The MIT Bag Model as an infinite mass limit

The MIT Bag Model as an infinite mass limit Tome 6 (2019), p. 329-365.

<http://jep.centre-mersenne.org/item/JEP_2019__6__329_0>

© Les auteurs, 2019.

Certains droits réservés.

Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D’ATTRIBUTIONCREATIVECOMMONSBY 4.0.

https://creativecommons.org/licenses/by/4.0/

L’accès aux articles de la revue « Journal de l’École polytechnique — Mathématiques » (http://jep.centre-mersenne.org/), implique l’accord avec les conditions générales d’utilisation (http://jep.centre-mersenne.org/legal/).

Publié avec le soutien

du Centre National de la Recherche Scientifique

Publication membre du

Centre Mersenne pour l’édition scientifique ouverte

THE MIT BAG MODEL AS AN INFINITE MASS LIMIT

by Naiara Arrizabalaga, Loïc Le Treust, Albert Mas

& Nicolas Raymond

Abstract. — The Dirac operator, acting in three dimensions, is considered. Assuming that a large massm >0lies outside a smooth enough and bounded open setΩ⊂R³, it is proved that its spectrum approximates the one of the Dirac operator onΩwith the MIT bag boundary condition. The approximation, modulo an error of ordero(1/√

m), is carried out by introduc- ing tubular coordinates in a neighborhood of∂Ωand analyzing one dimensional optimization problems in the normal direction.

Résumé(Le modèle MIT bag obtenu comme une limite de masse grande)

Nous considérons l’opérateur de Dirac en dimension3dont la massem >0est supposée grande à l’extérieur d’un ouvert borné et régulierΩ⊂R³. Nous démontrons que son spectre approche celui de l’opérateur de Dirac surΩqui intègre dans son domaine les conditions au bord dites « MIT bag ». L’analyse asymptotique est réalisée grâce à l’usage de coordonnées tubulaires et à l’analyse d’un problème d’optimisation unidimensionnel dans la direction normale.

Contents

1. Introduction. . . 330

2. About the exterior optimization problem. . . 336

3. A vectorial Laplacian with Robin-type boundary conditions . . . 351

4. Proof of the main theorem. . . 356

Appendix. Sketch of the proof of Lemma 4.3. . . 363

References. . . 364

2010Mathematics Subject Classification. — 35J60, 35Q75, 49J45, 49S05, 81Q10, 81V05, 35P15, 58C40.

Keywords. — Dirac operator, relativistic particle in a box, MIT bag model, spectral theory.

N.A. was partially supported by ERCEA Advanced Grant 669689-HADE, MTM2014-53145-P (MICINN, Gobierno de España) and IT641-13 (DEUI, Gobierno Vasco). L.LT. was partially sup- ported by ANR DYRAQ ANR-17-CE40-0016-01. A.M. was partially supported by MTM2017- 84214 and MTM2017-83499 projects of the MCINN (Spain), 2017-SGR-358 project of the AGAUR (Catalunya), and ERC-2014-ADG project HADE Id. 669689 (European Research Council).

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 massmoutside a smooth and bounded open setΩ. From physical considerations, see [8, 10], it is expected that, whenmbecomes large, the eigenfunc- tions of low energy do not visit R³rΩ and tend to satisfy the so-called MIT bag condition on∂Ω. 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 60⁰s [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 70⁰s, 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

Ω7−→λ1(Ω) +b|Ω|,

whereλ1(Ω)is the first nonnegative eigenvalue of the Dirac operator with the bound- ary condition introduced by Bogolioubov, |Ω| is the volume of Ω ⊂ R³ and b > 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 justifica- tion 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∂Ω. 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,Ωdeno- tes a fixed bounded domain ofR³ withC^2,1 boundary.

Let us recall the definition of the Dirac operator associated with the energy of a relativistic particle of mass m₀ ∈ Rand spin 1/2, see [17]. The Dirac operator is a first order differential operator (H,Dom(H)), acting on L²(R³;C⁴) in the sense of distributions, defined by

(1.1) H =cα·D+m0c²β, D=−i~∇,

whereDom(H) =H¹(R³;C⁴),c >0is the velocity of light,~>0is Planck’s constant, α= (α1, α2, α3)andβ are the4×4Hermitian and unitary matrices given by

β= 12 0

0 −12

, α_k = 0 σk

σk 0

fork= 1,2,3.

Here, the Pauli matricesσ1, σ2 andσ3are defined by σ1=

0 1 1 0

, σ2= 0 −i

i 0

, σ3= 1 0

0 −1

,

and α·X denotesP3

j=1α_jX_j for any X = (X₁, X₂, X₃). 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 (H,Dom(H))is self-adjoint, and satisfies

H²= 14(m²₀−∆),

(see for instance [17, §1.1]). Let us also mention that its spectrum is (−∞,−|m₀|]∪[|m₀|,+∞).

In this paper, we consider particles with large effective massm m0 outsideΩ.

