Averaging of magnetic fields and applications
aa r X i v : . [ m a t h . SP ] D ec AVERAGING OF MAGNETIC FIELDS AND APPLICATIONS
AYMAN KACHMAR AND MOHAMMAD WEHBE
Abstract.
We estimate the magnetic Laplacian energy norm in appropriate planar domainsunder a weak regularity hypothesis on the magnetic field. Our main contribution is an averagingestimate, valid in small cells, allowing us to pass from non-uniform to uniform magnetic fields.As a matter of application, we derive new upper and lower bounds of the lowest eigenvalue of theDirichlet Laplacian which match in the regime of large magnetic field intensity. Furthermore,our averaging technique allows us to estimate the non-linear Ginzburg-Landau energy, and as abyproduct, yields a non-Gaussian trial state for the Dirichlet magnetic Laplacian.
Contents
1. Introduction 11.1. The magnetic field 21.2. The averaging estimate 21.3. The Dirichlet magnetic Laplacian 31.4. The Ginzburg-Landau functional 41.5. Organization of the paper 62. Preliminaries and notation 6Asymptotic order 6The averaged magnetic field 63. Averaging of the magnetic field 74. Approximation of the quadratic form 85. Magnetic Laplacian 105.1. Upper bound 105.2. Lower bound 115.3. Proof of Theorem 1.2 116. The Ginzburg-Landau model 116.1. Lower bound of GL energy 126.2. Upper bound of GL energy 136.3. Proof of Theorem 1.3 146.4. Further remarks 156.5. Application: The Dirichlet Laplacian 16Acknowledgments 17References 171.
Introduction
The spectral properties of magnetic Schrödinger operators with minimal regularity assump-tions on the magnetic field, magnetic potential and electrical potential, have been central sincedecades [14]. Averaging of magnetic fields was also a valuable tool to study such spectral prop-erties, notably the question of existence of a compact resolvent [13].
Date : December 2, 2020.Mathematics Subject Classification (2010): 35B40, 35P15, 35Q56.
In this paper, we study the averaging of magnetic fields in the context of spectral asymptotics(large field/semi-classical asymptotics). Our estimates will allow us to capture the leading orderterm in the large field asymptotics for the ground state energy of the magnetic Laplacian with aDirichlet condition, via the essential infimum of the scalar magnetic field, under a weak regularityhypothesis.1.1.
The magnetic field.
Consider a real-valued function B ∈ H ( R ) . (1.1)The function B stands for a magnetic field (more precisely this is the vertical magnetic fieldwith non-uniform intensity B , i.e. B~z ). We introduce the corresponding magnetic potential A as follows A ( x ) = ( A ( x ) , A ( x )) := 2 Z B ( sx ) A ( sx ) ds ( x ∈ R ) , (1.2)where A is the canonical magnetic potential, satisfying curl A = 1 and defined as follows A ( x ) = 12 ( − x , x ) ( x = ( x , x ) ∈ R ) . (1.3)Clearly, A ∈ H ( R ) and curl A := ∂ x A − ∂ x A = B in R . (1.4)There are many other reasonable choices for the magnetic potential generating the magnetic field B , e.g. A + ∇ χ for any smooth function χ .The aim of this paper is to estimate quantities of the form Z U | ( ∇ − iσ A ) u | dx (1.5)where σ ∈ R , U is an appropriate convex subset of R , typically a square or a disc of smalldiameter compared to the parameter σ , and u ∈ H ( U ) . Such questions naturally occur inmany problems of mathematical physics, such as superconductivity [6], liquid crystals [8] andthe theory of Schrödinger operators [18]. The case of a smooth A is well developed in theliterature, so our aim here is to address this question for the less regular case where B ∈ H ( R ) (i.e. A ∈ H ( R ) ). This is related to [6, Sec. 16.6.1, Open Problem 9] and [16, Problem 2.2.9].Our approach to approximate the quantity in (1.5) is through an averaging technique whichwill allow us to pass from A generating the non-smooth field B , to A av generating a constantfield B av . The approximation will be valid in the regime of large field intensity, σ → + ∞ , andsmall domain U , diam( U ) → ( diam( U ) stands for the diameter of U ). The precise statementwill be given in Theorem 1.1 and Proposition 4.1 below.1.2. The averaging estimate.
Assume that x ∈ U ⊂ R and U is open and convex . (1.6)We denote by | U | the area of U , and by diam( U ) , the diameter of U . We introduce the newmagnetic potential A U new ( x ) = 2 Z B ( s ( x − x ) + x ) A (cid:0) s ( x − x ) (cid:1) ds , (1.7)where A is the canonical magnetic potential introduced in (1.3).Note that, on U , curl A U new = B = curl A , where A is the magnetic potential in (1.2). Sothere exists a function ϕ U ∈ H ( U ) such that A = A U new − ∇ ϕ U on U . (1.8)We introduce the average of the magnetic field B in U as follows B U av = 1 | U | Z U B ( x ) dx . (1.9) AGNETIC LAPLACIAN 3
It is then natural to introduce the average magnetic potential A U av ( x ) = B U av A ( x − x ) = 2 B U av Z A (cid:0) s ( x − x ) (cid:1) ds . (1.10)which generates the constant averaged magnetic field, curl A U av = B U av . Theorem 1.1 belowestablishes that the magnetic potential A U av is a good approximation of A new in the convexdomain U . Theorem 1.1.
For every s ∈ (0 , , and every domain U ⊂ R satisfying (1.6) , the followinginequality holds, Z U | A U new ( x ) − A U av ( x ) | | v ( x ) | dx ≤ δ k∇ B k L ( U ) | U | Z U | v ( x ) | dx , where δ = diam( U ) , B ∈ H ( R ) and v ∈ H ( U ) are arbitrary functions. The Dirichlet magnetic Laplacian.
As a consequence of Theorem 1.1, we can estimatethe lowest eigenvalue, λ ( σ, A ; Ω) , of the Dirichlet magnetic Laplacian − ( ∇ − iσ A ) in L (Ω) , fora domain Ω with a smooth C boundary. Studying the strong field asymptotics, the essentialinfimum of the function B in Ω shows up; this is the quantity introduced as follows m ( B ; Ω) := ess inf x ∈ Ω B ( x ) = sup { c ∈ R : B ( x ) ≥ c a . e . on Ω } . (1.11)The variational min-max principle allows us to express the eigenvalue as follows (when Ω isbounded) λ ( σ, A ; Ω) = inf u ∈ H (Ω) \{ } k ( ∇ − iσ A ) u k L (Ω) k u k L (Ω) . (1.12)Now we state our new estimates on the eigenvalue λ ( σ, A ; Ω) . Theorem 1.2.
