A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities
Pedro R.S. Antunes, Rafael D. Benguria, Vladimir Lotoreichik, Thomas Ourmières-Bonafos
AA VARIATIONAL FORMULATION FOR DIRAC OPERATORS INBOUNDED DOMAINS. APPLICATIONS TO SPECTRALGEOMETRIC INEQUALITIES.
PEDRO R. S. ANTUNES, RAFAEL D. BENGURIA, VLADIMIR LOTOREICHIK,AND THOMAS OURMI`ERES-BONAFOS
Abstract.
We investigate spectral features of the Dirac operator with infinitemass boundary conditions in a smooth bounded domain of R . Motivated byspectral geometric inequalities, we prove a non-linear variational formulationto characterize its principal eigenvalue. This characterization turns out to bevery robust and allows for a simple proof of a Szeg¨o type inequality as wellas a new reformulation of a Faber-Krahn type inequality for this operator.The paper is complemented with strong numerical evidences supporting theexistence of a Faber-Krahn type inequality. Contents
1. Introduction 21.1. Motivations and state of the art 21.2. Structure of the paper 52. Preliminaries 52.1. Sobolev spaces on ∂ Ω 52.2. Periodic pseudo-differential operators 62.3. Cauchy singular integral operators 73. Maximal Wirtinger operators 74. Bergman and Hardy spaces on Ω 104.1. Potential theory of the Wirtinger derivatives 114.2. Explicit description of the Bergman and Hardy spaces 134.3. Explicit description of the domain of the maximal Wirtinger operators 155. Variational characterization of the principal eigenvalue 165.1. The quadratic form q Ω E and its associated self-adjoint operator H Ω E a r X i v : . [ m a t h . SP ] M a r Introduction
Motivations and state of the art.
In the past few years there has been agrowing interest in the study of Dirac operators among the mathematical physicscommunity; the main reason being that low-energy electrons in a single-layeredsheet of graphene are driven by an effective hamiltonian being a two-dimensionalmassless Dirac operator.Various mathematical studies have been undertaken, starting with a rigorousmathematical derivation of such hamiltonians, see e.g. [20] for the effective hamil-tonian derivation or [3, 8, 30, 37] for the justification of the so-called infinite massboundary conditions. Many properties of such operators have been investigatedas their self-adjointness in bounded domains with specified boundary conditions orcoupled with the so-called δ -interactions, see [9, 11]. Let us also mention recentworks on spectral properties and asymptotics of Dirac-type operators in specificasymptotic regimes (see [4, 23]).In this work, we are interested in finding geometrical bounds on the eigenvalues ofone of the simplest Dirac operator relevant in physics: the two-dimensional masslessDirac operator with infinite mass boundary conditions.To set the stage, let Ω ⊂ R be a C ∞ simply connected domain and let n =( n , n ) (cid:62) be the outward pointing normal field on ∂ Ω. The Dirac operator withinfinite mass boundary conditions in L (Ω , C ) is defined as D Ω := (cid:18) − ∂ z − ∂ ¯ z (cid:19) , dom( D Ω ) := { u = ( u , u ) (cid:62) ∈ H (Ω , C ) : u = i n u on ∂ Ω } , where we have set n := n + i n and with the Wirtinger operators defined as usualby ∂ z = 12 ( ∂ − i ∂ ) , ∂ ¯ z = 12 ( ∂ + i ∂ ) . The Dirac operator with infinite mass boundary conditions D Ω is known to beself-adjoint (see [11, Thm. 1.1.]), moreover its spectrum is symmetric with respectto the origin and constituted of eigenvalues of finite multiplicity satisfying · · · ≤ − E k (Ω) ≤ · · · ≤ − E (Ω) < < E (Ω) ≤ · · · ≤ E k (Ω) ≤ · · · . In the recent paper [12], the following geometrical lower bound is obtained E (Ω) ≥ (cid:115) π | Ω | , (1)where | Ω | denotes the area of the domain Ω. However, this lower bound is neverattained among Euclidean domains and by analogy with the famous Faber-Krahninequality [19, 26], a natural conjecture for the optimal lower-bound is the following. Conjecture 1.
There holds E (Ω) ≥ (cid:114) π | Ω | E ( D ) , where D is the unit disk. There is equality in the above inequality if and only if Ωis a disk. Remark 2.
As explained in [12, Remark 2] (see also [28, Appendix]), the eigen-structure of the unit disk is explicit. Indeed, E ( D ) (cid:39) . . . . is the first non-negative root of the equation J ( E ) = J ( E ) where J and J are the Bessel func-tions of the first kind of order 0 and of order 1, respectively. Moreover, an associated eigenfunction is given for ( x , x ) ∈ D by (cid:18) J ( | x | )i x +i x | x | J ( | x | ) (cid:19) . Conjecture 1 motivated part of this paper and is still an open question. However,in Section 8 we provide strong numerical evidences supporting it and in Section 7we show how Conjecture 1 is intimately connected to the famous Bossel-Danersinequality for the Robin Laplacian (see [14, 16]).The quest for a geometrical upper-bound has also attracted attention recentlyas for instance in [28]. In this work, the given geometrical upper-bound is sharp inthe sense that it is an equality if and only if the considered domain is a disk. Never-theless, this upper-bound depends in a complicated fashion of different geometricalparameters and may be hard to compute in practice.Let us also mention that similar questions are dealt with in the differential ge-ometry literature for lower bounds and upper bounds for Dirac operators on spin-manifolds (see for instance [1, 6, 7, 33]).One of the main result of this paper is the following theorem which gives ageometrical upper-bound in term of simple geometric quantities: | Ω | the area of Ω, | ∂ Ω | the perimeter of Ω as well as r i the inradius of Ω. Theorem 3.
Let Ω ⊂ R be a C ∞ simply connected domain. There holds E (Ω) ≤ | ∂ Ω | ( πr i + | Ω | ) E ( D ) , with equality if and only if Ω is a disk. The proof is by combining a new variational characterization of E (Ω), inspiredby min-max techniques for operators with gaps introduced in [17] and the classicalproof of Szeg¨o about the eigenvalues of membranes of fixed area [38].It turns out this new variational characterization is of interest by itself becauseit also allows for numerical simulations and we believe that it could be an adequatestarting point to prove Conjecture 1 as discussed further on in Section 7. Tointroduce it, consider the quadratic form q Ω E, ( u ) := 4 (cid:90) Ω | ∂ ¯ z u | dx − E (cid:90) Ω | u | dx + E (cid:90) ∂ Ω | u | ds, dom( q Ω E, ) := C ∞ (Ω , C ) . (2)For E > q E, is bounded below with dense domain and we consider q Ω E the closurein L (Ω) of q Ω E, . Then, we define the first min-max level µ Ω ( E ) := inf u ∈ dom( q Ω E ) \{ } (cid:82) Ω | ∂ ¯ z u | dx − E (cid:82) Ω | u | dx + E (cid:82) ∂ Ω | u | ds (cid:82) Ω | u | dx . (3)The second main result of this paper is the following non-linear variational charac-terization of E (Ω). Theorem 4.