Their kinetic energy is associated with the self-adjoint operator (H_m,Dom(H_m)) defined by

H_m=α·D+ (m₀+mχ_Ω0)β,

where Ω⁰ is the complementary set ofΩ,χΩ⁰ is the characteristic function of Ω⁰ and Dom(Hm) =H¹(R³;C⁴). The essential spectrum of(Hm,Dom(Hm))is

(−∞,−|m₀+m|]∪[|m₀+m|,+∞).

In this paper, the massm0is not assumed to be positive since this assumption is not used in the proofs (see also Remark 1.10).

Notation1.1. — In the following,Γ :=∂Ωand for all x∈Γ,n(x)is the outward- pointing unit normal vector to the boundary,L(x) =dn_xdenotes the second funda- mental form of the boundary, and

κ(x) =TrL(x) and K(x) = detL(x) are the mean curvature and the Gauss curvature of Γ, respectively.

Definition1.2. — The MIT bag Dirac operator (H^Ω,Dom(H^Ω)) is defined on the domain

Dom(H^Ω) ={ψ∈H¹(Ω;C⁴)|Bψ=ψ onΓ}, with B=−iβ(α·n), by H^Ωψ = Hψ for all ψ ∈ Dom(H^Ω). Observe that the trace is well-defined by a classical trace theorem.

If Γ is C², the operator (H^Ω,Dom(H^Ω)) is self-adjoint with compact resolvent [15, 3, 9, 6, 4].

Notation1.3. — We denote byh·,·i theC⁴ scalar product (antilinear with respect to the left argument) and byh·,·i_U theL²scalar product on the set U ⊂R³.

Notation1.4. — We define, for every n∈S², the orthogonal projections

(1.2) Ξ^±= 14±B

2 associated with the eigenvalues±1of the matrix B.

1.3. Squared operators, heuristics, and main results. — The aim of this paper is to relate the spectra ofH_mandH^Ωin the limitm→+∞.

Notation1.5. — Let(λk)_k∈N^∗and(λk,m)_k∈N^∗be the increasing sequences defined by λk= inf

V⊂Dom(H^Ω) dimV=k

sup

ϕ∈V kϕk_{L2 (Ω)}=1

H^Ωϕ _L2(Ω)

= sup

{ψ1,...,ψk−1}⊂Dom(H^Ω)

inf

ϕ∈span(ψ1,...,ψ_k−1)^⊥ kϕk_{L2 (Ω)}=1

H^Ωϕ _L2(Ω),

and

λk,m= inf

V⊂H¹(R³;C⁴) dimV=k

sup

ϕ∈V kϕk_{L2 (R3 )}=1

kHmϕk_L2(R³)

= sup

{ψ1,...,ψk−1}⊂H¹(R³;C⁴)

inf

ϕ∈span(ψ1,...,ψk−1)^⊥ kϕk_{L2 (R3 )}=1

kH_mϕk_L2(R³),

fork∈N^∗andm >0. Here,N^∗:=N r{0}. By themin-maxcharacterization and the properties given in Definition 1.2, the sequence(λk)_k∈N^∗is made of all the eigenvalues of the operator|H^Ω|, each one being repeated according to its multiplicity. Similarly, the terms of the sequence(λ_k,m)_k∈N^∗ that satisfy

λk,m<|m0+m|

are the eigenvalues of |Hm|lying below its essential spectrum[|m0+m|,+∞), 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 that λk and λ_k,mare related, especially because of the boundary condition ofH^Ω. It becomes less surprising when computing the squares of the operators. This is the purpose of the following lemma.

Lemma1.6. — Let ϕ∈Dom(H^Ω)andψ∈H¹(R³;C⁴). Then (1.3) kH^Ωϕk²_L2(Ω)=Q^int(ϕ) :=k∇ϕk²_L2(Ω)+

Z

Γ

κ 2 +m0

|ϕ|²dΓ +m²₀kϕk²_L2(Ω), whereκis defined in Notation 1.1, and

kHmψk²_L2(R³)=k∇ψk²_L2(Ω)+k∇ψk²_L2(Ω⁰)+k(m0+mχ_Ω⁰)ψk²_L2(R³)

−mRehBψ, ψiΓ

=k∇ψk²_L2(Ω)+k∇ψk²_L2(Ω⁰)+k(m₀+mχ_Ω⁰)ψk²_L2(R³)

+mkΞ⁻ψk²_L2(Γ)−mkΞ⁺ψk²_L2(Γ). (1.4)

Proof. — The equality (1.3) is proved for instance in [2, §A.2]. Letψ∈H¹(R³;C⁴).

By integrations by parts,

kHmψk²_L2(R³)=kα·Dψk²_L2(R³)+k(m0+mχ_Ω⁰)ψk²_L2(R³)+ 2mRehα·Dψ, βψiΩ⁰

=k∇ψk²_L2(R³)+k(m0+mχΩ⁰)ψk²_L2(R³)−mRehBψ, ψiΓ. Then, note that, for allψ∈H¹(R³;C⁴),

RehBψ, ψiΓ =kΞ⁺ψk²_L2(Γ)− kΞ⁻ψk²_L2(Γ). Considering (1.4) leads to the following minimization problem, forv∈H¹(Ω), (1.5) Λm(v) = inf{Qm(u), u∈Vv}, Qm(u) =k∇uk²_L2(Ω⁰)+m²kuk²_L2(Ω⁰), where

Vv={u∈H¹(Ω⁰,C⁴)|u=v onΓ}.