Assume that
Ω = N S i =1 Ω i where N ≥ is a positive integer, the sets Ω i arepairwise disjoint, and each Ω i is a bounded connected domain of R such that ∂ Ω i consists of afinite number of smooth C closed curves.If B ∈ H (Ω) and the essential infimum in (1.11) is positive , then the lowest eigenvalue in (1.12) satisfies m ( B ; Ω) σ ≤ λ ( σ, A ; Ω) ≤ m ( B ; Ω) σ + o ( σ ) (cid:0) σ → + ∞ (cid:1) . The content of Theorem 1.2 is consistent with the known estimates for a smooth magneticfield (see [11]), in which case the essential infimum becomes m ( B ; Ω) = min x ∈ Ω B ( x ) , and the remainder term o ( σ ) can be explicitly controlled.The non-asymptotic lower bound, λ ( σ, A ; Ω) ≥ σ m ( B ; Ω) , follows by a standard argument.The matching upper bound, λ ( σ, A ; Ω) ≤ m ( B ; Ω) σ + o ( σ ) , follows by constructing a trial state;the produced errors are controlled by the averaging estimate of Theorem 1.1.The novelty in Theorem 1.2 is establishing its validity in the weakly regular situation when(1.1) holds. This prevents us of deducing it from other works treating non-uniform magneticfields, like smooth magnetic fields [2, 11, 15, 18], B ∈ C ,α ( R ) , or step magnetic fields [3, 12].It would be desirable to establish Theorem 1.2 under the much weaker hypothesis, B ∈ L (Ω) .This is motivated by the current Theorem 1.2 and the existing results when B is a step function The definition of the eigenvalue λ ( σ, A ; Ω) requires a vector field A (and consequently a magnetic field B )defined on Ω , not the whole space R . Our assumption on the domain Ω allows us to extend functions in theSobolev space H (Ω) to functions in the space H ( R ) , so that starting with B ∈ H ( R ) is not really a restriction.Our proofs require to deal with the value of the magnetic field outside the set Ω . AYMAN KACHMAR AND MOHAMMAD WEHBE [3, 12]. However, knowing B ∈ L (Ω) without further regularity, our averaging estimate inTheorem 1.1 will be out of reach, thereby preventing us from deducing Theorem 1.2.1.4. The Ginzburg-Landau functional.
One more setting where our averaging mechanismis robust is the study of the non-linear Ginzburg-Landau functional (see Theorem 1.3 below).Under the regularity assumption (1.1) on B , our contribution adds to the mainstream of under-standing the role of non-uniform magnetic fields in the Ginzburg-Landau model [2, 5, 9, 10, 17].Handling the particularities of our regularity hypothesis in (1.1) would not be possible withoutthe averaging estimate of Theorem 1.1.We restrict our study to a bounded domain Ω ⊂ R which we assume connected and with asmooth boundary consisting of a finite number of smooth curves of class C . More precisely, weassume that Ω = ˜Ω \ n S k =1 ω k , where ω , · · · , ω n , ˜Ω are simply connected domains with smooth C boundaries, each ω k ⊂ ˜Ω , and the sets ω k are pair-wise disjoint.A central role will be played by the magnetic potential F ∈ H ( ˜Ω) satisfyingcurl F = B , div F = 0 in ˜Ω , ν · F = 0 on ∂ ˜Ω , (1.13)where ν is the unit interior normal vector of ∂ ˜Ω . Since the domain ˜Ω is simply connected, A − F is a gradient field on ˜Ω and we can find a function ϑ ∈ H ( ˜Ω) such that (see [6, Prop. D.1.1]) A = F + ∇ ϑ on ˜Ω . (1.14) The functional & critical configurations.
The GL functional is defined for configurations ( ψ, A ) in the space H (Ω; C ) × H ( ˜Ω; R ) asfollows G ( ψ, A ) = Z Ω (cid:18) | ( ∇ − iκH A ) ψ | − κ | ψ | + κ | ψ | (cid:19) dx + ( κH ) Z ˜Ω | curl ( A − F ) | dx , (1.15)where F is the magnetic potential introduced in (1.13). We introduce the ground state energy E( κ, H ) = inf {G ( ψ, A ) : ( ψ, A ) ∈ H (Ω; C ) × H ( ˜Ω; R ) } , (1.16)where A ∈ H ( ˜Ω; R ) means A ∈ H ( ˜Ω; R ) , div A = 0 in ˜Ω , ν · A = 0 on ∂ ˜Ω , (1.17)and ν is the inward normal vector of ∂ ˜Ω . The property of gauge invariance yields [6, Sec. 10.1.2] E( κ, H ) = inf {G ( ψ, A ) : ( ψ, A ) ∈ H (Ω; C ) × H ( ˜Ω; R ) } . Every minimizing configuration ( ψ, A ) κ,H is a critical point of the GL functional, that is itsatisfies the following equations: − ( ∇ − iκH A ) ψ = κ (1 − | ψ | ) ψ in Ω , −∇ ⊥ (cid:16) curl ( A − F ) (cid:17) = 1 κH Ω Im (cid:16) ψ ( ∇ − iκH A ) ψ (cid:17) in ˜Ω ,ν · · ( ∇ − iκH A ) ψ = 0 on ∂ Ω , curl ( A − F ) = 0 on ∂ ˜Ω , (1.18)where ∇ ⊥ = ( ∂ x , − ∂ x ) is the Hodge gradient. Bulk energy function.
The GL ground state energy E( κ, H ) in (1.16) is closely related to a simplified effective energy,which we will call the bulk energy function. This is the convex function g : [0 , + ∞ ) → [ − , that we will introduce below. First, we set g (0) = − and g ( b ) = 0 for all b ≥ ; the definitionof g ( b ) when b ∈ (0 , is implicit through the large area limit of a certain non-linear energy[1, 7, 20]. AGNETIC LAPLACIAN 5
Let
R > and Q R = ( − R/ , R/ × ( − R/ , R/ . We define the following Ginzburg-Lamdauenergy with the constant magnetic field on H ( Q R ) by G b,Q R ( u ) = Z Q R (cid:18) b | ( ∇ − i A ) u | − | u | + 12 | u | (cid:19) dx. Here A is the vector field introduced in (1.3). We introduce the two ground state energies m ( b, R ) = inf u ∈ H ( Q R ) G σb,Q R ( u ) , and m ( b, R ) = inf u ∈ H ( Q R ) G σb,Q R ( u ) . We gather the following remarkable properties (see [7, Thm. 2.1]): • If b ≥ and R > , then m ( b, R ) = 0 . • m (0 , R ) = − R . • Every minimizer u b,R of m ( b, R ) or m ( b, R ) satisfies the uniform bound | u b,R | ≤ . • For all b ∈ [0 , ∞ ) , the following limits exist g ( b ) = lim R →∞ m ( b, R ) R = lim R →∞ m ( b, R ) R . • There exist positive constants C and R , such that, for all R ≥ R and b ∈ [0 , , g ( b ) ≤ m ( b, R ) R ≤ g ( b ) + CR and g ( b ) − CR ≤ m ( b, R ) R ≤ g ( b ) + CR . (1.19)
The leading order energy.