E > is the first non-negative eigenvalue of D Ω if and only if µ Ω ( E ) = 0 . The advantage of the quadratic form q Ω E is two-fold. First, functions in theconsidered variational space are now scalar valued and, second, the infinite massboundary conditions does not appear in the variational formulation. However,the first drawback is that dom ( q Ω E ) contains the Hardy space H (Ω), constitutedof holomorphic functions with traces in L ( ∂ Ω). In particular, dom ( q Ω E ) is nota usual Sobolev space and a special care is needed in order to prove Theorem 4.In particular, it asks for a precise description of the domain dom ( q Ω E ) as well as the domain of the associated self-adjoint operator via Kato’s first representationtheorem (see [25, Chap. VI, Thm. 2.1]). It is done using convolution operatorsreminiscent of what is done in [5, 31], elliptic regularity properties of the maximalWirtinger operators as well as using Cauchy singular integral operators on ∂ Ω,seen as periodic pseudo-differential operators.Theorem 4 is reminiscent of [17, 18], where a similar strategy is used to dealwith the Dirac-Coulomb operator. To our knowledge, this is the first time this ideais extended to boundary value problems and now, we describe its heuristic.Let ( u, v ) (cid:62) ∈ dom ( D Ω ) be an eigenfunction associated with the eigenvalue E >
0. In Ω, the eigenvalue equation reads − ∂ z v = Eu, − ∂ ¯ zu = Ev. (4)If we assume that this identity is true up to the boundary ∂ Ω, we obtain thefollowing boundary condition for u : n∂ ¯ z u + E u = 0 on ∂ Ω . (5)Now, Equation (4) gives − ∂ z ∂ ¯ z u = E u in Ω . (6)Hence, a weak formulation is obtained taking the scalar product by u , integratingby parts and taking into account the boundary condition (5). This formally gives q Ω E ( u ) = 0 and this is the reason for introducing the quadratic form q Ω E in (2).Let us add two remarks. The first one explains that (5)-(6) can be recast into anon-linear eigenvalue problem for a Laplace operator with oblique boundary con-ditions. The second remark, explains how Theorem 4 could be extended to handlethe next eigenvalues. Remark 5.
Note that (6) is an eigenvalue equation for the Laplace operator andreads − ∆ u = E u . The boundary condition (5) is a relation between the normalderivative, the tangential derivative and the value of the function on ∂ Ω. If we let t be the tangent field on ∂ Ω such that ( n , t ) is a direct frame, the problem can bere-interpreted as an oblique problem (cid:26) − ∆ u = E u in Ω ,∂ n u + i ∂ t u + Eu = 0 on ∂ Ω , (7)where ∂ n and ∂ t are the normal and tangential derivatives, respectively.Note that Problem (7) is non-linear because the parameter E >
Remark 6.
For j ≥
1, one can consider the j -th min-max level of q Ω E defined as µ Ω j ( E ) := inf F ⊂ dom ( q Ω E )dim F = j sup u ∈ F \{ } (cid:82) Ω | ∂ ¯ z u | dx − E (cid:82) Ω | u | dx + E (cid:82) ∂ Ω | u | ds (cid:82) Ω | u | dx . As in [17], Theorem 4 could be extended as follows:
E > j -th non-negativeeigenvalue of D Ω if and only if µ Ω j ( E ) = 0. We do not discuss it here because weare concerned only with the principal eigenvalue E (Ω).Finally, let us comment the hypothesis on Ω. First, one would like to lower thesmoothness hypothesis to be able to handle, for instance, Lipschitz domains. Thisis a natural question but there is no reason for the Dirac operator with infinite massboundary to be self-adjoint on such a domain dom ( D Ω ) (see the case of polygo-nal domains in [27]). Moreover, as part of the proof relies on pseudo-differentialtechniques, we prefer to keep the C ∞ smoothness assumption on ∂ Ω because itallows for a more efficient treatment of singular integral operators on the boundary.
Second, the simply connectedness assumption may be an unnecessary hypothesisfor Theorem 4 to hold. Nevertheless, we are not able to drop it in Theorem 3because the proof relies on the Riemann mapping theorem to build an admissibletest function for q Ω E .1.2. Structure of the paper.
In Section 2, we gather several results on Sobolevspaces on ∂ Ω, periodic pseudo-differential operators on ∂ Ω and deduce variousmapping properties of the Cauchy singular integral operators.Section 3 contains a description of the domain of the maximal Wirtinger op-erators. In particular, we discuss the existence of a trace operator for functionsbelonging to these domains and state a fundamental elliptic regularity result.Section 4 deals with the description of the Bergman and Hardy spaces on Ωthanks to integral operators. This is done by introducing the Szeg¨o projectors onthe Sobolev spaces on the boundary H s ( ∂ Ω) ( s ∈ {− , , } ). As a byproduct ofthis analysis we are able to describe explicitly the domains of the maximal Wirtingeroperators.Theorem 4 is proved in Section 5. We start by describing the domain of thequadratic form q Ω E in terms of the first-order Sobolev space H (Ω) and the Hardyspace on Ω. Then, the analysis is pushed forward to study the domain of the self-adjoint operator associated with q Ω E via Kato’s first representation theorem (see[25, Chap. VI, Thm. 2.1]). Combining these tools, we prove Theorem 4.Then, we apply Theorem 4 in Section 6 to prove Theorem 3. The proof is byadapting the well-known proof of Szeg¨o [38] to our setting, constructing an adequatetest function for the new variational formulation.In Section 7, we show that Conjecture 1 can be reformulated and that it is relatedto the famous Bossel-Daners inequality.We conclude in Section 8 illustrating by numerical experiments the validity ofConjecture 1 and several theoretical results discussed all along the paper.2.
Preliminaries
Sobolev spaces on ∂ Ω . In the following, T is the torus T := R / Z , D ( T ) = C ∞ ( T ) is the space periodic smooth functions on the torus T and D ( T ) (cid:48) the spaceof periodic distributions on the torus T . Let f ∈ D ( T ) (cid:48) we define its Fouriercoefficients using the duality pairing by (cid:98) f ( n ) := (cid:104) f, e − n (cid:105) D ( T ) (cid:48) , D ( T ) , e n := t ∈ T (cid:55)→ e πnt . For s ∈ R , the Sobolev space of order s on T is defined as H s ( T ) := { f ∈ D ( T ) (cid:48) : + ∞ (cid:88) n = −∞ (1 + | n | ) s | (cid:98) f ( n ) | < + ∞} . Set (cid:96) := | ∂ Ω | and let γ : R (cid:14) [0 , (cid:96) ] → ∂ Ω be a smooth arc-length parametrization of ∂ Ω. Consider the map U ∗ : D ( T ) → D ( ∂ Ω) , ( U ∗ g )( x ) := (cid:96) − g ( (cid:96) − γ − ( x )) , x ∈ ∂ Ω , where we have set D ( ∂ Ω) := C ∞ ( ∂ Ω). We define the map U : D ( ∂ Ω) (cid:48) → D ( T ) (cid:48) as (cid:104) U f, g (cid:105) D ( T ) (cid:48) , D ( T ) := (cid:104) f, U ∗ g (cid:105) D ( ∂ Ω) (cid:48) , D ( ∂ Ω) . (8)The Sobolev space of order s ∈ R on ∂ Ω is defined as H s ( ∂ Ω) := { f ∈ D ( ∂ Ω) (cid:48) : U f ∈ H s ( T ) } . Periodic pseudo-differential operators.