A classical extension theorem (see [12, §5.4]) ensures thatVv is non-empty.

1.3.2. Heuristics. — In this paper, we will analyze the behavior ofΛm(v)and prove in particular (see Proposition 2.1) that there existsC >0 such that formlarge, and allv∈H¹(Ω;C⁴)

(1.6) o(1)>Λ_m(v)−

mkvk²_L2(Γ)+ Z

Γ

κ 2 |v|²dΓ

>−C

mkvk²_H1(Γ). Replacingmbym0+min (1.6), we get, for allψ∈H¹(R³;C⁴),

(1.7) kH_mψk²_L2(R³)>k∇ψk²_L2(Ω)+m²₀kψk²_L2(Ω)

+ Z

Γ

κ 2 +m0

|ψ|²dΓ + 2mkΞ⁻ψk²_L2(Γ)−C

mkψk²_L2(Γ). Take any eigenfunctionϕofH^Ωand consider a minimizeruϕof (1.5) forv=ϕandm replaced bym+m0. Then, lettingψ=1^Ωϕ+1^Ω0uϕ∈H¹(R³;C⁴), we get

kHmψk²_L2(R³)=k∇ϕk²_L2(Ω)+m²₀kψk²_L2(Ω)+ Λm+m₀(ϕ)−mkΞ⁺ϕk²_L2(Γ). With (1.6) at hand, we deduce that, for allj∈N^∗,

λ²_j,m6λ²_j+o(1).

Conversely, if we are interested in the eigenvalues of (H_m)² that are of order 1 whenm→+∞, we see from (1.7) that the corresponding normalized eigenfunctions must satisfy Ξ⁻ψ = O(m⁻¹) and, in particular, Bψ = ψ+O(m⁻¹). Thus, we get formally, for allj∈N^∗,

λ²_j,m>λ²_j+o(1).

The aim of this paper is to make this heuristics rigorous. We now state our main theorem.

Theorem1.7. — Let Ω⊂R³ be a bounded domain of class C^2,1 (i.e., the derivative of the curvatures is bounded). The singular values ofHm can be estimated as follows:

(i) limm→+∞λk,m=λk, for all k∈N^∗.

(ii) Let k1 ∈ N^∗ be the multiplicity of the first eigenvalue λ1 of |H^Ω|. For all k∈ {1, . . . , k1}, we have

λk,m= λ²₁+νk

m +o(1/m)1/2

, where

(1.8) νk = inf

V⊂ker(|H^Ω|−λ1) dimV=k

sup

u∈V kuk_{L2 (Ω)}=1

η(u),

with

η(u) = Z

Γ

|∇su|²

2 −|(∂n+κ/2 +m0)u|²

2 +K

2 −κ² 8 −λ²₁

2 |u|²

dΓ.

Here,(λk)_k∈N^∗ and(λk,m)_k∈N^∗ are defined in Notation 1.5, andκandK are defined in Notation 1.1. We denote by∂nthe outward pointing normal derivative and by ∇s

the tangential gradient on Γ.

Remark 1.8. — The max-min formula (1.8) makes sense since ker(|H^Ω| −λId) ⊂ H²(Ω;C⁴)for any eigenvalueλof|H^Ω|.

Remark1.9. — HmandH^Ωanticommute with the charge conjugationCdefined, for allψ∈C⁴, by

Cψ=iβα₂ψ,

whereψ∈C⁴is the vector obtained after complex conjugations of each of the compo- nents ofψ(see for instance [17, §1.4.6] and [2, §A.1]). As a consequence, the spectrum ofH_mand H^Ω are symmetric with respect to0, and Theorem 1.7 may be rewritten as a result on the eigenvalues ofHmandH^Ω.

Remark1.10. — Let us define the operator(gH^Ω,Dom(gH^Ω))on Dom(gH^Ω) ={ψ∈H¹(Ω;C⁴)|Bψ=−ψ onΓ}

bygH^Ωψ=Hψfor allψ∈Dom(gH^Ω). It is the MIT bag Dirac operator withreversed boundary condition (see Definition 1.2). The singular values ofHg^Ωare approximated by the singular values of H_m as m tends to −∞. This follows immediately from Theorem 1.7, conjugating all the operators by the chirality matrix

γ5=

0 12

12 0

, and by using the algebraic properties

βγ5=−γ5β, γ5(α·x) = (α·x)γ5, γ5Bγ5=−B, for allx∈R³.

Remark 1.11. — Our proof of Theorem 1.7 also provides the convergence of the eigenprojectors associated with the first eigenvalues of |Hm|. They converge towards the eigenprojectors associated with the first eigenvalues of|H^Ω|, see Lemma 4.1 and Remark 4.2, and [16, Th. 1] for the two-dimensional case.