The approximation of the energy E( κ, H ) will require the decomposition of the domain Ω intosmall cells, which we describe below and eventually define the leading order energy in (1.27).We fix two positive constants c and c such that < c < c , and we let ℓ be a parameterthat varies in the following manner c κ − / ≤ ℓ ≤ c κ − / , (1.20)so that ℓ approaches in the regime of large GL parameter κ .Now we set x ℓm,n := ( ℓm, ℓn ) (cid:0) ( m, n ) ∈ Z (cid:1) , (1.21) J ℓ = { x ℓm,n : ( m, n ) ∈ Z & Q ℓ ( x ℓm,n ) ⊂ Ω } , (1.22)and Ω ℓ := [ x ∈J ℓ Q ℓ ( x ) , (1.23)where Q ℓ ( · ) is the open square introduced in (2.1). The definition of the set J ℓ yields that thesquares (cid:0) Q ℓ ( x ) (cid:1) x ∈J ℓ are pairwise disjoint, and Ω ℓ ⊂ Ω . Consequently the set J ℓ is finite, sincethe domain Ω is bounded, and its cardinal N ( ℓ ) := Card( J ℓ ) (1.24)satisfies the obvious upper bound N ( ℓ ) ≤ | Ω | ℓ − . (1.25)Furthermore, by smoothness and boundedness of the boundary ∂ Ω , we can write the followinglower bound on the number N ( ℓ ) , N ( ℓ ) ≥ | Ω | ℓ − − O ( ℓ − ) ( ℓ → + ) . (1.26)We demonstrate in Theorem 1.3 below that the GL ground state energy, E( κ, H ) , introducedin (1.16), is to leading order given by the following energy E asy ( b, ℓ ) = ℓ X x ∈J ℓ g (cid:0) bB ℓ av ( x ) (cid:1) , (1.27)where ℓ and J ℓ are introduced in (1.20) and (1.22) respectively, and g ( · ) is the bulk energyfunction introduced in (1.19). AYMAN KACHMAR AND MOHAMMAD WEHBE
Theorem 1.3.
Assume that there exists a positive real number c such that B ≥ c > a.e. in Ω .Given ǫ ∈ (0 , and c > c > , there exist constants C, κ such that, for all κ ≥ κ , H = bκ , ℓ satisfying (1.20) , and b ∈ ( ǫ, ǫ − ) , the following holds (cid:12)(cid:12) E( κ, H ) − κ E asy ( b, ℓ ) (cid:12)(cid:12) ≤ Cκ / . Remark . Since g ( · ) ≥ − , we get by (1.25), − | Ω | ≤ ℓ X x ∈J ℓ g (cid:0) bB ℓ av ( x ) (cid:1) ≤ . Furthermore, since g ( · ) is convex, Jensen’s inequality yields ℓ X x ∈J ℓ g (cid:0) bB ℓ av ( x ) (cid:1) ≤ X x ∈J ℓ Z Q ℓ ( x ) g (cid:0) bB ( y ) (cid:1) dy = Z Ω g (cid:0) bB ( y ) (cid:1) dy + O ( ℓ ) . Consequently, we see that, if b > is a fixed constant (independent from the parameters κ, H, ℓ ),the asymptotic energy in (1.27), satisfies E asy ( b, ℓ ) = ℓ → + o (1) ⇐⇒ (cid:12)(cid:12) { y ∈ Ω : bB ( y ) < } (cid:12)(cid:12) = 0 . (1.28) Remark . We can deduce Theorem 1.2 from Theorem 1.3, by using the GL order parameteras a trial state for the Dirichlet eigenvalue. We present this construction in Sec. 6.5, whichhighlights the possibility of extracting spectral asymototics from the study of the GL model,despite the many existing results that go in the opposite direction, namely studying the GLmodel starting from eigenvalue estimates of the magnetic Laplacian.1.5.
Organization of the paper.
The paper is organized as follows. Section 2 contains somestandard material that we are going to use through the paper. Section 3 contains the proof of theaveraging estimate, Theorem 1.1. The estimate of the energy in (1.5) occupies Section 4. Theproof of Theorem 1.2 is given in Section 5. Section 6 is devoted to the study of the Ginzburg-Landau model and ends up by an alternative proof of the eigenvalue upper bound for the Dirichletmagnetic Laplacian (Sec. 6.5).2.
Preliminaries and notation
The purpose of this section is to introduce the necessary material for the statement of themain theorems in the subsequent sections.
Asymptotic order.
We will use the standard Landau notation to denote bounded quanti-ties, O (1) , and vanishingly small quantities, o (1) , with respect to a parameter σ living in aneighborhood of + ∞ . Additionally, we use the notation ≈ in the following context; given twofunctions a ( σ ) and b ( σ ) , writing a ≈ b means that there exist positive constants σ , c , c suchthat c b ( σ ) ≤ a ( σ ) ≤ c b ( σ ) . We use the letter C to denote constants. The value of C mightchange from one inequality to another without mentioning this explicitly. The averaged magnetic field.
For all x ∈ R and ℓ > , we introduce the open square ofcenter x and side-length ℓ as follows Q ℓ ( x ) = ( x − ℓ/ , x + ℓ/ × ( x − ℓ/ , x + ℓ/ . (2.1)We introduce the averaged magnetic field in the square Q ℓ ( x ) , B ℓ av ( x ) = 1 ℓ Z Q ℓ ( x ) B ( y ) dy . (2.2)Note that, if B satisfies the following condition in some open set Ω ⊂ R , ∃ c ∈ R , B ≥ c a . e . (2.3) AGNETIC LAPLACIAN 7 then the averaged magnetic field satisfies B ℓ av ( x ) ≥ c whenever Q ℓ ( x ) ⊂ Ω . (2.4)Assuming (1.1), we will prove that B ℓ av ( x ) can have only slow growth in the small length limit. Lemma 2.1.
For all ζ ∈ (0 , , there exist C, ℓ > such that, for all ℓ ∈ (0 , ℓ ) , B ∈ H ( R ) and x ∈ R , the following holds, | B ℓ av ( x ) | ≤ Cℓ − ζ k B k H ( R ) . Proof.
Notice that, (cid:12)(cid:12) B ℓ av ( x ) (cid:12)(cid:12) ≤ ℓ Z Q ℓ ( x ) | B ( y ) | dy . Let p = ζ and q = pp − the Hölder conjugate of p . By Hölder’s inequality Z Q ℓ ( x ) | B ( y ) | dy ≤ | Q ℓ ( x ) | /q k B k L p ( R ) . Consequently, (cid:12)(cid:12) B ℓ av ( x ) (cid:12)(cid:12) ≤ ℓ q − k B k L p ( R ) = ℓ − ζ k B k L p ( R ) . To finish the proof, we use the Sobolev embedding of H ( R ) in L p ( R ) . (cid:3) Averaging of the magnetic field
The proof of Theorem 1.1 relies on the following proposition.
Proposition 3.1.
