq “ dnx denotes the second fun- damental form of the boundary, and
κpxq “TrLpxq and Kpxq “det Lpxq
are the mean curvature and the Gauss curvature of Γ, respectively.
Definition 1.2. The MIT bag Dirac operator pHΩ,DompHΩqq is defined on the domain
DompHΩq “ tψ PH1pΩ;C4q : Bψ “ψ on Γu, with B “ ´iβpα¨nq, by HΩψ “ Hψ for all ψ P DompHΩq. Observe that the trace is well-defined by a classical trace theorem.
If Γ is C2, the operator pHΩ,DompHΩqq is self-adjoint with compact resolvent [15, 3, 9, 6, 4].
Notation 1.3. We denote by x¨,¨y the C4 scalar product (antilinear w.r.t. the left argument) and by x¨,¨yU the L2 scalar product on the setU ĂR3.
Notation 1.4. We define, for every nPS2, the orthogonal projections
(1.2) Ξ˘ “ 14˘B
2 associated with the eigenvalues˘1 of the matrix B.
1.3. Squared operators, heuristics, and main results. The aim of this paper is to relate the spectra of Hm and HΩ in the limit mÑ `8.
Notation 1.5. LetpλkqkPN˚ and pλk,mqkPN˚ be the increasing sequences defined by λk “ inf
V ĂDompHΩq, dimV “k
sup
ϕPV, kϕkL2pΩq“1
HΩϕ
L2pΩq
“ sup
tψ1,...,ψk´1uĂDompHΩq
inf
ϕPspanpψ1, . . . , ψk´1qK, kϕkL2pΩq“1
HΩϕ
L2pΩq
,
and
λk,m “ inf
V ĂH1pR3;C4q, dimV “k
sup
ϕPV, kϕkL2pR3q“1
kHmϕkL2pR3q
“ sup
tψ1,...,ψk´1uĂH1pR3;C4q
inf
ϕPspanpψ1, . . . , ψk´1qK, kϕkL2pR3q“1
kHmϕkL2pR3q,
for k PN˚ and mą 0. Here,N˚ :“Nzt0u. By the min´max characterization and the properties given in Definition 1.2, the sequencepλkqkPN˚ is made of all the eigen- values of the operator |HΩ|, each one being repeated according to its multiplicity.
Similarly, the terms of the sequence pλk,mqkPN˚ that satisfy λk,mă |m0`m|
are the eigenvalues of |Hm|lying below its essential spectrum r|m0`m|,`8q, each one being repeated according to its multiplicity. For k large enough, this sequence may become stationary at |m0 `m|.
1.3.1. The quadratic forms. At first sight, it might seem surprising thatλk andλk,m are related, especially because of the boundary condition of HΩ. It becomes less surprising when computing the squares of the operators. This is the purpose of the following lemma.
Lemma 1.6. Let ϕP DompHΩq and ψ PH1pR3;C4q. Then (1.3) }HΩϕ}2L2pΩq“Qintpϕq:“ }∇ϕ}2L2pΩq`
ż
Γ
ˆκ 2 `m0
˙
|ϕ|2dΓ`m20kϕk2L2pΩq, where κ is defined in Notation 1.1, and
}Hmψ}2L2pR3q“ }∇ψ}2L2pΩq` }∇ψ}2L2pΩ1q` }pm0`mχΩ1qψ}2L2pR3q
´mRexBψ, ψyΓ
“ }∇ψ}2L2pΩq` }∇ψ}2L2pΩ1q` }pm0`mχΩ1qψ}2L2pR3q
`m}Ξ´ψ}2L2pΓq´m}Ξ`ψ}2L2pΓq. (1.4)
Proof. The equality (1.3) is proved for instance in [2, Section A.2].
Let ψ PH1pR3;C4q. By integrations by parts,
}Hmψ}2L2pR3q “ }α¨Dψ}2L2pR3q` }pm0`mχΩ1qψ}2L2pR3q`2mRexα¨Dψ, βψyΩ1
“ }∇ψ}2L2pR3q` }pm0`mχΩ1qψ}2L2pR3q´mRexBψ, ψyΓ. Then, note that, for all ψ PH1pR3;C4q,
RexBψ, ψyΓ “ }Ξ`ψ}2L2pΓq´ }Ξ´ψ}2L2pΓq.