Remark1.12. — In view of Theorem 1.7, it is natural to ask if one has convergence of Hm to H^Ω in some resolvent sense when m→ +∞. On one hand, in the recent work [5] it is shown the convergence in the norm resolvent sense for the bidimensional analogues of H_m andH^Ω. On the other hand, in [14] the authors study interactions of the free Dirac operator in R³ with potentials that shrink towards ∂Ω, proving convergence in the strong resolvent sense toδ-shell interactions with precise coupling constants. As m→ +∞, our operator H_m 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 resultingδ-shell interaction leads to the operatorH^Ω.

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 Theo- rem 1.7 and that may be of interest on its own. We consider the vectorial Laplacian associated with the quadratic form

Q^intm(u) =k∇uk²_L2(Ω)+m²₀kuk²_L2(Ω)+ Z

Γ

κ 2 +m0

|u|²dΓ + 2m Ξ⁻u

2 L²(Γ)

(1.9)

foru∈Dom(Q^intm) =H¹(Ω;C⁴)andm >0, whereΞ⁻,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.

Lemma1.13. — The self-adjoint operator associated withQ^intm is defined by Dom(L^int_m) =

u∈H²(Ω;C⁴)

Ξ⁻(∂_n+κ/2 +m₀+ 2m)u= 0 onΓ, Ξ⁺(∂n+κ/2 +m0)u= 0 onΓ

L^int_mu= −∆ +m²₀

u for allu∈Dom(L^int_m).

(1.10)

It has compact resolvent and its spectrum is discrete.

Using an integration by parts and the identities (1.2), we get hu, L^int_mui_Ω=Q^intm(u),

for allu∈Dom(L^int_m).

Notation1.14. — Let(λ^int_k,m)_k∈N^∗ denote the sequence of eigenvalues, each one being repeated according to its multiplicity and such that

(1.11) λ^int_1,m6λ^int_2,m6· · ·

The asymptotic behavior of the eigenvalues of L^int_m is detailed in the following theorem.

Theorem1.15. — The following holds:

(i) For every k∈N^∗,lim_m→+∞λ^int_k,m=λ²_k.

(ii) Let λ be an eigenvalue of |H^Ω| of multiplicity k1 ∈ N^∗. Consider the unique integer k0∈Nsuch that for all k∈ {1, . . . , k1},λk₀+k =λ.

Then, for allk∈ {1,2, . . . , k₁}, we have λ^int_k0+k,m=λ²+µλ,k

m +o(1/m), where

(1.12) µλ,k:= inf

V⊂ker(|H^Ω|−λ) dimV=k

sup

v∈V kvk_{L2 (Ω)}=1

−k(∂n+κ/2 +m0)vk²_L2(Γ)

2 .

Here, (λk)_k∈N^∗ is defined in Notation 1.5, (λ^int_k,m)_k∈N^∗ 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 Dom(H^Ω) Q^int −

(1.5) Ω⁰ Vv Qm Λm(v)

(1.9) Ω H¹(Ω;C⁴) Q^intm −

(2.1) Vδ Vv,δ Qm Λm,δ(v)

(2.12) Vbm Vbm Qbm Λ_m,m−1/2(v)

(2.14) (0,√

m) Vb_m,κ,K Qbm,κ,K Λ_m,κ,K

2. About the exterior optimization problem

The aim of this section is to study the minimizers of (1.5) and their properties whenmtends to+∞. These properties are gathered in the following proposition.

Proposition2.1. — For allv∈H¹(Ω), there exists a unique minimizeru_m(v)asso- ciated with Λm(v), and it satisfies, for allu∈Vv,

Qm(u) = Λ_m(v) +Qm(u−u_m(v)).

Moreover, the following holds:

(i) Assume that Γ is C². There exist C, m1 > 0 such that, for every m > m1, v∈H¹(Ω),

Cmkvk²_H1(Ω)>Λm(v)>

mkvk²_L2(Γ)+ Z

Γ

κ 2|v|²dΓ

−C

mkvk²_L2(Γ). Assume that ΓisC^2,1. There existsC >0 such that, for everym>m1,

(ii) forv∈H¹(Ω),

mkvk²_L2(Γ)+ Z

Γ

κ 2 |v|²dΓ

+o(1)>Λm(v).

Here, the termo(1) depends onv (not only on theH¹ norm ofv).

(iii) for allv∈H²(Ω),

Λm(v)−Λem(v) 6 C

m^3/2kvk²_H3/2(Γ), (iv) for all v∈H²(Ω),

kum(v)k²_L2(Ω⁰)−kvk²_L2(Γ)

2m 6 C

m²kvk²_H3/2(Γ),

Λem(v) =m Z

Γ

|v|²dΓ + Z

Γ

κ

2|v|²dΓ +m⁻¹ Z

Γ

n|∇sv|² 2 +K

2 −κ² 8

|v|²o 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 +∞, the minimizers are exponentially localized near the interface Γ. This allows to replace our optimization problem on Ω⁰ by the same optimization problem on a thin (of sizem^−1/2)neighborhood ofΓ.

– In Section 2.4, we study the optimization problem in the tubular neighborhood.

In this “tube”, we can use the classical tubular coordinates, called(s, t), wheres∈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 tot) with parameters involving the curvature of the boundary. Then, explicit compu- tations provide the asymptotics of the1D-minimizers.