For every s ∈ (0 , , and every domain U ⊂ R satisfying (1.6) , the followinginequality holds, Z U | B ( m ( x )) − B U av | | v ( x ) | dx ≤ δ k∇ B k L ( U ) | U | Z U | v ( x ) | dx , where δ = diam( U ) , m ( x ) = s ( x − x ) + x , B ∈ H ( R ) and v ∈ H ( U ) are arbitrary functions.Proof. We will prove Proposition 3.1 in the special case where B ∈ C ( R ) . The general casefollows then by a density argument, using the density of C ∞ ( R ) in H ( R ) and the Sobolevembedding of H ( R ) in L ( R ) .First, notice that B ( m ( x )) − B U av = 1 | U | Z U (cid:0) B ( m ( x )) − B ( y ) (cid:1) dy , and for all x, y ∈ U , | m ( x ) − y | ≤ δ .Let w ( x, y ) = (cid:0) B ( m ( x )) − B ( y ) (cid:1) v ( x ) . Then, by Jensen’s inequality, Z U | B ( m ( x )) − B U av | | v ( x ) | dx ≤ | U | Z U (cid:18)Z U | w ( x, y ) | dy (cid:19) dx . Now, it is enough to prove the following inequality, for all z ∈ U , Z U | B ( z ) − B ( y ) | dy ≤ δ k∇ B k L ( U ) . Indeed, for y, z ∈ U with y = z , the convexity of U ensures that z + t y − z | y − z | ∈ U for t ∈ [0 , | y − z | ] ,hence, B ( z ) − B ( y ) = − Z | y − z | ddt B (cid:18) z + t y − z | y − z | (cid:19) dt = − Z | y − z | ∇ B (cid:18) z + t y − z | y − z | (cid:19) · y − z | y − z | dt AYMAN KACHMAR AND MOHAMMAD WEHBE
Consequently, | B ( z ) − B ( y ) | ≤ Z | y − z | (cid:12)(cid:12)(cid:12)(cid:12) ∇ B (cid:18) z + t y − z | y − z | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dt = | y − z | Z |∇ B ( z + τ ( y − z )) | dτ after performing the change of variable τ = t/ | y − z | . By Jensen’s inequality, we get further | B ( z ) − B ( y ) | ≤ | y − z | Z |∇ B ( z + τ ( y − z )) | dτ ≤ δ Z |∇ B ( z + τ ( y − z )) | dτ . Now we integrate over U and perform the change of variable y a = z + τ ( y − z ) ; we obtain Z U | B ( z ) − B ( y ) | dy ≤ δ Z Z U |∇ B ( z + τ ( y − z )) | dydτ ≤ δ Z Z U z,τ |∇ B ( a ) | da dτ , where U z,τ = { z + τ ( y − z ) : y ∈ U } ; since U is convex, U z,τ ⊂ U for all z ∈ U and τ ∈ [0 , . (cid:3) Proof of Theorem 1.1.
Collecting (1.7) and (1.10), we write, for all x ∈ U , A U new ( x ) − A U av ( x ) = 2 Z (cid:16) B ( s ( x − x ) + x ) − B U av (cid:17) A (cid:0) s ( x − x ) (cid:1) ds . Since (cid:12)(cid:12) A (cid:0) s ( x − x ) (cid:1)(cid:12)(cid:12) ≤ s | x − x | ≤ s diam( U ) on U , we get by using Jensen’s inequality, ∀ x ∈ U , | A U new ( x ) − A U av ( x ) | ≤ δ Z | B ( s ( x − x ) + x ) − B U av | s ds . (3.1)We apply Proposition 3.1 to the estimate the term in the r.h.s. in (3.1). This finishes the proofof Theorem 1.1. (cid:3) Remark . If we perform the change of variable y = s ( x − x ) + x and note that U is convex(which guarantees that y ∈ U , for all x ∈ U ), we deduce from (3.1), Z U | A U new ( x ) − A U av ( x ) | dx ≤ δ Z Z U | B ( s ( x − x ) + x ) − B U av | s dxds ≤ δ Z U | B ( y ) − B U av | dy . (3.2)4. Approximation of the quadratic form
Given a bounded open set U ⊂ R , a function u ∈ H ( U ) , a vector field a ∈ H ( U ; R ) and areal number σ , we introduce q σ (cid:0) u, a ; U (cid:1) = Z U | ( ∇ − iσ a ) u | dx . (4.1) Proposition 4.1.
Given η, ρ ∈ (0 , ) and < c < c , there exist constants C ′ , σ > such thatthe following is true. If • σ ≥ σ ; • U ⊂ R is open and convex ; • c σ − ρ ≤ diam( U ) , | U | / ≤ c σ − ρ • u ∈ H ( U ) , B ∈ H ( R ) & A defined by (1.2) , AGNETIC LAPLACIAN 9 then there exists a function ϕ := ϕ U ∈ H ( U ) such that (1 − σ − η ) q σ ( v, A U av ; U ) − C ′ σ − ρ + η k∇ B k L ( U ) Z U | u | dx ≤ q σ ( u, A ; U ) ≤ (1 + σ − η ) q σ ( v, A U av ; U ) + C ′ σ − ρ + η k∇ B k L ( U ) Z U | u | dx where A U av is introduced in (1.10) and v = e iσϕ u .Remark . The condition ρ ∈ (0 , ) is a consequence of a scaling argument. Since x ∈ U and diam( U ) ≈ σ − ρ , we have U ⊂ {| x − x | ≤ O ( σ − ρ ) } . The change of variable, y = σ / ( x − x ) yields (see (1.10)) q σ ( v, A U av ; U ) = σ Z ˜ U σ | ( ∇ − B U av A )˜ v | dy , where ˜ U σ = { y = σ / ( x − x ) , x ∈ U } ⊂ {| y | ≤ O ( σ − ρ ) } and ˜ v ( y ) = v ( x ) . To ensure that ˜ U σ approaches R (which is a fixed domain), we impose the condition ρ ∈ (0 , ) . Remark . Earlier works on the magnetic Laplacian [6, Sec. 1.4.2, p. 11] suggest that q σ ( u, A ; U ) behaves like O ( σ ) R U | u | dx , when σ → + ∞ . Such a behavior is consistent with the estimatesin Proposition 4.1 if we manage to work with a trial-state u satisfying Z U | u | dx ≤ O ( | U | ) Z Ω | u | dx , (4.2)which ensures that the error term satisfies σ − ρ + η k∇ B k L ( U ) Z U | u | dx = O ( σ − ρ + η ) k∇ B k L ( U ) Z Ω | u | dx = o ( σ ) k∇ B k L ( U ) Z Ω | u | dx for a specific choice of ( ρ, η ) ∈ (0 , ) × (0 , . A trial-state having the rich property in (4.2) isprovided by the non-linear model studied in Sec. 6. Proof of Proposition 4.1.