Considering (1.4) leads to the following minimization problem, for v P H1pΩq, (1.5) Λmpvq “inftQmpuq , uPVvu, Qmpuq “ }∇u}2L2pΩ1q`m2}u}2L2pΩ1q, where
Vv “ tuP H1pΩ1,C4q s.t. u“v on Γu.
A classical extension theorem (see [12, Section 5.4]) ensures that Vv is non-empty.
1.3.2. Heuristics. In this paper, we will analyse the behavior of Λmpvqand prove in particular (see Proposition 2.1) that there exists C ą0 such that for m large, and allv P H1pΩ;C4q
(1.6) op1q ě Λmpvq ´ ˆ
m}v}2L2pΓq` ż
Γ
κ
2|v|2dΓ
˙
ě ´C
m}v}2H1pΓq. Replacing m by m0`m in (1.6), we get, for allψ P H1pR3;C4q,
}Hmψ}2L2pR3q ě }∇ψ}2L2pΩq`m20kψk2L2pΩq
` ż
Γ
ˆκ 2 `m0
˙
|ψ|2dΓ`2m}Ξ´ψ}2L2pΓq´ C
m}ψ}2L2pΓq. (1.7)
Take any eigenfunctionϕof HΩ and consider a minimizeruϕ of (1.5) for v “ϕand m replaced by m`m0. Then, letting ψ “1Ωϕ`1Ω1uϕ PH1pR3;C4q, we get
}Hmψ}2L2pR3q“ }∇ϕ}2L2pΩq`m20kψk2L2pΩq`Λm`m0pϕq ´m}Ξ`ϕ}2L2pΓq. With (1.6) at hand, we deduce that, for all j PN˚,
λ2j,m ďλ2j `op1q.
Conversely, if we are interested in the eigenvalues of pHmq2 that are of order 1 whenmÑ `8, we see from (1.7) that the corresponding normalized eigenfunctions must satisfy Ξ´ψ “ Opm´1q and, in particular, Bψ “ ψ `Opm´1q. Thus, we get formally, for all j PN˚,
λ2j,m ěλ2j `op1q.
The aim of this paper is to make this heuristics rigorous. We now state our main theorem.
Theorem 1.7. Let ΩĂR3 be a bounded domain of class C2,1 (i.e. the derivative of the curvatures is bounded). The singular values of Hm can be estimated as follows:
(i) limmÑ`8λk,m “λk, for all k PN˚.
(ii) Let k1 P N˚ be the multiplicity of the first eigenvalue λ1 of |HΩ|. For all k P t1, . . . , k1u, we have
λk,m “
˜
λ21`νk
m `o ˆ 1
m
˙¸1{2
, where
(1.8) νk “ inf
V Ăkerp|HΩ| ´λ1q, dimV “k
sup
uPV, kukL2pΩq“1
ηpuq,
with
ηpuq “ ż
Γ
¨
˝
|∇su|2
2 ´ |pBn`κ{2`m0qu|2
2 `
˜ K
2 ´ κ2 8 ´ λ21
2
¸
|u|2
˛
‚dΓ. Here, pλkqkPN˚ andpλk,mqkPN˚ are defined in Notation 1.5, and κ and K are defined in Notation 1.1.
Bn is the outward pointing normal derivative and ∇s is the tangential gradient on Γ.
Remark 1.8. The max-min formula (1.8) makes sense since kerp|HΩ| ´ λIdq Ă H2pΩ;C4qfor any eigenvalue λ of |HΩ|.
Remark 1.9. Hm and HΩ anticommute with the charge conjugation C defined, for all ψ PC4, by
Cψ “iβα2ψ,
where ψ P C4 is the vector obtained after complex conjugations of each of the components of ψ (see for instance [17, Section 1.4.6] and [2, Section A.1]). As a consequence, the spectrum of Hm and HΩ are symmetric with respect to 0, and Theorem 1.7 may be rewritten as a result on the eigenvalues of Hm and HΩ. Remark 1.10. Let us define the operator pHĂΩ,DompHĂΩqqon
DompHĂΩq “ tψ P H1pΩ;C4q : Bψ “ ´ψ on Γu
byHĂΩψ “Hψ for allψ PDompHĂΩq. It is the MIT bag Dirac operator withreversed boundary condition (see Definition 1.2). The singular values ofHĂΩare approximated by the singular values of Hm as m tends to ´8. This follows immediately from Theorem 1.7, conjugating all the operators by the chirality matrix
γ5 “
ˆ 0 12 12 0
˙ , and by using the algebraic properties
βγ5 “ ´γ5β , γ5pα¨xq “ pα¨xqγ5, γ5Bγ5 “ ´B, for all xPR3.