– In Section 2.6, we establish Proposition 2.1. In particular, we use the projec- tion 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 C^2,1. Indeed, we need at leastC^2,1 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 ofΛ_m(v)(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 elementary properties. We will see later that, in the limitm→+∞, this minimization problem on Ω⁰ is closely related to the same problem on a tubular neighborhood in Ω⁰ ofΓ.

Forδ >0, m >0, andv∈H¹(Ω), we define

(2.1) Λm,δ(v) = inf{Qm(u)|u∈Vv,δ},

whereQm(u) =k∇uk²_L2(Ω⁰)+m²kuk²_L2(Ω⁰) is defined in (1.5) and Vδ ={x∈Ω⁰ |dist(x,Γ)< δ},

V_v,δ=

u∈H¹(Vδ,C⁴)|u=v onΓandu(x) = 0if dist(x,Γ) =δ .

Remark2.2. — Note that, sinceΩis a smooth set, there existsδ₀>0such that, for allδ∈(0, δ0), the set Vδ has the same regularity asΩ.

2.2.1. Existence and uniqueness of minimizers

Lemma2.3. — For δ ∈ (0, δ0), m > 0, and v ∈ H¹(Ω), the minimizers associated with (1.5)and (2.1)exist and are unique.

Proof. — Let (un) and (uδ,n) be minimizing sequences for Λm(v) and Λm,δ(v) re- spectively. These two sequences are uniformly bounded in H¹ so that, up to sub- sequences, they converge weakly to u ∈ H¹(Ω⁰) and v_δ ∈ H¹(Vδ), respectively. By Rellich-Kondrachov compactness Theorem and the interpolation inequality, the se- quences converges strongly inH_loc^s for anys∈[0,1). The trace theorem ensures then that the convergence also holds inL²_loc(Γ)andL²_loc(∂Vδ), so thatu∈V_vandu_δ∈V_v,δ. Since

Λ_m(v) = lim

n→+∞Qm(u_n)>Qm(u)>Λ_m(v) and

Λm,δ(v) = lim

n→+∞Qm(uδ,n)>Qm(uδ,n)>Λm,δ(v), uandu_δ are minimizers.

Finally, since V and Vδ are convex sets and the quadratic form Qm is a strictly

convex function, the uniqueness follows.

Notation2.4. — The unique minimizers associated withΛ_m(v)andΛ_m,δ(v)will be denoted byum(v)andum,δ(v), respectively, or byumandum,δwhen the dependence onv is clear.

2.2.2. Euler-Lagrange equations. — The following lemma gathers some properties re- lated to the Euler-Lagrange equations.

Lemma2.5. — For all δ∈(0, δ₀),m >0, andv∈H¹(Ω), the following holds:

(i) (−∆ +m²)u_m= 0 and(−∆ +m²)u_m,δ= 0,

(ii) Λm(v) =− h∂num, umi_Γ andΛm,δ(v) =− h∂num,δ, um,δi_Γ,

(iii) Qm(u) = Λm(v) +Qm(u−um)for allu∈Vv,Qm(u) = Λm,δ(v) +Qm(u−um,δ) for all u∈Vv,δ,

whereΛm(v)andVvare defined in(1.5),Λm,δ(v)andVv,δ are defined in(2.1), andδ0

is defined in Remark 2.2.

Proof. — Letv∈H₀¹(Ω⁰). The function

R3t7−→Qm(um+tv)

has a minimum at t= 0. Hence, the Euler-Lagrange equation is (−∆ +m²)um= 0.

The same proof holds forum,δ. The second point follows from integrations by parts.

And for the last point, let u∈V_v. We have, by an integration by parts,

Qm(u−um) =Qm(u) +Qm(um)−2 Rehu,(−∆ +m²)umi_Ω0+ 2hum, ∂numi_Γ

=Qm(u)−Λm(v),

and the result follows. The same proof works forΛm,δ(v).

2.3. Agmon estimates. — This section is devoted to the decay properties of the min- imizers in the regimem→+∞.

As an intermediate step, we will need the following localization formulas.

Lemma2.6. — Let m >0andχ be any real bounded Lipschitz function onΩ⁰. Then, (2.2) Qm(umχ) =− h∂num, χ²umi_Γ+k(∇χ)umk²_L2(Ω⁰).

The same holds for u_m,δ. Proof. — By definition, we have

Qm(umχ) =m²kχumk²_L2(Ω⁰)+k(∇χ)um+χ(∇um)k²_L2(Ω⁰)

=m²kχumk²_L2(Ω⁰)+k(∇χ)umk²_L2(Ω⁰)+kχ(∇um)k²_L2(Ω⁰)

+ 2 Rehumχ,∇χ· ∇umi_Ω0. Then, by an integration by parts,

kχ(∇u_m)k²_L2(Ω⁰)=− h∂_nu_m, χ²u_mi_Γ−2 Rehu_mχ,∇χ· ∇u_mi_Ω0

+ Reh−∆um, χ²umi_Ω0. It remains to use Lemma 2.5 to get

Qm(umχ) =− h∂num, χ²umi_Γ+k(∇χ)umk²_L2(Ω⁰).