Note that the following holds:i. (Gauge transformation) if v = e − iσφ u , then q σ ( v, A U av , U ) = q σ ( v, A U av − ∇ φ, U ) ;ii. (Cauchy’s inequality) for every a, b, σ > , ( a + b ) ≤ (1 + σ − η ) a + (1 + σ η ) b ;iii. Proposition 1.1 ;iv. δ := diam( U ) satisfies δ ≤ c σ − ρ , and | U | ≤ c σ ρ .Now we write q σ ( u, A ; U ) := Z U | ( ∇ − iσ A ) u | dx = Z U (cid:12)(cid:12)(cid:0) ∇ − iσ ( A U new − A U av + A U av − ∇ φ ) (cid:1) u (cid:12)(cid:12) dx i . = Z U (cid:12)(cid:12) ( ∇ − i A U av ) v − iσ ( A U new − A U av ) u (cid:12)(cid:12) dx ii . ≤ (1 + σ − η ) Z U (cid:12)(cid:12) ( ∇ − iσ A U av ) v (cid:12)(cid:12) dx + (1 + σ η ) σ Z U (cid:12)(cid:12) ( A U new − A U av ) u (cid:12)(cid:12) dx iii . ≤ (1 + σ − η ) Z U (cid:12)(cid:12) ( ∇ − iσ A U av ) v (cid:12)(cid:12) dx + σ (1 + σ η ) C δ | U | k∇ B k L ( U ) Z U | u | dx iv . ≤ (1 + σ − η ) q σ ( v, A U av ; U ) + σ (1 + σ η ) Cc c σ − ρ k∇ B k L ( U ) Z U | u | dx = (1 + σ − η ) q σ ( v, A U av ; U ) + σ (1 + σ η ) σ − ρ C ′ k∇ B k L ( U ) Z U | u | dx . A similar argument yields (1 − σ − η ) q σ ( v, A U av ; U ) − C ′ σ − ρ + η k∇ B k L ( U ) Z U | u | dx ≤ q σ ( u, A ; U ) . (cid:3) Magnetic Laplacian
The aim of this section is to prove Theorem 1.2, which is concerned with the principal eigen-value of the magnetic Laplacian ∆ σ A = − ( ∇ − iσ A ) (5.1)with domain (when Ω ⊂ R is bounded and with a smooth C boundary) D = H (Ω) ∩ H (Ω) . (5.2)The operator ∆ σ A is self-adjoint in the Hilbert space L (Ω) and its principal eigenvalue isintroduced in (1.12).5.1. Upper bound.
We will construct a trial state by means of a Gaussian function, but lo-calized near a point x ε ∈ Ω such that the Lebesgue differentiation theorem holds for B ( x ) and |∇ B ( x ) | at x ε , and as ε → + , B ( x ε ) = m ( B ; Ω) + O ( ε ) , where m ( B ; Ω) is the essentialinfimum introduced in (1.11).By the Lebesgue differentiation theorem, the two sets N = { u ∈ Ω , lim ℓ → | D ( u, ℓ ) | Z D ( u,ℓ ) |∇ B ( x ) | dx = |∇ B ( u ) | } ˜ N = { u ∈ Ω , lim ℓ → | D ( u, ℓ ) | Z D ( u,ℓ ) B ( x ) dx = B ( u ) } have zero Lebesgue measure, where D ( u, ℓ ) denotes the open disk of center u and radius ℓ .We assume that m ( B ; Ω) > . For all ε ∈ (0 , , we introduce the set M ε = { x ∈ Ω , m ( B ; Ω) ≤ B ( x ) ≤ m ( B ; Ω) + ε } . Since the set M ε has a non-zero Lebesgue measure, we get by the Lebesgue differentiation theorem ∃ x ε ∈ M ε , | D ( x ε , ℓ ) | Z D ( x ε ,ℓ ) |∇ B ( x ) | dx −→ ℓ → + |∇ B ( x ε ) | < + ∞ and 1 | D ( x ε , ℓ ) | Z D ( x ε ,ℓ ) B ( x ) dx −→ ℓ → + B ( x ε ) < + ∞ . (5.3)In the sequel, ρ ∈ (0 , ) and U := D ( x ε , σ − ρ ) ⊂ Ω for σ sufficiently large. Let ϕ := ϕ U bethe gauge function in Proposition 4.1. Consider the trial state u ( x ) = e − iσϕ v ( x ) , with v thefollowing Gaussian, v ( x ) = π − / (cid:0) B U av (cid:1) / σ / χ (cid:0) σ ρ ( x − x ) (cid:1) exp (cid:18) − (cid:0) B U av (cid:1) / σ | x − x | (cid:19) , where χ ∈ C ∞ c (cid:0) R ; [0 , (cid:1) is supported in the unit disk and equal to on {| x | ≤ } . By a changeof variable, we see that k u k L ( U ) = k u k L ( U ) = 1 + o ( σ ) and q σ ( v, A U av ; U ) = B U av σ + o ( σ ) . Consequently, we deduce from Proposition 4.1, q σ ( u, A ; U ) k u k L ( U ) ≤ (1 + σ − η ) σB U av + O (cid:0) k∇ B k L ( U ) σ − ρ + η (cid:1) . AGNETIC LAPLACIAN 11
Since U = D ( x ε , σ − ρ ) , we infer from (5.3) that q σ ( u, A ; U ) k u k L ( U ) ≤ (1 + σ − η ) σB ( x ε ) + O (cid:0) |∇ B ( x ε ) | σ − ρ + η (cid:1) . We choose ρ = 3 / and η = 1 / . Since u is supported in U , we deduce from the min-maxprinciple (1.12), λ ( σ, A ; Ω) ≤ B ( x ε ) σ + O ( σ / ) ≤ ( m ( B ; Ω) + ε ) σ + O ( σ / ) . Taking the successive limits, as σ → + ∞ then as ε → + , we get lim sup σ → + ∞ λ ( σ, A ; Ω) σ ≤ m ( B ; Ω) . (5.4)5.2. Lower bound.
As mentioned earlier, the lower bound in Theorem 1.2 is non-asymptoticand does not require the hypothesis that the essential infimum is strictly positive.
Proposition 5.1.
Let A ∈ H ( R ; R ) and B = curl A . For all u ∈ C ∞ c (Ω) and σ > , thefollowing lower bound holds Z Ω | ( ∇ − iσ A ) u | dx ≥ Z Ω B ( x ) | u ( x ) | dx . Proof.
Consider a sequence ( A n ) n ≥ ⊂ C ∞ ( R ; R ) such that A n → A in H ( R ; R ) . For all n ≥ , let B n = curl A n . Note that B n → B in L ( R ) .Fix u ∈ C ∞ c (Ω) . Since A n is smooth, we have (see [6, Lem. 1.4.1]) Z Ω | ( ∇ − iσ A n ) u | dx ≥ Z Ω B n ( x ) | u ( x ) | dx . It is easy to check that lim n → + ∞ Z Ω | ( ∇ − iσ A n ) u | dx = Z Ω | ( ∇ − iσ A ) u | dx and lim n → + ∞ Z Ω B n ( x ) | u ( x ) | dx = Z Ω B ( x ) | u ( x ) | dx . In fact, (cid:12)(cid:12)(cid:12) k ( ∇ − iσ A n ) u k L (Ω) − k ( ∇ − iσ A ) u k L (Ω) (cid:12)(cid:12)(cid:12) ≤ σ k A n − A k L (Ω) k u k L (Ω) and (cid:12)(cid:12)(cid:12)(cid:12)Z Ω (cid:0) B n ( x ) − B ( x ) (cid:1) | u ( x ) | dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k B n − B k L (Ω) k u k L (Ω) . (cid:3) Proof of Theorem 1.2.
Collect (5.4) and Proposition 5.1.6.