Let us start by defining periodicpseudo-differential operators on T . Definition 7.
A linear operator H on C ∞ ( T ) is a periodic pseudo-differentialoperator on T if there exists h : T × Z → C such that:(1) for all n ∈ Z , h ( · , n ) ∈ C ∞ ( T ),(2) H acts as Hf = (cid:80) n ∈ Z h ( · , n ) (cid:98) f ( n ) e n ,(3) there exists α ∈ R such that for all p, q ∈ N there exists c p,q > (cid:12)(cid:12)(cid:12)(cid:0) d p dt p ( ω q h ) (cid:1) ( t, n ) (cid:12)(cid:12)(cid:12) ≤ c p,q (1 + | n | ) α − q , where the operator ω is defined for all ( t, n ) ∈ T × Z by ( ωh )( t, n ) := h ( t, n + 1) − h ( t, n ). α is called the order of the pseudo-differential operator H . The set of pseudo-differential operators of order α on T is denoted Ψ α and we defineΨ −∞ := (cid:92) α ∈ R Ψ α . Example 8.
For further use, we introduce the example of multiplication operators.Consider H : C ∞ ( T ) → C ∞ ( T ) defined as( Hf )( t ) := h ( t ) f ( t ) , h ∈ C ∞ ( T ) . Decomposing in Fourier series, one immediately obtains( Hf ) = (cid:88) n ∈ Z h (cid:98) f ( n ) e n . There holds ω q h = 0 for all q ≥ h ∈ C ∞ ( T ), for all t ∈ T we obtain (cid:12)(cid:12)(cid:12)(cid:0) d p hdt p (cid:1) ( t ) (cid:12)(cid:12)(cid:12) ≤ c p , for some c p > H ∈ Ψ .Using the map U defined in (8), we define periodic pseudo-differential operatorson ∂ Ω as follows.
Definition 9.
A linear operator H on C ∞ ( ∂ Ω) is a periodic pseudo-differentialoperator on ∂ Ω of order α ∈ R if the operator H := U HU − ∈ Ψ α . The set ofpseudo differential operators on ∂ Ω of order α is denoted Ψ α∂ Ω and we setΨ −∞ ∂ Ω := (cid:92) α ∈ R Ψ α∂ Ω . We will need the following properties of pseudo-differential operators on ∂ Ω.They can be found in [34, § Proposition 10.
Let s, α, β ∈ R and H ∈ Ψ α∂ Ω , G ∈ Ψ β∂ Ω .(1) H extends uniquely to a bounded linear operator, also denoted H , from H s ( ∂ Ω) to H s − α ( ∂ Ω) .(2) There holds H + G ∈ Ψ max( α,β ) ∂ Ω , HG ∈ Ψ α + β∂ Ω , [ H, G ] ∈ Ψ α + β − ∂ Ω . Cauchy singular integral operators.
For f ∈ C ∞ ( ∂ Ω), the Cauchy singu-lar integral operator is defined as a principal value by S h ( f )( z ) := 1 iπ p . v . (cid:90) ∂ Ω f ( ξ ) ξ − z dξ, z ∈ ∂ Ω . We define its anti-holomorphic counterpart as S ah ( f )( z ) := S h ( f )( z ) = − iπ p . v . (cid:90) ∂ Ω f ( ξ ) ξ − ¯ z dξ, z ∈ ∂ Ω . It turns out S h and S ah are periodic pseudo-differential operators on ∂ Ω. This isthe purpose of the following proposition.
Proposition 11.
The linear maps S h and S ah are periodic pseudo-differential op-erators of order . In particular, they are bounded linear operators from H s ( ∂ Ω) onto itself for all s ∈ R . Proof . This is proved in [13, Prop 2.9.] where the operators S h and S ah are denoted C Σ and − C (cid:48) Σ respectively (with Σ := ∂ Ω). (cid:3)
We will also need the following property.
Proposition 12.
Let H n be the multiplication operator by the normal n in C ∞ ( ∂ Ω) . There holds:(1) H n is a periodic pseudo-differential operator of order .(2) Let (cid:93) ∈ { h , ah } we have [ H n , S (cid:93) ] ∈ Ψ − ∂ Ω .(3) There holds S ah + S h ∈ Ψ −∞ ∂ Ω . Proof . Point (1) is proved remarking that the operator
U H n U − is a multiplicationoperator in T . Thanks to Example 8, we know that U H n U − ∈ Ψ hence bydefinition we get H n ∈ Ψ ∂ Ω .Let us deal with Point (2). Let (cid:93) ∈ { h , ah } , by Proposition 11, S (cid:93) ∈ Ψ ∂ Ω and byPoint (1) H n ∈ Ψ ∂ Ω . Hence, by (2) Proposition 10, we obtain Point (2).Finally, we prove Point (3). By [13, Proposition 2.9.] there exists L ∈ Ψ ∂ Ω and R , R ∈ Ψ −∞ ∂ Ω such that S h = L + R , S ah = − L + R . Hence, S h + S ah = R + R ∈ Ψ −∞ ∂ Ω by (2) Proposition 10. (cid:3) Maximal Wirtinger operators
In this section we describe elemental properties of the maximal Wirtinger oper-ators defined as ∂ h u = ∂ ¯ z u, dom( ∂ h ) := { u ∈ L (Ω) : ∂ ¯ z u ∈ L (Ω) } ,∂ ah u = ∂ z u, dom( ∂ ah ) := { u ∈ L (Ω) : ∂ z u ∈ L (Ω) } . For (cid:93) ∈ { h , ah } , consider the operator norms (cid:107) · (cid:107) (cid:93) defined as (cid:107) u (cid:107) (cid:93) := (cid:107) ∂ (cid:93) u (cid:107) L (Ω) + (cid:107) u (cid:107) L (Ω) , u ∈ dom( ∂ (cid:93) ) . In particular, dom( ∂ (cid:93) ) endowed with the scalar product defined for u, v ∈ dom( ∂ (cid:93) )by (cid:104) u, v (cid:105) (cid:93) = (cid:104) ∂ (cid:93) u, ∂ (cid:93) v (cid:105) L (Ω) + (cid:104) u, v (cid:105) L (Ω) is a Hilbert space.The first lemma is obtained by a simple integration by parts. Lemma 13.