The conclusion follows.

We can now establish the following important proposition.

Proposition2.7. — Let γ∈(0,1). There existC1, C2>0such that, for allδ∈(0, δ0), m >0, andv∈H¹(Ω),

(2.3) ke^{mγdist(·,Γ)}u_mk²_L2(Ω⁰)6C₁ku_mk²_L2(Ω⁰), and

(2.4) (1−e^−γm1/2C₂m⁻¹)Λ_m,m−1/2(v)6Λ_m(v)6Λ_m,δ(v).

Here,δ0 is defined in Remark 2.2.

Proof. — Let us first prove (2.3). Givenε >0, we define Φ :x7−→min(γdist(x,Γ), ε⁻¹), χm:x7−→e^mΦ(x),

and

ξ1: [0,1]−→[0,1] ξ2: [0,1]−→[0,1]

r7−→ 1−r

pr²+ (1−r)² r7−→ r

pr²+ (1−r)²,

so thatξ₁²+ξ₂²= 1. We denotec=kξ1kL^∞([0,1])=kξ2kL^∞([0,1]) >0. Let R >0. Let χ_1,m,R, χ_2,m,R be the Lipschitz quadratic partition of the unity defined by

χ1,m,R(x) =











1 if dist(x,Γ)6R/2m,

ξ1(2m/R dist(x,Γ)−1) ifR/2m6dist(x,Γ)6R/m,

0 if dist(x,Γ)>R/m,

and

χ2,m,R(x) =











0 if dist(x,Γ)6R/2m,

ξ2(2m/R dist(x,Γ)−1) ifR/2m6dist(x,Γ)6R/m,

1 if dist(x,Γ)>R/m.

We get, fork∈ {1,2},

k∇χk,m,Rk_L∞(Ω⁰)62mc R .

Sinceχmis a bounded, Lipschitz function and is equal to1onΓ, we getumχm∈Vv. By definition and using (2.2), we get

Λ_m(v) =Qm(u_m) =− h∂nu_m, u_mi_Γ=Qm(u_mχ_m)− k(∇χm)u_mk²_L2(Ω⁰). Then, we use the fact that∇(χ²_1,m,R+χ²_2,m,R) = 0to get

Qm(um) =Qm(umχmχ1,m,R) +Qm(umχmχ2,m,R)− k(∇χm)umk²_L2(Ω⁰)

− k(∇χ_1,m,R)χ_mu_mk²_L2(Ω⁰)− k(∇χ_2,m,R)χ_mu_mk²_L2(Ω⁰). SinceQm(umχmχ1,m,R)>Λm(v)and

Qm(umχmχ2,m,R)>m²kumχmχ2,m,Rk²_L2(Ω⁰)

=m²kumχmk²_L2(Ω⁰)−m²kumχmχ1,m,Rk²_L2(Ω⁰), we get that

m²

1−γ²−8c² R²

kumχmk²_L2(Ω⁰)6m²kumχmχ1,m,Rk²_L2(Ω⁰)

6m²e^2mmin(γR/m,1/ε)ku_mk²_L2(Ω⁰)

6m²e^2γRkumk²_L2(Ω⁰). TakingR >0 big enough so that1−γ²−8c²/R²>0, we have

kumχmk²_L2(Ω⁰)6Ckumk²_L2(Ω⁰),

where C does not depend onε. Taking the limit ε→0 and using the Fatou lemma we obtain (2.3).

Let us now prove (2.4). We have for everyδ∈(0, δ0)that Vv,δ ⊂Vv, so that Λm(v)6Λm,δ(v).

Let us consider a Lipschitz functionχem: Ω⁰→[0,1]defined for allx∈Ω⁰ by χem(x) =

(1 if dist(x,Γ)61/2m^1/2, 0 if dist(x,Γ)>1/m^1/2, withk∇χemk_L^∞_(Ω⁰₎62cm^1/2. Thanks to (2.2), we find

(2.5) Λ_m,m−1/2(v)6Qm(umχem) = Λm(v) +kum∇χemk²_L2(Ω⁰). Then, by (2.3) we have

kum∇χemk²_L2(Ω⁰)6e^−γm1/24c²mke^{mγdist(·,Γ)}umk²_L2(Ω⁰)6C1e^−γm1/24c²mkumk²_L2(Ω⁰). Observing that

mkumk²_L2(Ω⁰)6m⁻¹Λ_m(v),

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 estimateΛm(v), it is sufficient to estimateΛ_m,m−1/2(v). For that purpose, we will use tubular coordinates.

2.4.1. Tubular coordinates. — Letιbe the canonical embedding ofΓinR³andgthe induced metric onΓ.(Γ, g)is aC²Riemannian manifold, which we orientate according to the ambient space. Let us introduce the mapΦ : Γ×(0, δ)→Vδ defined by the formula

Φ(s, t) =ι(s) +tn(s),

whereVδ is defined below (2.1). The transformationΦis a C¹diffeomorphism for all δ∈(0, δ0)provided that δ0 is sufficiently small. The induced metric onΓ×(0, δ) is given by

G=g◦(Id +tL(s))²+ dt²,

where L(s) = dns is the second fundamental form of the boundary at s ∈ Γ, see Notation 1.1.