The Ginzburg-Landau model
This section is devoted to the proof of Theorem 1.3. Also, in Sec. 6.5, we use Theorem 1.3 togive a new proof of Theorem 1.2.
Lower bound of GL energy.
In the sequel, ( ψ, A ) κ,H denotes a configuration in the space H (Ω; C ) × H (Ω; R ) such that G ( ψ, A ) = E( κ, H ) . Our aim is to prove the following proposition.
Proposition 6.1.
Given ǫ ∈ (0 , , there exist C, κ > such that the following inequality holds G ( ψ, A ; Q ℓ ( x )) ≥ g (cid:0) bB ℓ av ( x ) (cid:1) κ ℓ − C (cid:16) κ / ℓ + κ / k∇ B k L ( Q ℓ ( x ) (cid:17) , where • x ∈ J ℓ ; • ℓ = κ − / ; • ( ψ, A ) κ,H is a minimizer of the GL functional ; • H = bκ and b ∈ ( ǫ, ǫ ) ; • G ( ψ, A ; Q ℓ ( x )) = Z Q ℓ ( x ) (cid:18) | ( ∇ − iκH A ) ψ | − κ | ψ | + κ | ψ | (cid:19) dx .Proof. First we notice the useful inequalities (see [4, Prop. 4.1 & Thm. 4.2]) k ψ k L ∞ (Ω) ≤ , k ( ∇ − iκH A ) ψ k L (Ω) ≤ | Ω | κ , kA − F k C ,α (Ω) ≤ C α κ , (6.1)where α ∈ (0 , can be chosen in an arbitrary manner.We set φ x := (cid:16) A ( x ) − F ( x ) (cid:17) · ( x − x ) and A new = A − ∇ φ x . (6.2)It is easy to check that G ( ψ, A ; Q ℓ ( x )) = G ( u, A new ; Q ℓ ( x )) (6.3)where u ( x ) = e − iκHφ x ψ ( x ) . (6.4)Writing A new = F + A new − F , we get by Cauchy’s inequality, | ( ∇ − iκH A new ) u | ≥ (1 − κ − / ) | ( ∇ − iκH F ) u | − κ / ( κH ) | ( A new − F ) u | . Consequently, we infer from the foregoing inequality and the third inequality in (6.1), G (cid:0) u, A new ; Q ℓ ( x ) (cid:1) ≥ (1 − κ − / ) G (cid:0) u, F , Q ℓ ( x ) (cid:1) + κ / Z Q ℓ ( x ) (cid:18) − κ | u | + κ | u | (cid:19) dx − Cb ℓ α κ / Z Q ℓ ( x ) | u | dx. (6.5)Using that | u | = | ψ | ≤ by (6.1), we can estimate the remainder terms in (6.5) as follows κ − / Z Q ℓ ( x ) (cid:18) − κ | u | + κ | u | (cid:19) dx − Cb ℓ α κ / Z Q ℓ ( x ) | u | dx ≥ − κ ℓ (cid:0) κ − / + Cb κ / ℓ α (cid:1) . (6.6)In order to estimate the term G (cid:0) u, F , Q ℓ ( x ) (cid:1) in (6.5), we will go from the potential F to thepotential A introduced in (1.2). Let ϑ be the function in (1.14) and set v = e − iκHϑ u . (6.7) AGNETIC LAPLACIAN 13
Then G (cid:0) u, F ; Q ℓ ( x ) (cid:1) = G (cid:0) v, A ; Q ℓ ( x ) (cid:1) ≥ (1 − κ − / ) G (cid:16) w, A Q ℓ ( x )av ; Q ℓ ( x ) (cid:17) − ˆ Cκ / k∇ B k L ( Q ℓ ( x )) + κ − / Z Q ℓ ( x ) (cid:18) − κ | w | + κ | w | (cid:19) dx (6.8)where we used Proposition 4.1, with σ = κH = bκ and η = 1 / , to estimate the L -norm of | ( ∇ − iκH A ) v | ; the function w is expressed in terms of v and the gauge function ϕ Q ℓ ( x ) ofProposition 4.1 as follows w ( x ) = v ( x ) exp (cid:16) iκHϕ Q ℓ ( x ) ( x ) (cid:17) . (6.9)Since | v | = | u | = | ψ | ≤ by (6.1), we infer from (6.8), G (cid:0) u, F ; Q ℓ ( x ) (cid:1) ≥ (1 − κ − / ) G (cid:16) w, A Q ℓ ( x )av ; Q ℓ ( x ) (cid:17) − ˆ Cκ / k∇ B k L ( Q ℓ ( x )) − κ / ℓ . (6.10)Note that curl A Q ℓ ( x )av = B ℓ av ( x ) introduced in (1.9). We write now a lower bound of the energy G (cid:16) w, A Q ℓ ( x )av ; Q ℓ ( x ) (cid:17) using the bulk energy function g ( · ) . To that end, we introduce • ˆ b = Hκ B ℓ av ( x ) = bB ℓ av ( x ) ; • R = ℓ p κHB ℓ av ( x ) ; • h ( x ) = w (cid:0) ℓR x + x (cid:1) for x ∈ Q R := ( − R/ , R/ ; • The change of variable y = Rℓ ( x − x ) .It is then easy to check that G (cid:16) w, A Q ℓ ( x )av ; Q ℓ ( x ) (cid:17) = 1ˆ b G ˆ b,Q R ( h ) ≥ m (ˆ b, R ) ≥ b (cid:0) g (ˆ b ) R − ˜ CR (cid:1) , by (1.19). Inserting the foregoing inequality into (6.10), then remembering the definition of ˆ b ,choosing α = , and collecting the inequalities in (6.8), (6.6), (6.5), and (6.3), we eventually getthe following inequality, G ( ψ, A ; Q ℓ ( x )) ≥ g (cid:16) bB ℓ av ( x ) (cid:17) κ ℓ − ˇ C (cid:18) κ ℓ (cid:16) κ − / + κ − / q B ℓ av ( x ) (cid:17) − κ / k∇ B k L ( Q ℓ ( x )) (cid:19) . Finally, we apply Lemma 2.1 with ζ = . (cid:3) Upper bound of GL energy.Proposition 6.2.