Remark 1.11. Our proof of Theorem 1.7 also provides the convergence of the eigen- projectors associated with the first eigenvalues of |Hm|. They converge towards the eigenprojectors associated with the first eigenvalues of |HΩ|, see Lemma 4.1 and Remark 4.2, and [16, Theorem 1] for the two-dimensional case.
Remark 1.12. In view of Theorem 1.7, it is natural to ask if one has convergence of Hm toHΩ in some resolvent sense whenmÑ `8. On one hand, in the recent work [5] it is shown the convergence in the norm resolvent sense for the bidimensional analogues of Hm and HΩ. On the other hand, in [14] the authors study interactions of the free Dirac operator in R3 with potentials that shrink towards BΩ, proving convergence in the strong resolvent sense toδ-shell interactions with precise coupling constants. As m Ñ `8, our operator Hm may be seen as a degenerate case of the interactions with shrinking potentials considered in [14] and, at a formal level, in this case the resulting δ-shell interaction leads to the operatorHΩ.
The above-mentioned results suggest that convergence in the norm (or at least strong) resolvent sense may also hold in our three dimensional setting.
1.3.3. A vectorial Laplacian with Robin-type boundary conditions. Let us also men- tion an intermediate spectral problem whose study is needed in our proof of Theorem 1.7 and that may be of interest on its own. We consider the vectorial Laplacian as- sociated with the quadratic form
Qintmpuq “k∇uk2L2pΩq`m20kuk2L2pΩq` ż
Γ
ˆκ 2 `m0
˙
|u|2dΓ`2m
Ξ´u
2 L2pΓq
(1.9)
for u P DompQintmq “ H1pΩ;C4q and m ą0, where Ξ´,Ξ` are defined by (1.2). By a classical trace theorem, this form is bounded from below. More precisely, we have the following result whose proof is sketched in Section 3.1.
Lemma 1.13. The self-adjoint operator associated with Qintm is defined by
DompLintmq “
"
uPH2pΩ;C4q: Ξ´`
Bn`κ{2`m0`2m˘
u“0 on Γ, Ξ``
Bn`κ{2`m0˘
u“0 on Γ
*
Lintmu“`
´∆`m20˘
u for all uPDompLintmq.
(1.10)
It has compact resolvent and its spectrum is discrete.
Using an integration by parts and the identities (1.2), we get xu, LintmuyΩ“Qintmpuq,
for all uPDompLintmq.
Notation 1.14. Let pλintk,mqkPN˚ denote the sequence of eigenvalues, each one being repeated according to its multiplicity and such that
(1.11) λint1,mďλint2,m ď. . .
The asymptotic behavior of the eigenvalues of Lintm is detailed in the following theorem.
Theorem 1.15. The following holds:
(i) For every k PN˚, limmÑ`8λintk,m“λ2k.
(ii) Let λ be an eigenvalue of |HΩ| of multiplicity k1 P N˚. Consider k0 P N the unique integer such that for all k P t1, . . . , k1u, λk0`k “λ.
Then, for all k P t1,2, . . . , k1u, we have λintk
0`k,m“λ2`µλ,k m `o
ˆ 1 m
˙ , where
(1.12) µλ,k :“ inf
V Ăkerp|HΩ| ´λq, dimV “k
sup
vPV, kvkL2pΩq“1
´
kpBn`κ{2`m0qvk2L2pΓq
2 .
Here, pλkqkPN˚ is defined in Notation 1.5, pλintk,mqkPN˚ in Notation 1.14, and κ in Notation 1.1.
1.4. Organization of the paper. In Section 2, we discuss the asymptotic prop- erties of the minimizers associated with the exterior optimization problem (1.5). In Section 3, we investigate the interior problem given by (1.9). Finally, in Section 4, we prove Theorem 1.7.