Let us now describe how our optimization problem is transformed under the change of coordinates. For allu∈L²(Vδ), we define the pull-back function

(2.6) eu(s, t) :=u(Φ(s, t)).

For allu∈H¹(Vδ), we have (2.7)

Z

Vδ

|u|²dx= Z

Γ×(0,δ)

|eu(s, t)|²eadΓ dt and

(2.8)

Z

Vδ

|∇u|²dx= Z

Γ×(0,δ)

hh∇seu,eg⁻¹∇suie +|∂tu|e²i

eadΓ dt, where

eg= Id +tL(s)2

,

and ea(s, t) = |eg(s, t)|^1/2. Here h·,·i is the Euclidean scalar product and ∇s is the differential on Γseen through the metric. Since L(s) is self-adjoint onTsΓ, we have the exact formula

(2.9) ea(s, t) = 1 +tκ(s) +t²K(s),

whereκandK 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 introduce the rescaling

(s, τ) = (s, mt), and the new weights

(2.11) bam(s, τ) =ea(s, m⁻¹τ), bgm(s, τ) =eg(s, m⁻¹τ).

Remark2.8. — Note that there existsm₁ >1 such that for all m>m₁,s∈Γ and τ∈[0, m^1/2), we haveba_m(s, τ)>1/2.

We set

bVm= Γ×(0,√ m),

Vbm={u∈H¹(bVm,C⁴;bamdΓ dτ) : u(·,√

m) = 0},

Qbm(u) =m⁻¹ Z

Vbm

h∇su,bg⁻¹_m∇sui+m²|∂τu|²

bamdΓ dτ +m

Z

Vbm

|u|²bamdΓ dτ, Lcm=−m⁻¹ba⁻¹_m∇_s(ba_mbg_m⁻¹∇_s) +m −ba⁻¹_m∂_τba_m∂_τ+ 1

. (2.12)

Notation2.9. — Givenm>m₁, andκ, K ∈R, we define a_m,κ,K: (0,√

m)−→R τ7−→1 + τ κ

m +τ²K m² . We let

(2.13) A=kκk_L∞(Γ) and B =kKk_L∞(Γ).

Remark2.10. — We can assume (up to taking a largerm1) that for any (m, κ, K)∈[m1,+∞)×[−A, A]×[−B, B],

we have am,κ,K(τ)>1/2 for allτ ∈(0,√ m).

In the following, we assume that(m, κ, K)∈[m1,+∞)×[−A, A]×[−B, B].

2.5. One dimensional optimization problem with parameters. — We denote by Qbm,κ,K the “transverse” quadratic form defined for u∈H¹((0,√

m), am,κ,Kdτ)by Qbm,κ,K(u) =

Z

√m

0

|∂_τu|²+|u|²

a_m,κ,Kdτ.

We let

(2.14) Λ_m,κ,K= inf{Qbm,κ,K(u)|u∈Vb_m,κ,K}, where

Vbm,κ,K=

u∈H¹((0,√

m), am,κ,Kdτ)|u(0) = 1, u(√

m) = 0 . The following lemma follows from the same arguments as for Lemma 2.3.

Lemma2.11. — There is a unique minimizer um,κ,K for the optimization problem (2.14).

Lemma2.12. — Let u∈H²((0,√

m), a_m,κ,Kdτ) andv∈H¹((0,√

m), a_m,κ,Kdτ) be such that u(√

m) =v(√

m) = 0. We have

(2.15) Z

√m

0

h∂τu, ∂_τvia_m,κ,Kdτ+ Z

√m

0

hu, via_m,κ,Kdτ

= Z

√m

0

Lbm,κ,Ku, v

a_m,κ,Kdτ− h∂_τu(0), v(0)i, where

Lbm,κ,K=−a⁻¹_m,κ,K∂_τa_m,κ,K∂_τ+ 1 =−∂²_τ− m⁻¹κ+m⁻²2Kτ

1 +m⁻¹κτ+m⁻²Kτ²∂_τ+ 1.

Proof. — The lemma follows essentially by integration by parts and Notation 2.9.

Lemma2.13. — We have thatum,κ,K∈C^∞([0,√

m])and

Lbm,κ,Kum,κ,K= 0, Λm,κ,K =−∂τum,κ,K(0), whereum,κ,K is defined in Lemma 2.11.

Moreover, for allu∈Vb_m,κ,K,

Qbm,κ,K(u) = Λ_m,κ,K+Qbm,κ,K(u−u_m,κ,K).

Proof. — This follows from Lemma 2.12.

The aim of this section is to establish an accurate estimate ofΛm,κ,K. Proposition2.14. — There exists a constantC >0 such that for all

(m, κ, K)∈[m1,+∞)×[−A, A]×[−B, B], we have

Λ_m,κ,K− 1 + κ

2m + 1 m²

K 2 −κ²

8

6Cm⁻³,

and

Z

√m

0

|um,κ,K|²am,κ,Kdτ−1 2

6Cm⁻¹.