Given ǫ ∈ (0 , and c > c > , there exist C, κ > such that, for all κ ≥ κ , the following holds.For every x ∈ J ℓ , with ℓ satisfying (1.20) , there exists a function v x ,ℓ ∈ H ( Q ℓ ( x )) suchthat G ( v x ,ℓ , F ; Q ℓ ( x )) ≤ g (cid:0) bB ℓ av ( x ) (cid:1) κ ℓ + C (cid:16) κ / ℓ + κ / k∇ B k L ( Q ℓ ( x ) (cid:17) , where • F is the magnetic potential introduced in (1.13) ; • H = bκ and b ∈ ( ǫ, ǫ ) ; • the functional G ( · , · ; Q ℓ ( x )) is introduced in Proposition 6.1 .Proof. We choose b ∈ ( ǫ, ǫ − ) and an arbitrary point x ∈ J ℓ , with ℓ ≈ κ − / . We introduce thetwo parameters (that depend on x and ℓ ) ˆ b = Hκ B ℓ av ( x ) = bB ℓ av ( x ) and R = ℓ q κHB ℓ av ( x ) . Let u ˆ b,R ∈ H (cid:0) ( − R/ , R/ (cid:1) be a minimizer of the energy functional m (ˆ b, R ) . For all x ∈ Q ℓ ( x ) , we introduce the function v := v x ,ℓ ∈ H ( Q ℓ ( x )) as follows v ( x ) = exp (cid:16) iκH ( ϕ + ϑ ) (cid:17) u ˆ b,R (cid:18) Rℓ ( x − x ) (cid:19) , where ϑ is the function introduced in (1.14) and ϕ := ϕ Q ℓ ( x ) is the function introduced inProposition 4.1. Setting h = exp (cid:16) − iκH ( ϕ + ϑ ) (cid:17) v , it is easy to check that G (cid:16) h, A Q ℓ ( x )av ; Q ℓ ( x ) (cid:17) = 1ˆ b m (ˆ b, R ) . Using (1.19), we get further G (cid:16) h, A Q ℓ ( x )av ; Q ℓ ( x ) (cid:17) ≤ g (cid:0) bB ℓ av ( x ) (cid:1) κ ℓ + O ( κℓ ) . (6.11)Setting u = exp (cid:16) − iκHϑ (cid:17) v , we get by (1.14), G (cid:0) v, F ; Q ℓ ( x ) (cid:1) = G (cid:0) u, A ; Q ℓ ( x ) (cid:1) . (6.12)Now we apply Proposition 4.1 with σ = κH = bκ , ρ = 3 / and η = 1 / ; eventually we get G (cid:0) u, A ; Q ℓ ( x ) (cid:1) ≤ (1 + κ − / ) G (cid:0) h, A Q ℓ ( x )av ; Q ℓ ( x ) (cid:1) − κ − / Z Q ℓ ( x ) (cid:18) − κ | h | + κ | h | (cid:19) dx + O (cid:16) κ k∇ B k L ( Q ℓ ( x ) (cid:17) Z Q ℓ ( x ) | h | dx . Since | h | ≤ , we get further G (cid:0) u, A ; Q ℓ ( x ) (cid:1) ≤ (1 + κ − / ) G (cid:0) h, A Q ℓ ( x )av ; Q ℓ ( x ) (cid:1) + O ( κ / ℓ ) + O ( κ / ) k∇ B k L ( Q ℓ ( x )) . (6.13)Collecting (6.13), (6.12) and (6.11), we finish the proof of Proposition 6.2. (cid:3) Proof of Theorem 1.3.
Now we work under the assumptions of Theorem 1.3. We fix ǫ ∈ (0 , and assume that H = bκ with b varying in ( ǫ, ǫ − ) . Recall that ℓ ≈ κ − / by (1.20). Step 1:
Denote by ( ψ, A ) κ,H a minimizing configuration such that G ( ψ, A ) = E( κ, H ) . Dropping theterm κ H R Ω | curl ( A − F ) | dx from the energy G ( ψ, A ) , we get the obvious lower bound E( κ, H ) = G ( ψ, A ) ≥ G (cid:0) ψ, A ; Ω (cid:1) = G (cid:0) ψ, A ; Ω ℓ (cid:1) + G (cid:0) ψ, A ; Ω \ Ω ℓ (cid:1) where G is energy introduced in Proposition 6.1, and Ω ℓ is the domain introduced in (1.23).Using the uniform bounds | ψ | ≤ and | Ω \ Ω ℓ | = O ( ℓ ) , we get G (cid:0) ψ, A ; Ω \ Ω ℓ (cid:1) ≥ − κ Z Ω \ Ω ℓ = O ( ℓκ ) = O ( κ / ) . Now, we use the obvious decomposition G (cid:0) ψ, A ; Ω ℓ (cid:1) = P x ∈J ℓ G (cid:0) ψ, A ; Q ℓ ( x ) (cid:1) and apply Proposi-tion 6.1. Eventually, we get E( κ, H ) ≥ κ ℓ X x ∈J ℓ g (cid:0) bB ℓ av ( x ) (cid:1) − Cκ / − C X x ∈J ℓ (cid:16) κ / ℓ + κ / k∇ B k L ( Q ℓ ( x )) (cid:17) . AGNETIC LAPLACIAN 15
Since the squares ( Q ℓ ( x )) x ∈J ℓ are pairwise disjoint, P x ∈J ℓ k∇ B k L ( Q ℓ ( x )) = k∇ B k L (Ω ℓ ) ≤ k∇ B k L (Ω) .Using (1.25), P x ∈J ℓ ℓ = N ( ℓ ) ℓ = O (1) . Consequently, E( κ, H ) ≥ κ ℓ X x ∈J ℓ g (cid:0) bB ℓ av ( x ) (cid:1) + O ( κ / ) ( κ → + ∞ ) . Step 2:
We introduce the function ψ trial ∈ H (Ω) as follows ψ trial ( y ) = X x ∈J ℓ Q ℓ ( x ) v x,ℓ ( y ) ( y ∈ Ω) , (6.14)where, for x ∈ J ℓ , v x,ℓ ∈ H ( Q ℓ ( x )) is the function introduced in Proposition 6.2 and extendedby on Ω \ Q ℓ ( x ) . Clearly, E( κ, H ) ≤ G (cid:0) ψ trial , F (cid:1) = G (cid:0) ψ trial , F ; Ω (cid:1) . Using Proposition 6.2 andthat the squares ( Q ℓ ( x )) x ∈J ℓ are pairwise disjoint, we write G (cid:0) ψ trial , F (cid:1) = X x ∈J ℓ G (cid:0) v x,ℓ , F ; Q ℓ ( x ) (cid:1) ≤ κ ℓ X x ∈J ℓ g (cid:0) bB ℓ av ( x ) (cid:1) + C X x ∈J ℓ (cid:16) κ / ℓ + κ / ||∇ B || L ( Q ℓ ( x )) (cid:17) ≤ κ ℓ X x ∈J ℓ g ( bB ℓ av ( x )) + CN ( ℓ ) κ / ℓ + κ / k∇ B k L (Ω) = κ ℓ X x ∈J ℓ g ( bB ℓ av ( x )) + O ( κ / ) . Further remarks.