The following identities hold. H ( R ) = { f ∈ L ( R ) : ∂ z f ∈ L ( R ) } = { f ∈ L ( R ) : ∂ ¯ z f ∈ L ( R ) } Proof . Let f ∈ C ∞ ( R ). Integrating by parts several times we obtain: (cid:107)∇ f (cid:107) L ( R ) = (cid:104) f, − ∆ f (cid:105) L ( R ) = 4 (cid:104) f, − ∂ z ∂ ¯ z f (cid:105) L ( R ) = 4 (cid:107) ∂ ¯ z f (cid:107) L ( R ) = 4 (cid:104) f, − ∂ ¯ z ∂ z f (cid:105) L ( R ) = 4 (cid:107) ∂ z f (cid:107) L ( R ) As C ∞ ( R ) is dense in H ( R ), we obtain the expected result. (cid:3) The next lemma is a density result.
Lemma 14.
Let (cid:93) ∈ { h , ah } . The space C ∞ (Ω) := C ∞ (Ω , C ) is dense in dom( ∂ (cid:93) ) . Proof . Let u ∈ dom( ∂ h ) and assume that for all ϕ ∈ C ∞ (Ω) there holds0 = (cid:104) u, ϕ (cid:105) h = (cid:104) ∂ ¯ z u, ∂ ¯ z ϕ (cid:105) L (Ω) + (cid:104) u, ϕ (cid:105) L (Ω) . In particular, if ϕ ∈ C ∞ (Ω), we obtain − ∆ u = − u first in D (Ω) (cid:48) then in L (Ω).Define v = ∂ ¯ z u and denote by v its extension to the whole R by 0. For ϕ ∈ C ∞ ( R ) there holds (cid:104) ∂ z v , ϕ (cid:105) D (cid:48) ( R ) , D ( R ) = −(cid:104) v , ∂ ¯ z ϕ (cid:105) D (cid:48) ( R ) , D ( R ) = −(cid:104) v, ∂ ¯ z ϕ (cid:105) L (Ω) = −(cid:104) ∂ ¯ z u, ∂ ¯ z ϕ (cid:105) L (Ω) = (cid:104) u, ϕ (cid:105) L (Ω) = (cid:104) u , ϕ (cid:105) L ( R ) = (cid:104) u , ϕ (cid:105) D (cid:48) ( R ) , D ( R ) , where u denotes the extension by zero of u to the whole R . It gives ∂ z v = u ∈ L ( R ). By Lemma 13, v is in H ( R ) and by [15, Prop. IX.18.] we get v ∈ H (Ω). Remark that in D (cid:48) (Ω), there holds ∂ ¯ z ∂ z v = v . Indeed, we have ∂ ¯ z ∂ z v = ∂ ¯ z ∂ z ∂ ¯ z u = ∂ ¯ z u = v. In particular this identity also holds true in L (Ω). Now, pick a sequence v n ∈ C ∞ (Ω) converging to v in the H (Ω)-norm. There holds (cid:104) v, v n (cid:105) L (Ω) = (cid:104) ∂ z ∂ ¯ z v, v n (cid:105) L (Ω) = −(cid:104) ∂ ¯ z v, ∂ ¯ z v n (cid:105) D (cid:48) (Ω) , D (Ω) = −(cid:104) ∂ ¯ z v, ∂ ¯ z v n (cid:105) L (Ω) Letting n → + ∞ one obtains (cid:107) v (cid:107) L (Ω) = −(cid:107) ∂ ¯ z v (cid:107) L (Ω) which implies v = 0. In D (cid:48) (Ω) we have ∂ z v = ∂ z ∂ ¯ z u = u . As v = 0, u = 0 which concludes the proof for (cid:93) = h. The case (cid:93) = ah is handled similarly. (cid:3) In order to describe precisely the domains dom( ∂ (cid:93) ) ( (cid:93) ∈ { h , ah } ) we need toprove the existence of traces on ∂ Ω for functions in dom( ∂ (cid:93) ). To this aim, definethe following Dirichlet trace operatorsΓ + : H (Ω) → H ( ∂ Ω) , Γ − : H loc ( R \ Ω) → H ( ∂ Ω) . (9)These linear operators are known to be bounded (see [29, Thm. 3.37]) and thereexists continuous extension operators such that for f ∈ H ( ∂ Ω) there holds E + f ∈ H (Ω) , E − f ∈ H ( R \ Ω) and Γ ± E ± f = f. Actually, the operator Γ + can be extended to functions in dom( ∂ (cid:93) ) ( (cid:93) ∈ { h , ah } ).This is the purpose of the following proposition. Lemma 15.
Let (cid:93) ∈ { h , ah } . The operator Γ + defined in (9) extends into a linearbounded operator between dom( ∂ (cid:93) ) and H − ( ∂ Ω) . Proof . Let ( v n ) n ∈ N ∈ C ∞ (Ω) N be a sequence that converges to v in the (cid:107) · (cid:107) h -normwhen n → + ∞ . Let us prove that (Γ + v n ) n ∈ N has a limit in H − ( ∂ Ω). First recallthe integration by part formula12 (cid:104) Γ + u, n Γ + w (cid:105) L ( ∂ Ω) = (cid:104) ∂ ¯ z u, w (cid:105) L (Ω) + (cid:104) u, ∂ z w (cid:105) L (Ω) valid for any u, w ∈ H (Ω). Second, pick f ∈ H ( ∂ Ω) and consider w = E + ( n f ) ∈ H (Ω). There holds (cid:104) Γ + ( v n − v m ) , f (cid:105) L ( ∂ Ω) = 2 (cid:104) ∂ ¯ z ( v n − v m ) , w (cid:105) L (Ω) + 2 (cid:104) v n − v m , ∂ z w (cid:105) L (Ω) . In particular, we have (cid:12)(cid:12) (cid:104) Γ + ( v n − v m ) , f (cid:105) L ( ∂ Ω) (cid:12)(cid:12) ≤ (cid:107) ∂ ¯ z ( v n − v m ) (cid:107) L (Ω) (cid:107) w (cid:107) L (Ω) + 2 (cid:107) v n − v m (cid:107) L (Ω) (cid:107) ∂ z w (cid:107) L (Ω) ≤ (cid:107) w (cid:107) H (Ω) (cid:107) v n − v m (cid:107) h ≤ c Ω (cid:107) f (cid:107) H ( ∂ Ω) (cid:107) v n − v m (cid:107) h (for some c Ω > , where we have used that E + is a continuous linear map and that the multiplicationoperator by n is bounded from H ( ∂ Ω) onto itself. When n, m → + ∞ we obtain (cid:107) Γ + ( v n − v m ) (cid:107) H − ( ∂ Ω) →
0. In particular (Γ + v n ) n ∈ N is a Cauchy sequence in H − ( ∂ Ω) thus converges to an element g ∈ H − ( ∂ Ω) and we define Γ + v := g .Remark that the definition of Γ + v does not depend on the chosen sequence ( v n ) n ∈ N and that we have (cid:107) Γ + v n (cid:107) H − ( ∂ Ω) ≤ c Ω (cid:107) v n (cid:107) h which implies, when n → + ∞ , that Γ + is bounded from dom( ∂ h ) to H − ( ∂ Ω).The proof for dom( ∂ ah ) is handled similarly. (cid:3) Remark 16.