Proof. — By Lemmas 2.11 and 2.13, the unique solutionum,κ,K of the problem sat- isfies

−∂_τ²− m⁻¹κ+m⁻²2Kτ

1 +m⁻¹κτ +m⁻²Kτ²∂_τ+ 1

u_m,κ,K= 0.

We expand formallyum,κ,K as u0+m⁻¹u1+m⁻²u2+O(m⁻³):

(i) For thezero order term, we get

(−∂_τ²+ 1)u₀= 0 and u₀(1) = 1, lim

τ→∞u₀(τ) = 0, so thatu₀(τ) =e^−τ.

(ii) At thefirst order,

(−∂_τ²+ 1)u₁=κ∂_τu₀=−κe^−τ and u₁(1) = 0, lim

τ→∞u₁(τ) = 0, so thatu₁(τ) =−(κ/2)τ e^−τ.

(iii) At thesecond order,

(−∂_τ²+ 1)u2=κ∂τu1+ (κ²−2K)τ ∂τu0=−κ²

2 e^−τ+3κ²

2 −2K τ e^−τ, u2(0) = 0 and lim

τ→∞u2(τ) = 0, so thatu₂(τ) = ^κ₈² −^K₂

τ e^−τ+ ^3κ₈² −^K₂ τ²e^−τ.

This formal construction leads to define a possible approximation ofum,κ,K. Consider vm,κ,K(τ) : =χm(τ) u0(τ) +m⁻¹u1(τ) +m⁻²u2(τ)

, χm(τ) =χ(τ /√

(2.16) m),

whereχ:R+7→[0,1]is a smooth function such that χ(τ) =

(1 ifτ ∈[0,1/2], 0 ifτ >1.

In the following, we denotevm≡vm,κ,K to shorten the notation.

We immediately get thatvm belongs toVbm,κ,K. Note that (2.17) −∂_τv_m(0) = 1 + κ

2m+m⁻²K 2 −κ²

8

and

(2.18) kLbm,κ,Kvmk_L2((0,√

m),a_m,κ,Kdτ)=O(m⁻³).

Using Lemmas 2.12 and 2.13, we have Λm,κ,K=

Z

√m

0

∂τum,κ,K, ∂τvm

am,κ,Kdτ+ Z

√m

0

um,κ,K, vm

am,κ,Kdτ

and

Λm,κ,K = Z

√m

0

Lbm,κ,Kvm, um,κ,K

am,κ,Kdτ−∂τvm(0).

By Lemma 2.12, the Cauchy-Schwarz inequality, (2.17), and (2.18), we see that

Λm,κ,K− 1 + κ

2m+m⁻²K 2 −κ²

8

= Z

√m

0

Lbm,κ,Kv_m, u_m,κ,K

a_m,κ,Kdτ 6kLbm,κ,Kv_mk_L2((0,√

m),a_m,κ,Kdτ)· kum,κ,Kk_L2((0,√

m),a_m,κ,Kdτ)

6Λ^1/2_m,κ,KkbLm,κ,Kvmk_L2((0,√

m),a_m,κ,Kdτ)

6Cm⁻³Λ^1/2_m,κ,K.

From this, it follows first thatΛm,κ,K =O(1)uniformly in(κ, K), and then the first estimate of the proposition is established. Using Lemmas 2.12 and 2.13, the fact that vm(0)−um,κ,K(0) = 0, and Cauchy-Schwarz inequality, we have

Qbm,κ,K(vm−um,κ,K)

6kLbm,κ,K(v_m−u_m,κ,K)k_L2((0,√

m),a_m,κ,Kdτ)kvm−u_m,κ,Kk_L2((0,√

m),a_m,κ,Kdτ)

6Cm⁻³kvm−um,κ,Kk_L2((0,√

m),am,κ,Kdτ). The second estimate of the proposition follows since

kvm−um,κ,Kk²_L2((0,√

m),a_m,κ,Kdτ)6Qbm,κ,K(vm−um,κ,K) andkv_mk²_L2((0,^√_m),a

m,κ,Kdτ)= (1/2) +O(m⁻¹).

2.6. Asymptotic study of Λ_m,m−1/2(v). — From Proposition 2.14 and (2.12), we deduce the following lower bound.

Corollary2.15. — The following holds:

(i) Assume thatΓisC². There existsC >0such that, for everym>m1,v∈H¹(Ω), Cmkvk²_H1(Ω)>Λ_m,m−1/2(v)>

mkvk²_L2(Γ)+ Z

Γ

κ 2|v|²dΓ

−C

mkvk²_L2(Γ). (ii) Assume that Γ is C^2,1. There exists C > 0 such that, for every m > m1, v∈H¹(Ω),

mkvk²_L2(Γ)+ Z

Γ

κ 2|v|²dΓ

+o(1)>Λ_m,m−1/2(v).

Here, the termo(1) depends onv (not only on theH¹ norm ofv).

Proof. — By Proposition 2.14, the lower bound of Point (i) follows. Let us focus on Point (ii).