In order to ease the reading, we provide here a list of notation regarding the spaces and the quadratic forms, as well as the equation number where they are introduced, that we will use in the sequel:
Key Space domain Variational space Quadratic form Infimum
(1.3) Ω DompHΩq Qint ´
(1.5) Ω1 Vv Qm Λmpvq
(1.9) Ω H1pΩ;C4q Qintm ´
(2.1) Vδ Vv,δ Qm Λm,δpvq
(2.12) Vpm Vpm Qpm Λm,m´1{2pvq
(2.14) p0,?
mq Vpm,κ,K Qpm,κ,K Λm,κ,K
2. About the exterior optimization problem
The aim of this section is to study the minimizers of (1.5) and their properties when m tends to `8. These properties are gathered in the following proposition.
Proposition 2.1. For all v P H1pΩq, there exists a unique minimizer umpvq asso- ciated with Λmpvq, and it satisfies, for all uPVv,
Qmpuq “Λmpvq `Qmpu´umpvqq. Moreover, the following holds:
(i) Assume that Γ is C2. There exist C, m1 ą 0 such that, for every m ě m1, v PH1pΩq,
Cmkvk2H1pΩqěΛmpvq ě ˆ
m}v}2L2pΓq` ż
Γ
κ
2|v|2dΓ
˙
´ C
m}v}2L2pΓq. Assume that Γ is C2,1. There exists C ą0 such that, for every měm1,
(ii) for v PH1pΩq, ˆ
m}v}2L2pΓq` ż
Γ
κ
2|v|2dΓ
˙
`op1q ě Λmpvq.
Here, the term op1q depends on v (not only on the H1 norm of v).
(iii) for all v PH2pΩq, ˇ ˇ
ˇΛmpvq ´Λrmpvq ˇ ˇ ˇď C
m3{2}v}2H3{2pΓq, (iv) for all v PH2pΩq,
ˇ ˇ ˇ ˇ ˇ
}umpvq}2L2pΩ1q´
}v}2L2pΓq
2m ˇ ˇ ˇ ˇ ˇ
ď C
m2}v}2H3{2pΓq,
Λrmpvq “ m ż
Γ
|v|2dΓ` ż
Γ
κ
2|v|2dΓ`m´1 ż
Γ
!|∇sv|2
2 `
´K 2 ´κ2
8
¯
|v|2 )
dΓ.
2.1. Organization of the section. Since there are many steps in the proof of Proposition 2.1, let us briefly describe the strategy:
— In Section 2.2, we explain why the minimizers exist, are unique, and we describe their Euler-Lagrange equations.
— In Section 2.3, we prove Proposition 2.7. This proposition states that, when m goes to `8, the minimizers are exponentially localized near the interface Γ.
This allows to replace our optimization problem on Ω1 by the same optimization problem on a thin (of size m´1{2qneighborhood of Γ.
— In Section 2.4, we study the optimization problem in the tubular neighborhood.
In this “tube”, we can use the classical tubular coordinates, called ps, tq, where s P Γ and t represents the distance to Γ. In these coordinates, we are led to consider a “transverse” optimization problem, that is a problem in one dimension (with respect to t) with parameters involving the curvature of the boundary.
Then, explicit computations provide the asymptotics of the 1D-minimizers.
— In Section 2.6, we establish Proposition 2.1. In particular, we use the projection on the 1D-minimizers to give the asymptotics of the minimizers in the tubular neighborhood. Note that our refined bounds are proved under the assumption that the boundary is of class C2,1. Indeed, we need at least C2,1 regularity to control the tangential derivative of the transverse optimizers (which depend on the curvature, see Lemma 2.20) when establishing, for instance, the accurate upper bound of Λmpvq (see Corollary 2.15).
2.2. Existence, uniqueness and Euler-Lagrange equations. Let us discuss here the existence of the minimizers announced in Proposition 2.1 and their elemen- tary properties. We will see later that, in the limit m Ñ `8, this minimization problem on Ω1 is closely related to the same problem on a tubular neighborhood in Ω1 of Γ. For δą0,m ą0, and v PH1pΩq, we define
(2.1) Λm,δpvq “ inf Qmpuq, uP Vv,δ( , where Qmpuq “ }∇u}2L2pΩ1q`m2}u}2L2pΩ1q is defined in (1.5) and
Vδ “ txPΩ1 : distpx,Γq ăδu,
Vv,δ “ uPH1pVδ,C4q s.t. u“v on Γ andupxq “ 0 if distpx,Γq “ δ( . Remark 2.2. Note that, since Ω is a smooth set, there exists δ0 ą 0 such that, for allδ P p0, δ0q, the setVδ has the same regularity as Ω.