If one picks
R > ⊂ B (0 , R ) := { x ∈ R : (cid:107) x (cid:107) < R } , onecan prove that for (cid:63) ∈ { z, z } , Γ − extends into a linear bounded operator betweenthe space { u ∈ L ( B (0 , R ) \ Ω) : ∂ (cid:63) u ∈ L ( B (0 , R ) \ Ω) } and H − ( ∂ Ω). The proofgoes along the same lines as the one of Lemma 15, using an extension operator E − : H ( ∂ Ω) → H ( B (0 , R ) \ Ω) constructed such that for all f ∈ H ( ∂ Ω), E − ( f ) | ∂B (0 ,R ) = 0. Remark 17.
Pick u ∈ dom( ∂ ah ) and w ∈ H (Ω). Note that by definition, thefollowing Green’s Formula holds (cid:104) ∂ z u, w (cid:105) L (Ω) = −(cid:104) u, ∂ ¯ z w (cid:105) L (Ω) + 12 (cid:104) n Γ + u, Γ + w (cid:105) H − ( ∂ Ω) ,H ( ∂ Ω) . (10)The following elliptic regularity result is rather well known (see the analogousstatement [11, Lemma 2.4.]). Lemma 18.
Let (cid:93) ∈ { h , ah } and u ∈ dom( ∂ (cid:93) ) . If Γ + u ∈ H ( ∂ Ω) then u ∈ H (Ω) . Proof . Let u ∈ dom( ∂ h ) be such that Γ + u ∈ H ( ∂ Ω) and set v = u − E + (Γ + u ).Then, Γ + v = 0 and if v ∈ H (Ω) the result is proved. If v n ∈ C ∞ (Ω) is a sequenceconverging to v in the (cid:107) · (cid:107) h -norm there holds Γ + v n → H − ( ∂ Ω) by Lemma15. In particular, it gives for any w ∈ H (Ω) (cid:104) v, ∂ z w (cid:105) L (Ω) = lim n → + ∞ (cid:16) − (cid:104) ∂ ¯ z v n , w (cid:105) L (Ω) + 12 (cid:104) Γ + v n , n Γ + w (cid:105) L ( ∂ Ω) (cid:17) = −(cid:104) ∂ ¯ z v, w (cid:105) L (Ω) . Let v (resp. h ) be the extension of v (resp. h := ∂ ¯ z v ) by zero to the whole R . If ϕ ∈ C ∞ ( R ), there holds −(cid:104) h , ϕ (cid:105) D (cid:48) ( R ) , D ( R ) = −(cid:104) h, ϕ (cid:105) L (Ω) = (cid:104) v, ∂ z ϕ (cid:105) L (Ω) = (cid:104) v , ∂ ¯ z ϕ (cid:105) D (cid:48) ( R ) , D ( R ) = −(cid:104) ∂ ¯ z v , ϕ (cid:105) D (cid:48) ( R ) , D ( R ) . Thus ∂ ¯ z v = h ∈ L ( R ) and by Lemma 13, v ∈ H ( R ) and v ∈ H (Ω). Theproof for u ∈ dom( ∂ ah ) is handled similarly. (cid:3) Bergman and Hardy spaces on
ΩWe introduce A (Ω) and A (Ω) the holomorphic and anti-holomorphicBergman spaces on Ω, respectively. They are defined as A (Ω) := { u ∈ Hol (Ω) ∩ L (Ω) } , A (Ω) := { u : u ∈ A (Ω) } , where Hol (Ω) denotes the space of holomorphic functions in Ω. The holomorphicand anti-holomorphic Hardy spaces, denoted H (Ω) and H (Ω), respectively, aredefined as H (Ω) := { u ∈ A (Ω) : Γ + u ∈ L ( ∂ Ω) } , H (Ω) := { u : u ∈ H (Ω) } . (11)This section aims to describe explicitely the Bergman and Hardy spaces on Ω interms of Cauchy integrals and Szeg¨o projectors that we define now.For f ∈ C ∞ ( ∂ Ω) consider the Cauchy integrals defined for z ∈ C \ ∂ Ω byΦ h ( f )( z ) := 12i π (cid:90) ∂ Ω f ( ξ ) ξ − z dξ, Φ ah ( f )( z ) := − π (cid:90) ∂ Ω f ( ξ ) ξ − z dξ. It is well-known (see [34, § h ( f ) (resp. Φ ah ( f )) defines a holomorphicfunction (resp. anti-holomorphic function) in R \ ∂ Ω.The well-known Plemelj-Sokhotski formula (see [34, Thm. 4.1.1]) state that for f ∈ C ∞ ( ∂ Ω) the functions Φ h ( f ) and Φ ah ( f ) have an interior and an exteriorDirichlet trace, denoted respectively γ +0 and γ − , such that: γ ± Φ h ( f ) = ± f + 12 S h f, γ ± Φ ah ( f ) = ± f + 12 S ah f. (12)Let (cid:93) ∈ { h , ah } , note that by [10, Theorem 3.1.], for f ∈ C ∞ ( ∂ Ω) we know thatΦ (cid:93) ( f ) | Ω ∈ C ∞ (Ω) as well as Φ (cid:93) ( f ) | R \ Ω ∈ C ∞ ( R \ Ω). In particular, the traces γ ± Φ (cid:93) ( f ) coincide with Γ ± Φ (cid:93) ( f ), where Γ ± are the trace operators defined in Lemma15 and Remark 16. Definition 19.
We define the Szeg¨o projectors in C ∞ ( ∂ Ω) byΠ ± h := ± Γ ± Φ h , Π ± ah := ± Γ ± Φ ah . (13) Proposition 20.
Let s ∈ R and (cid:93) ∈ { h , ah } . The Szeg¨o projectors Π ± (cid:93) extenduniquely into bounded linear operators from H s ( ∂ Ω) onto itself. Moreover, Π ± (cid:93) areprojectors and Π + (cid:93) + Π − (cid:93) = 1 . Proof . Remark that for (cid:93) ∈ { h , ah } and f ∈ C ∞ ( ∂ Ω), there holdsΠ ± (cid:93) f = 12 f ± S (cid:93) f. By Proposition 11, Π ± (cid:93) extends into a bounded linear operator from H s ( ∂ Ω) ontoitself for all s ∈ R . Let s ∈ R and f ∈ H s ( ∂ Ω). A fundamental fact is that S f = f (see [34, Eqn.(4.10)]), in particular it implies that S f = f . Hence, we obtain(Π ± (cid:93) ) = ( 12 ± S (cid:93) )( 12 ± S (cid:93) )= 14 + 14 S (cid:93) ± S (cid:93) = 12 ± S (cid:93) = Π ± (cid:93) Hence Π ± (cid:93) are projectors and one easily checks that Π + (cid:93) + Π − (cid:93) = 1. (cid:3) The main goal of this section is to prove the following description of the Bergmanand Hardy spaces. As we will see further on in Proposition 22, this descriptionrelies on an extension of the operators Φ (cid:93) to Sobolev spaces on the boundary ∂ Ω( (cid:93) ∈ { h , ah } ). Theorem 21.