We collect here some additional properties for later use. In the sequel, ( ψ, A ) κ,H denotes a minimizing configuration of the energy in (1.15).We start by a rough estimate of A − F . By dropping the positive terms in the inequality G ( ψ, A ) ≤ G (0 , F ) = 0 we get the following estimate k curl ( A − F ) k L (˜Ω) ≤ H − k ψ k L (Ω) . Also, ( ψ, A ) being a critical point of the GL energy (see (1.18)), we know that curl ( A− F ) = 0 on ∂ ˜Ω (see [6, Eq. (10.8b)]); hence, the curl-div inequality [6, Prop. D.2.1] yields that A− F ∈ H ( ˜Ω) ;we deduce then by the Sobolev embedding of H ( ˜Ω) in L ( ˜Ω) that kA − F k L (˜Ω) ≤ C ∗ k curl ( A − F ) k L (˜Ω) ≤ C ∗ H − k ψ k L (Ω) , (6.15)where C ∗ depends on ˜Ω .We mention some additional properties that follow along the proof of Theorem 1.3 ( see e.g.[4, Thm 1.2 & p. 6636]). Firstly, we have the improved estimate for the magnetic energy k curl ( A − F ) k L (˜Ω) = O ( κ − / ) , and also for the energy of ψ , G ( ψ, A ) := Z Ω (cid:16) | ( ∇ − iκH A ) ψ | − κ | ψ | + κ | ψ | (cid:17) dx = E asy ( b, ℓ ) κ + o ( κ ) . We infer from (1.18) that G ( ψ, A ) = − κ k ψ k L (Ω) , which eventually yields the following formulafor the L -energy of the order parameter, k ψ k L (Ω) ≤ − asy ( b, ℓ ) + O ( κ − / ) . (6.16) Application: The Dirichlet Laplacian.
Assuming the hypothesis in Theorem 1.2 onthe domain Ω , we will derive an asymptotic upper bound on the eigenvalue λ ( σ, A ; Ω) , by con-structing a trial state related to the GL order parameter.Under the hypothesis in Theorem 1.2, it is sufficient to handle the case where the domain Ω consists of a single connected component. In fact, by the min-max principle, λ ( σ, A ; Ω) =min ≤ i ≤ N λ ( σ, A ; Ω i ) .In the sequel, we assume that Ω is connected and its boundary consists of a finite number ofconnected components (as in Sec. 6). Recall the divergence free magnetic potential, F , introducedin (1.13). In light of the relation (1.14), we observe that λ ( σ, A ; Ω) = λ ( σ, F ) := inf u ∈ H (Ω) \{ } k ( ∇ − iσ F ) u k L (Ω) k u k L (Ω) . (6.17)The hypothesis m ( B ; Ω) > yields that B ( x ) ≥ c > a.e. on Ω , where c = m ( B ; Ω) > isconstant. This allows us to benefit from the results and the analysis of Sec. 6.6.5.1. Link with the GL energy.
In the sequel, we set ℓ = ℓ σ := σ − / . (6.18)We fix a ∈ (0 , and introduce the parameters b = 1 − am ( B ; Ω) , κ = b − / σ / and H = bκ . (6.19)The conditions in (6.18) and (6.19) ensure that, as σ → + ∞ , the configuration ( κ, H, ℓ, b ) satisfiesthe requirements for using Theorem 1.3. In particular, E( κ, H ) = κ E asy ( b, ℓ ) + O ( κ / ) . (6.20)Furthermore, by Remark 1.4 and (1.28), there exist constants c ′ a > c a > such that − c a ≤ E asy ( b, ℓ ) ≤ − c ′ a (6.21)and c a , c ′ a = O ( a ) ( a → + ) . (6.22)Next we pick a minimizing configuration ( ψ, A ) κ,H . Collecting (6.21) and (6.16), we obtain c a ≤ k ψ k L (Ω) ≤ c ′ a . (6.23)Consequently, since | ψ | ≤ everywhere, we get Z Ω | ψ | dx ≥ Z Ω | ψ | dx ≥ c a > . (6.24)6.5.2. The trial state.
We introduce a cut-off function χ ℓ ∈ C ∞ c (Ω) in order to produce a trialstate in H (Ω) . We choose χ ℓ such that χ ℓ ( x ) = 1 for dist( x, ∂ Ω) > ℓ , and 0 ≤ χ ℓ ≤ , |∇ χ ℓ | ≤ C ℓ − in Ω . Using (1.18), we check that k ( ∇ − iκH A )( χ ℓ ψ ) k L (Ω) = Re h− ( ∇ − iκH A ) ψ, χ ℓ ψ i L (Ω) + k ψ ∇ χ ℓ k L (Ω) ≤ κ k χ ℓ ψ k L (Ω) + C ℓ − k ψ k L (Ω) . (6.25)By the simple identity A = F + ( A − F ) and Cauchy’s inequality, we write, for any δ ∈ (0 , , k ( ∇ − iκH A )( χ ℓ ψ ) k L (Ω) ≥ (1 − δ ) k ( ∇ − iκH F )( χ ℓ ψ ) k L (Ω) − δ − κ H k ( A − F ) ψ k L (Ω) . (6.26) AGNETIC LAPLACIAN 17
We estimate the term k ( A − F ) ψ k L (Ω) by using Hölder’s inequality, and the two estimates in(6.15) and (6.23); eventually, we get k ( A − F ) ψ k L (Ω) ≤ kA − F k L (Ω) k ψ k L (Ω) ≤ CH (2 c ′ a ) / Z Ω | ψ | dx . We insert this into (6.26) to get (note that (1 − δ ) − ≤ ) k ( ∇ − iκH A )( χ ℓ ψ ) k L (Ω) ≥ (1 − δ ) (cid:16) k ( ∇ − iκH F )( χ ℓ ψ ) k L (Ω) − κ δ − C (2 c ′ a ) / Z Ω | ψ | dx (cid:17) . Now we infer from (6.25), k ( ∇− iκH F )( χ ℓ ψ ) k L (Ω) ≤ − δ (cid:16) κ k χ ℓ ψ k L (Ω) + C ℓ − k ψ k L (Ω) (cid:17) +2 κ δ − C (2 c ′ a ) / Z Ω | ψ | dx . By (6.19), κH = σ . Then, in light of (6.17), we deduce that λ ( σ, F ) ≤ κ − δ + (cid:16) C − δ ℓ − + 2 κ δ − C (2 c ′ a ) / (cid:17) k ψ k L (Ω) k χ ℓ ψ k L (Ω) . (6.27)Since χ ℓ = 1 on { dist( x, ∂ Ω) > ℓ } , we get from (6.24) a constant M a > such that, k χ ℓ ψ k L (Ω) ≥ (1 − M a ℓ ) k ψ k L (Ω) ≥ k ψ k L (Ω) for ℓ close to 0 . Furthermore, by (6.19), κ = (1 − a ) − m ( B ; Ω) σ . And by (6.18), ℓ = σ − / . Therefore, wededuce from (6.27), λ ( σ, F ) ≤ m ( B ; Ω)(1 − δ )(1 − a ) σ + 2 C − δ σ / + 4 Cδ − (2 c ′ a ) / m ( B ; Ω)1 − a σ . Taking the successive limits, σ → + ∞ , a → + and δ → + , we get lim sup σ → + ∞ (cid:16) σ − λ ( σ, F ) (cid:17) ≤ m ( B ; Ω) . Note that we make use of (6.22) which ensures that c a vanishes as a approaches . Acknowledgments
The authors would like to thank B. Helffer and N. Raymond for the valuable comments on apreliminary version of this paper. A.K. is supported by the Lebanese University in the frameworkof the project “Analytical and Numerical Aspects of the Ginzburg-Landau Model”. Part of thiswork has been carried out at CAMS (
Center for Advanced Mathematical Sciences , Beirut). Theauthors acknowledge its hospitality.
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