2.2.1. Existence and uniqueness of minimizers.
Lemma 2.3. For δP p0, δ0q, m ą0, and v PH1pΩq,
the minimizers associated with (1.5) and (2.1) exist and are unique.
Proof. Let punq and puδ,nq be minimizing sequences for Λmpvq and Λm,δpvq respec- tively. These two sequences are uniformly bounded in H1 so that, up to subse- quences, they converge weakly to u P H1pΩ1q and vδ P H1pVδq, respectively. By Rellich - Kondrachov compactness Theorem and the interpolation inequality, the sequences converges strongly in Hlocs for any s P r0,1q. The trace theorem ensures then that the convergence also holds in L2locpΓq and L2locpBVδq, so that u P Vv and uδ PVv,δ. Since
Λmpvq “ lim
nÑ`8Qmpunq ěQmpuq ěΛmpvq and
Λm,δpvq “ lim
nÑ`8Qmpuδ,nq ě Qmpuδ,nq ěΛm,δpvq, u and uδ are minimizers.
Finally, since V and Vδ are convex sets and the quadratic form Qm is a strictly
convex function, the uniqueness follows.
Notation 2.4. The unique minimizers associated with Λmpvq and Λm,δpvq will be denoted byumpvqandum,δpvq, respectively, or byum andum,δ when the dependence onv is clear.
2.2.2. Euler-Lagrange equations. The following lemma gathers some properties re- lated to the Euler-Lagrange equations.
Lemma 2.5. For allδ P p0, δ0q, mą0, and v PH1pΩq, the following holds:
(i) p´∆`m2qum “0 and p´∆`m2qum,δ “0,
(ii) Λmpvq “ ´ xBnum, umyΓ and Λm,δpvq “ ´ xBnum,δ, um,δyΓ, (iii) Qmpuq “Λmpvq `Qmpu´umq for all uPVv,
Qmpuq “Λm,δpvq `Qmpu´um,δq for all uP Vv,δ,
where Λmpvq and Vv are defined in (1.5), Λm,δpvq and Vv,δ are defined in (2.1), and δ0 is defined in Remark 2.2.
Proof. Letv PH01pΩ1q. The function
RQtÞÑQmpum`tvq
has a minimum at t“0. Hence, the Euler-Lagrange equation is p´∆`m2qum “0.
The same proof holds forum,δ . The second point follows from integrations by parts.
And for the last point, let uPVv. We have, by an integration by parts,
Qmpu´umq “Qmpuq `Qmpumq ´2Rexu,p´∆`m2qumyΩ1 `2xum,BnumyΓ
“Qmpuq ´Λmpvq,
and the result follows. The same proof works for Λm,δpvq.
2.3. Agmon estimates. This section is devoted to the decay properties of the minimizers in the regime m Ñ `8.
As an intermediate step, we will need the following localization formulas.
Lemma 2.6. Let mą0 and χ be any real bounded Lipschitz function onΩ1. Then, (2.2) Qmpumχq “ ´ xBnum, χ2umyΓ` }p∇χqum}2L2pΩ1q.
The same holds for um,δ. Proof. By definition, we have
Qmpumχq “ m2}χum}2L2pΩ1q` }p∇χqum`χp∇umq}2L2pΩ1q
“m2}χum}2L2pΩ1q` }p∇χqum}2L2pΩ1q` }χp∇umq}2L2pΩ1q
`2Rexumχ,∇χ¨∇umyΩ1 . Then, by an integration by parts,
}χp∇umq}2L2pΩ1q “ ´ xBnum, χ2umyΓ´2Rexumχ,∇χ¨∇umyΩ1
`Rex´∆um, χ2umyΩ1 . It remains to use Lemma 2.5 to get
Qmpumχq “ ´ xBnum, χ2umyΓ` }p∇χqum}2L2pΩ1q.
The conclusion follows.
We can now establish the following important proposition.