Let (cid:93) ∈ { h , ah } . The Bergman spaces satisfy A (cid:93) (Ω) = { Φ (cid:93) ( f ) : f ∈ H − ( ∂ Ω) , Π − (cid:93) f = 0 } . The Hardy spaces verify H (cid:93) (Ω) = { Φ (cid:93) ( f ) : f ∈ L ( ∂ Ω) , Π − (cid:93) f = 0 } . Potential theory of the Wirtinger derivatives.
In this paragraph weprove the following proposition.
Proposition 22.
Let (cid:93) ∈ { h , ah } and s ∈ {− , , } . The operator Φ (cid:93) extendsuniquely into a bounded operator from H s ( ∂ Ω) to H s + (Ω) also denoted Φ (cid:93) . In order to prove Proposition 22, we will need a few lemma. Let us start bydefining fundamental solutions of the Wirtinger operators ∂ h and ∂ ah : ϕ h ( x ) = 1 π ( x + ix ) , ϕ ah ( x ) = 1 π ( x − ix ) . Lemma 23.
Let (cid:93) ∈ { h , ah } . The linear map N (cid:93) : u ∈ L (Ω) (cid:55)→ ϕ (cid:93) ∗ u is bounded from L (Ω) to H loc ( R ) . Here u denotes the extension of u by zero tothe whole R . Proof . Let us prove it for (cid:93) = h the proof for (cid:93) = ah being similar. In the space ofdistributions D (cid:48) ( R ), there holds ∂ ¯ z ϕ h = δ , (14)where δ is the delta-Dirac distribution.Now, for u in the Schwartz space S ( R ) recall that the Fourier transform of u isdefined as (cid:98) u ( k ) := (cid:90) R f ( x ) e − π (cid:104) x,k (cid:105) R dx, for all k ∈ R and (cid:98) u ∈ S ( R ). The Fourier transform extends to the space of tempered distribution S (cid:48) ( R ) and as δ ∈ S (cid:48) ( R ), the Fourier transform of (14) yields (cid:99) ϕ h ( k ) = 1 π i( k + i k ) , k = ( k , k ) ∈ R \ { (0 , } . Let K be a compact subset of R and take u ∈ L (Ω). We extend u by zero to R and denote this extension u ∈ L ( R ). (cid:107) ϕ h ∗ u (cid:107) H ( K ) ≤ (cid:107) ϕ h ∗ u (cid:107) L ( K ) + (cid:90) R | k | | ( (cid:92) ϕ h ∗ u )( k ) | dk = (cid:107) ϕ h ∗ u (cid:107) L ( K ) + (cid:90) R | k | | (cid:99) ϕ h ( k ) (cid:99) u ( k ) | dk = (cid:107) ϕ h ∗ u (cid:107) L ( K ) + 1 π (cid:90) R | (cid:99) u ( k ) | dk = (cid:107) ϕ h ∗ u (cid:107) L ( K ) + 1 π (cid:107) u (cid:107) L (Ω) . Now, let
R > K ⊂ { x ∈ R : | x | < R } and Ω ⊂ { x ∈ R : | x | < R } .Consider a cut-off function χ ∈ C ∞ ([0 , + ∞ )) such that χ ( ρ ) = 1 whenever 0 ≤ ρ < R, χ ( ρ ) = 0 whenever ρ > R. Define the function u χ as u χ ( x ) := (cid:90) R χ ( | x − y | ) ϕ h ( x − y ) u ( y ) dy. As defined, u χ | K ≡ ( ϕ h ∗ u ) | K . Hence, we get (cid:107) ϕ h ∗ u (cid:107) L ( K ) = (cid:107) u χ (cid:107) L ( K ) ≤ (cid:107) u χ (cid:107) L ( R ) ≤ (cid:107) χ ( | · | ) ϕ h (cid:107) L ( R ) (cid:107) u (cid:107) L (Ω) , where we have used Young’s inequality because χ ( | · | ) ϕ h ∈ L ( R ). Indeed, thereholds (cid:107) χ ( | · | ) ϕ h (cid:107) L ( R ) ≤ π (cid:90) B (0 , R ) | x | dx = 6 R. In particular, there exists c K >
0, such that (cid:107) ϕ h ∗ u (cid:107) H ( K ) ≤ c K (cid:107) u (cid:107) L (Ω) . Hence, for any compact K ⊂ R , N (cid:93) is a bounded linear operator from L (Ω) to H ( K ) and the proposition is proved. (cid:3) Next, we recall that the Dirichlet trace on ∂ Ω of a function in H loc ( R ) can bedefined as Γ : H loc ( R ) → H ( ∂ Ω)and is a bounded linear operator from H loc ( R ) to H ( ∂ Ω) (see [29, Thm. 3.37]).Moreover, for s ∈ [0 , (cid:96) ], we introduce t ( s ) := γ (cid:48) ( s ) + i γ (cid:48) ( s ) the expression of thetangent vector in the complex plane at the point γ ( s ) + i γ ( s ). Lemma 24.
The dual adjoints of ( t Γ N h ) and ( t Γ N ah ) , denoted ( t Γ N h ) (cid:48) and ( t Γ N ah ) (cid:48) respectively, are bounded linear maps from H − ( ∂ Ω) to L (Ω) . More-over if f ∈ C ∞ ( ∂ Ω) , in L (Ω) there holds: Φ ah ( f ) = i2 ( t Γ N h ) (cid:48) ( f ) , Φ h ( f ) = − i2 ( t Γ N ah ) (cid:48) ( f ) . Proof . Thanks to Lemma 23 and the mapping properties of Γ we know that Γ N (cid:93) is a bounded linear map from L (Ω) to H ( ∂ Ω) (for (cid:93) ∈ { h , ah } ). As Ω is smooth, t ∈ C ∞ ( ∂ Ω) and t ∈ C ∞ ( ∂ Ω). In particular the multiplication operators by t and t are bounded and invertible in H (Ω). Hence, their dual adjoints satisfy theexpected mapping property. Now, pick f ∈ C ∞ ( ∂ Ω) and v ∈ L (Ω). Denoting by v the extension of v byzero to the whole R and using Fubini’s theorem, there holds (cid:104) ( t Γ N h ) (cid:48) f, v (cid:105) L (Ω) = (cid:104) f, t Γ N h v (cid:105) H − ( ∂ Ω) ,H ( ∂ Ω) = (cid:104) f, t Γ N h v (cid:105) L ( ∂ Ω) = (cid:90) x ∈ ∂ Ω (cid:90) y ∈ R f ( x ) v ( y ) t ( x ) π (cid:0) ( x − i x ) − ( y − i y ) (cid:1) dyds ( x )= (cid:90) y ∈ R π (cid:32)(cid:90) (cid:96)s =0 f ( γ ( s ))( γ (cid:48) ( s ) − i γ (cid:48) ( s ))( γ ( s ) − i γ ( s )) − ( y − i y ) ds (cid:33) v ( y ) dy = (cid:90) y ∈ R (cid:16) π (cid:90) ξ ∈ ∂ Ω f ( ξ ) ξ − ( y − i y ) dξ (cid:17) v ( y ) dy = (cid:104)− ah ( f ) , v (cid:105) L (Ω) . The proof for ( t Γ N ah ) (cid:48) goes along the same lines, which concludes the proof of thislemma. (cid:3) For further use, we still denote Φ ah and Φ h the operators i2 ( t Γ N h ) (cid:48) and − i2 ( t Γ N ah ) (cid:48) . Now, for (cid:93) ∈ { h , ah } , when considering the operatorsΦ (cid:93) : (cid:0) C ∞ ( ∂ Ω) , (cid:107) · (cid:107) H − ( ∂ Ω) (cid:1) → (cid:0) dom( ∂ (cid:93) ) , (cid:107) · (cid:107) (cid:93) (cid:1) they are bounded operators because for any f ∈ C ∞ ( ∂ Ω), Φ h ( f ) and Φ ah ( f ) areholomorphic and anti-holomorphic in Ω, respectively. The density of C ∞ ( ∂ Ω) in H − ( ∂ Ω) yields for each operator a unique extension to H − ( ∂ Ω) which coincidewith the previous one. In particular, for any f ∈ H − ( ∂ Ω), Φ (cid:93) ( f ) ∈ dom( ∂ (cid:93) ) and ∂ (cid:93) Φ (cid:93) ( f ) = 0.Now, we have collected all the tools to prove Proposition 22. Proof of Proposition 22.
For s = − , Proposition 22 holds true, because of Lemma24 and the density of C ∞ ( ∂ Ω) in H − ( ∂ Ω). Let us prove it for s = . Remarkthat Φ (cid:93) ( f ) ∈ dom( ∂ (cid:93) ) so if f ∈ H ( ∂ Ω) we also have Γ + Φ (cid:93) ( f ) = Π + (cid:93) f ∈ H ( ∂ Ω)by Proposition 20. Hence, by Lemma 18, Φ (cid:93) ( f ) ∈ H (Ω).Let us use the closed graph theorem and take a sequence of functions f n ∈ H ( ∂ Ω) such that f n → f in the H ( ∂ Ω)-norm. Assume also that Φ (cid:93) ( f n ) → u ∈ H (Ω) where the convergence holds in the H (Ω)-norm.Because of the continuous embedding of H ( ∂ Ω) into H − ( ∂ Ω), f n → f also inthe H − ( ∂ Ω)-norm. In particular, by Proposition 22 for s = − , Φ (cid:93) ( f n ) → Φ (cid:93) ( f )in L (Ω). Consequently, the equality u = Φ (cid:93) ( f ) holds not only in L (Ω) but alsoin H (Ω) and by the closed graph theorem, Φ (cid:93) is a continuous linear map between H ( ∂ Ω) and H (Ω).The result for s = 0 holds by (real) interpolation theory (see [35, Prop. 2.1.62.& Prop. 2.3.11. & Prop. 2.4.3.]). (cid:3) Explicit description of the Bergman and Hardy spaces.
Let us proveTheorem 21, starting with the following proposition concerning the Bergman spaces.
Proposition 25.
Let (cid:93) ∈ { h , ah } . There holds: A (cid:93) (Ω) = { Φ (cid:93) ( f ) : f ∈ H − ( ∂ Ω) such that Π − (cid:93) f = 0 } , (cid:93) ∈ { h , ah } . (15) Moreover, for all f ∈ H − ( ∂ Ω) there holds Φ (cid:93) ( f ) = Φ (cid:93) (Π + (cid:93) f ) . Proof . Denote E (cid:93) the set on the right-hand side of (15). We prove it for (cid:93) = h, theproof for (cid:93) = ah being similar.Inclusion E h ⊂ A (Ω). Let u = Φ h ( f ) ∈ E h , with f ∈ H − ( ∂ Ω) such that Π − h f = 0.By Proposition 22, Φ h maps H − ( ∂ Ω) to L (Ω) thus u ∈ L (Ω). Moreover, thereholds ∂ ¯ z u = 0 which implies that u ∈ A (Ω).Inclusion A (Ω) ⊂ E h . For u ∈ C ∞ (Ω), x ∈ Ω and ε > π (cid:90) Ω \ B ( x,ε ) ∂ ¯ z (cid:16) x + i x ) − ( y + i y ) (cid:17) u ( y ) dy = − π (cid:90) Ω \ B ( x,ε ) ∂ ¯ z u ( y )( x + i x ) − ( y + i y ) dy + 12 π (cid:90) ∂ Ω u ( y )( x + i x ) − ( y + i y ) n ( y ) ds ( y )+ 12 π (cid:90) ∂B ( x,ε ) u ( y )( x + i x ) − ( y + i y ) ( x + i x ) − ( y + i y ) | y − x | ds ( y ):= − A + B + C. However, we have C = 12 π (cid:90) π u ( x + ε (cos t, sin t )) dt −→ u ( x ) , when ε → . By definition, if γ : [0 , (cid:96) ] → ∂ Ω is a smooth arc-length parametrization of ∂ Ω thereholds B = − i2 π (cid:90) (cid:96) u ( γ ( t ))( x + i x ) − ( γ ( t ) + i γ ( t )) ( γ (cid:48) ( t ) + i γ (cid:48) ( t )) dt = − π (cid:90) ∂ Ω u ( ξ ) ξ − ( x + i x ) dξ = − Φ h (Γ + u )( x ) . In particular, we obtainΦ h (Γ + u )( x ) = 12 π (cid:90) π u ( x + ε (cos( t ) , sin( t ))) dt − π (cid:90) Ω \ B ( x,ε ) ∂ ¯ z u ( y )( x + i x ) − ( y + i y ) dy = 12 π (cid:90) π u ( x + ε (cos( t ) , sin( t ))) dt − π (cid:90) R R \ B (0 ,ε ) ( x − y )( x + i x ) − ( y + i y ) ( ∂ ¯ z u ( y ) Ω ( y )) dy. (16)Note that the linear form on C ∞ ( R ) defined by p.v. (cid:16) x + i x (cid:17) := ϕ ∈ C ∞ ( R ) (cid:55)→ lim ε → (cid:90) R R \ B (0 ,ε ) ( x ) x + i x ϕ ( x ) dx ∈ C belongs to D (cid:48) ( R ). Remark that ( ∂ ¯ z u Ω ) ∈ D (cid:48) ( R ) and has compact support.Hence, p.v. ( x +i x ) ∗ ( ∂ ¯ z u Ω ) ∈ D (cid:48) ( R ) and taking the duality pairing with ϕ ∈ C ∞ (Ω) in (16) and ε → (cid:104) Φ h (Γ + u ) − u, ϕ (cid:105) D (cid:48) (Ω) , D (Ω) = 1 π (cid:104) p.v. (cid:16) x + i x (cid:17) ∗ ( ∂ ¯ z u Ω ) , ϕ (cid:105) D (cid:48) ( R ) , D ( R ) . (17)Now, remark that A (Ω) ⊂ dom( ∂ h ) and pick a sequence of C ∞ (Ω) functions( v n ) n ∈ N which converges to v ∈ A (Ω) in the norm of dom( ∂ h ) when n → + ∞ . In particular, ( v n ) n ∈ N converges to v and ( ∂ ¯ z v n ) Ω converges to 0 when n → + ∞ in D (cid:48) ( R ). Using (17) for u = v n and letting n → + ∞ we obtain that in D (cid:48) (Ω)there holds v = Φ h (Γ + v ) where we have used the continuity of the map Φ h ◦ Γ + :dom( ∂ h ) → L (Ω), and the continuity of the convolution in D (cid:48) ( R ). Now, remarkthat we also have v = Φ h (Γ + v ) in A (Ω) and taking the trace Γ + on both side ofthis identity we get Π +h Γ + v = Γ + v which implies v = Φ h (Π +h Γ + v ) and proves the other inclusion. (cid:3) We are now in a good position to prove Theorem 21.