Proposition 2.7. Letγ P p0,1q. There existC1, C2 ą0such that, for allδP p0, δ0q, m ą0, and v PH1pΩq,
(2.3) }emγdistp¨,Γqum}2L2pΩ1qďC1}um}2L2pΩ1q,
and
(2.4) p1´e´γm1{2C2m´1qΛm,m´1{2pvq ďΛmpvq ď Λm,δpvq. Here, δ0 is defined in Remark 2.2 .
Proof. Let us first prove (2.3). Given εą0, we define Φ :xÞÑminpγdistpx,Γq, ε´1q,
χm :xÞÑemΦpxq, and
ξ1: r0,1s Ñ r0,1s r ÞÑ ? 1´r
r2`p1´rq2
, ξ2: r0,1s Ñ r0,1s
r ÞÑ ? r
r2`p1´rq2
,
so that ξ12`ξ22 “1. We denote c“ }ξ1}L8pr0,1sq “ }ξ2}L8pr0,1sq ą0. Let R ą0. Let χ1,m,R, χ2,m,R be the Lipschitz quadratic partition of the unity defined by
χ1,m,Rpxq “
$
’&
’%
1 if distpx,Γq ďR{2m,
ξ1p2m{R distpx,Γq ´1q if R{2mďdistpx,Γq ďR{m,
0 if distpx,Γq ěR{m,
and
χ2,m,Rpxq “
$
’&
’%
0 if distpx,Γq ďR{2m,
ξ2p2m{R distpx,Γq ´1q if R{2mďdistpx,Γq ďR{m,
1 if distpx,Γq ěR{m.
We get, for k P t1,2u,
}∇χk,m,R}L8pΩ1q ď 2mc R .
Sinceχmis a bounded, Lipschitz function and is equal to 1 on Γ, we getumχm PVv. By definition and using (2.2), we get
Λmpvq “Qmpumq “ ´ xBnum, umyΓ “Qmpumχmq ´ }p∇χmqum}2L2pΩ1q. Then, we use the fact that ∇pχ21,m,R`χ22,m,Rq “ 0 to get
Qmpumq “ Qmpumχmχ1,m,Rq `Qmpumχmχ2,m,Rq ´ }p∇χmqum}2L2pΩ1q
´ }p∇χ1,m,Rqχmum}2L2pΩ1q´ }p∇χ2,m,Rqχmum}2L2pΩ1q. Since Qmpumχmχ1,m,Rq ě Λmpvqand
Qmpumχmχ2,m,Rq ě m2
umχmχ2,m,R
2 L2pΩ1q
“m2kumχmk2L2pΩ1q´m2
umχmχ1,m,R
2 L2pΩ1q , we get that
m2
´
1´γ2´ 8c2 R2
¯
kumχmk2L2pΩ1q ďm2
umχmχ1,m,R
2 L2pΩ1q
ďm2e2mminpγRm,1εqkumk2L2pΩ1q ďm2e2γRkumk2L2pΩ1q . TakingR ą0 big enough so that 1´γ2´8cR22 ą0, we have
kumχmk2L2pΩ1q ďCkumk2L2pΩ1q,
where C does not depend on ε. Taking the limit εÑ0 and using the Fatou lemma we obtain (2.3).
Let us now prove (2.4). We have for every δP p0, δ0qthat Vv,δ ĂVv, so that Λmpvq ďΛm,δpvq.
Let us consider a Lipschitz function ˜χm : Ω1 Ñ r0,1s defined for allxPΩ1 by
˜
χmpxq “
#
1 if distpx,Γq ď 2m11{2, 0 if distpx,Γq ě m11{2, with }∇χ˜m}L8pΩ1qď2cm1{2. Thanks to (2.2), we find
(2.5) Λm,m´1{2pvq ďQmpumχ˜mq “Λmpvq ` }um∇χ˜m}2L2pΩ1q. Then, by (2.3) we have
}um∇χ˜m}2L2pΩ1qďe´γm1{24c2m}emγdistp¨,Γqum}2L2pΩ1q ďC1e´γm1{24c2m}um}2L2pΩ1q. Observing that
m}um}2L2pΩ1qďm´1Λmpvq,
and using (2.5) we easily get (2.4).
2.4. Optimization problem in a tubular neighborhood. From Proposition 2.7, we see that, in order to estimate Λmpvq, it is sufficient to estimate Λm,m´1{2pvq. For that purpose, we will use tubular coordinates.
2.4.1. Tubular coordinates. Let ι be the canonical embedding of Γ in R3 and g the induced metric on Γ. pΓ, gq is a C2 Riemannian manifold, which we orientate according to the ambient space. Let us introduce the map Φ : Γˆp0, δq ÑVδdefined by the formula
Φps, tq “ιpsq `tnpsq,
where Vδ is defined below (2.1). The transformation Φ is a C1 diffeomorphism for allδ P p0, δ0q provided thatδ0 is sufficiently small. The induced metric on Γˆ p0, δq is given by
G“g ˝ pId`tLpsqq2` dt2,
where Lpsq “ dns is the second fundamental form of the boundary at s P Γ, see Notation 1.1.
Let us now describe how our optimization problem is transformed under the change of coordinates. For all uPL2pVδq, we define the pull-back function
(2.6) rups, tq:“upΦps, tqq. For all uPH1pVδq, we have
(2.7)
ż
Vδ
|u|2dx“ ż
Γˆp0,δq
|rups, tq|2˜adΓ dt and
(2.8)
ż
Vδ
|∇u|2dx“ ż
Γˆp0,δq
”
x∇sru,g˜´1∇sruy ` |Btru|2 ı
˜
adΓ dt , where
˜ g “`
Id`tLpsq˘2
,
and ˜aps, tq “ |˜gps, tq|12. Here x¨,¨y is the Euclidean scalar product and ∇s is the differential on Γ seen through the metric. Since
Lpsqis self-adjoint on TsΓ, we have the exact formula
(2.9) ˜aps, tq “1`tκpsq `t2Kpsq, whereκ and K are defined in Notation 1.1.
In the following, we assume that
(2.10) δ “m´1{2.
In particular, we will use (2.7) and (2.8) with this particular choice ofδ.
2.4.2. The rescaled transition optimization problem in boundary coordinates. We in- troduce the rescaling
ps, τq “ ps, mtq, and the new weights
(2.11) pamps, τq “ ˜aps, m´1τq, pgmps, τq “g˜ps, m´1τq.
Remark 2.8. Note that there exists m1 ě 1 such that for all m ě m1, s P Γ and τ P r0, m1{2q, we have pamps, τq ě1{2.
We set
Vpm “Γˆ p0,? mq,
Vpm “ tuPH1pVpm,C4;pamdΓ dτq: up¨,?
mq “0u, Qpmpuq “ m´1
ż
Vpm
´
x∇su,pgm´1∇suy `m2|Bτu|2
¯
pamdΓ dτ
`m ż
Vpm
|u|2pamdΓ dτ ,
Lxm “ ´m´1pa´1m ∇sppampgm´1∇sq `m`
´pa´1m BτpamBτ `1˘ . (2.12)
Notation 2.9. Given měm1, and κ, K PR, we define am,κ,K :p0,?
mq ÝÑR τ ÞÝÑ1`τ κ
m `τ2K m2 . We let
(2.13) A“ }κ}L8pΓq and B “ }K}L8pΓq.
Remark 2.10. We can assume (up to taking a larger m1) that for any pm, κ, Kq P rm1,`8q ˆ r´A, As ˆ r´B, Bs, we have am,κ,Kpτq ě1{2 for allτ P p0,?
mq.
In the following, we assume that pm, κ, Kq P rm1,`8q ˆ r´A, As ˆ r´B, Bs.
2.5. One dimensional optimization problem with parameters. We denote by Qpm,κ,K the “transverse” quadratic form defined for u P H1pp0,?
mq, am,κ,Kdτq by
Qpm,κ,Kpuq “ ż?m
0
´
|Bτu|2` |u|2
¯
am,κ,Kdτ.
We let
(2.14) Λm,κ,K “inftQpm,κ,Kpuq:uPVpm,κ,Ku, where
Vpm,κ,K “ uPH1pp0,?
mq, am,κ,Kdτq:up0q “ 1, up?
mq “0( . The following lemma follows from the same arguments as for Lemma 2.3.
Lemma 2.11. There is a unique minimizer um,κ,K for the optimization problem (2.14).
Lemma 2.12. Let u P H2pp0,?
mq, am,κ,Kdτq and v P H1pp0,?
mq, am,κ,Kdτq be such that up?
mq “vp?
mq “ 0. We have ż?m
0
xBτu,Bτvyam,κ,Kdτ` ż?m
0
xu, vyam,κ,Kdτ
“ ż?m
0
A
Lpm,κ,Ku, v E
am,κ,Kdτ ´ xBτup0q, vp0qy, (2.15)
where
Lpm,κ,K “ ´a´1m,κ,KBτam,κ,KBτ`1“ ´B2τ´ m´1κ`m´22Kτ
1`m´1κτ `m´2Kτ2Bτ`1. Proof. The lemma follows essentially by integration by parts and Notation 2.9.
Lemma 2.13. We have that um,κ,K PC8pr0,?
msq and
Lpm,κ,Kum,κ,K “0, Λm,κ,K “ ´Bτum,κ,Kp0q, where um,κ,K is defined in Lemma 2.11.
Moreover, for all uPVpm,κ,K,
Qpm,κ,Kpuq “ Λm,κ,K `Qpm,κ,Kpu´um,κ,Kq.
Proof. This follows from Lemma 2.12.
The aim of this section is to establish an accurate estimate of Λm,κ,K. Proposition 2.14. There exists a constant C ą0 such that for all
pm, κ, Kq P rm1,`8q ˆ r´A, As ˆ r´B, Bs, we have
ˇ ˇ ˇ ˇ ˇ ˇ
Λm,κ,K´
¨
˝1` κ 2m ` 1
m2
˜ K
2 ´κ2 8
¸˛
‚ ˇ ˇ ˇ ˇ ˇ ˇ
ďCm´3,
and ˇ
ˇ ˇ ˇ ˇ
ż?m 0
|um,κ,K|2am,κ,Kdτ´ 1 2 ˇ ˇ ˇ ˇ ˇ
ďCm´1.
Proof. By Lemmas 2.11 and 2.13, the unique solution um,κ,K of the problem satisfies
˜
´B2τ´ m´1κ`m´22Kτ
1`m´1κτ`m´2Kτ2Bτ`1
¸
um,κ,K “0.
We expand formallyum,κ,K as u0`m´1u1`m´2u2`Opm´3q: (i) For the zero order term, we get
p´Bτ2`1qu0 “0 and u0p1q “1, lim
τÑ8u0pτq “0, so that u0pτq “ e´τ.
(ii) At the first order,
p´Bτ2`1qu1 “κBτu0 “ ´κe´τ and u1p1q “ 0, lim
τÑ8u1pτq “0, so that u1pτq “ ´κ2τ e´τ.
(iii) At the second order,
p´Bτ2`1qu2 “κBτu1 ` pκ2´2KqτBτu0 “ ´κ2 2 e´τ `
˜ 3κ2
2 ´2K
¸ τ e´τ, u2p0q “0 and lim
τÑ8u2pτq “0, so that u2pτq “
´κ2 8 ´ K2
¯
τ e´τ`
´3κ2 8 ´ K2
¯ τ2e´τ.
This formal construction leads to define a possible approximation ofum,κ,K. Consider vm,κ,Kpτq:“χmpτq`
u0pτq `m´1u1pτq `m´2u2pτq˘ , χmpτq “χpτ{?
mq, (2.16)
where χ:R` ÞÑ r0,1s is a smooth function such that χpτq “
#
1 if τ P r0,1{2s, 0 if τ ě1.
In the following, we denotevm ”vm,κ,K to shorten the notation.
We immediately get thatvm belongs toVpm,κ,K. Note that (2.17) ´ Bτvmp0q “1` κ
2m `m´2
˜ K
2 ´ κ2 8
¸
and
(2.18) }Lpm,κ,Kvm}L2pp0,?
mq,am,κ,Kdτq“Opm´3q. Using Lemmas 2.12 and 2.13, we have
Λm,κ,K “ ż?m
0
A
Bτum,κ,K,Bτvm E
am,κ,Kdτ ` ż?m
0
A
um,κ,K, vm E
am,κ,Kdτ and
Λm,κ,K “ ż?m
0
A
Lpm,κ,Kvm, um,κ,K E
am,κ,Kdτ´ Bτvmp0q.