Proof of Theorem 21.
Proposition 25 is precisely the first statement of Theorem 21thus, the only thing left to prove is the statement for the Hardy spaces. Now, recallthat for (cid:93) ∈ { h , ah } , we have defined the Hardy spaces in (11) and that we want toprove H (cid:93) (Ω) = { Φ (cid:93) ( f ) : f ∈ L ( ∂ Ω) , Π − (cid:93) f = 0 } . Let E (cid:93) be the set on the right-hand side, we prove both inclusions.Inclusion E (cid:93) ⊂ H (cid:93) (Ω). Let u = Φ (cid:93) ( f ) ∈ E (cid:93) , by definition u ∈ A (cid:93) et Γ + u = f ∈ L ( ∂ Ω) ⊂ H − ( ∂ Ω) which proves this inclusion.Inclusion H (cid:93) (Ω) ⊂ E (cid:93) . Let u ∈ H (cid:93) (Ω). We know that in particular u = Φ (cid:93) ( f )for some f ∈ H − ( ∂ Ω) such that Π − (cid:93) f = 0. But we have Γ + u = f ∈ L ( ∂ Ω) whichproves this inclusion and concludes the proof. (cid:3)
Explicit description of the domain of the maximal Wirtinger oper-ators.
In this paragraph, we prove the following descrition of the domains of themaximal Wirtinger operators introduced in Section 3. This description involves theBergman spaces introduced in the beginning of Section 4.
Proposition 26.
Let (cid:93) ∈ { h , ah } . The following direct sum decomposition holds: dom( ∂ (cid:93) ) = { u ∈ H (Ω) : Π + (cid:93) Γ + u = 0 } (cid:117) A (cid:93) (Ω) . For (cid:93) ∈ { h , ah } , the range of the trace operator Γ + : dom ( ∂ (cid:93) ) → H − ( ∂ Ω) is ofcrucial importance to prove Proposition 26. We describe its range now, thanks tothe Szeg¨o projectors introduced in (13) but first, we prove a regularization result.
Lemma 27.
Let (cid:93) ∈ { h , ah } . The operator Π − (cid:93) ◦ Γ + is a bounded linear operatorfrom dom( ∂ (cid:93) ) to H ( ∂ Ω) . Proof . Let u ∈ dom( ∂ h ) and u n ∈ C ∞ (Ω) be a sequence converging to u in the (cid:107) · (cid:107) h -norm when n → + ∞ . Pick f ∈ C ∞ ( ∂ Ω), an integration by parts yields: (cid:104) Γ + u n , n Π +ah f (cid:105) L ( ∂ Ω) = 2 (cid:104) ∂ ¯ z u n , Φ ah ( f ) (cid:105) L (Ω) . It gives |(cid:104) Γ + u n , n Π +ah f (cid:105) L ( ∂ Ω) | ≤ c (cid:107) u n (cid:107) h (cid:107) f (cid:107) H − ( ∂ Ω) , for some c >
0, where we have used Lemma 15 and Proposition 22. As in L ( ∂ Ω)there holds S ∗ ah = − S h we get ( n Π +ah ) ∗ = Π − h n . In particular, there holds |(cid:104) Γ + u n , n Π +ah f (cid:105) L ( ∂ Ω) | = |(cid:104) (Π − h n Γ + u n , f (cid:105) L ( ∂ Ω) | ≤ c (cid:107) u n (cid:107) h (cid:107) f (cid:107) H − ( ∂ Ω) . Letting n → + ∞ , we get Π − h n Γ + u ∈ H ( ∂ Ω) and that Π − h ◦ H n ◦ Γ + is a linearbounded map from H − ( ∂ Ω) to H ( ∂ Ω). However, there holdsΠ − h Γ + u = (cid:16) n Π − h n − n [Π − h , n ] (cid:17) Γ + u = n Π − h n Γ + u + n [ S h , n ]Γ + u. By (2)Proposition 12, [ S h , n ] ∈ Ψ − ∂ Ω hence, it is a bounded operator from H − ( ∂ Ω)to H ( ∂ Ω). Finally, as the multiplication operator by n is bounded in H ( ∂ Ω) weobtain the expected result.The case u ∈ dom( ∂ ah ) is handled similarly. (cid:3) We are now in a good position to describe the range of the trace operator Γ + . Corollary 28.
Let (cid:93) ∈ { h , ah } . There holds ran(Γ + ) = { f ∈ H − ( ∂ Ω) : Π − (cid:93) f ∈ H ( ∂ Ω) } . Proof . Let us start by proving the reverse inclusion. Let f be in the set on right-hand side, there holds f = Π + (cid:93) f + Π − (cid:93) f . We know that there exists an extensionoperator E + from H ( ∂ Ω) to H (Ω) such that Γ + E + g = g for all g ∈ H ( ∂ Ω).Now, if Π − (cid:93) f ∈ H ( ∂ Ω), we set u := Φ (cid:93) (Π + (cid:93) f ) + E + (Π − (cid:93) f ) . It is easily seen that u ∈ dom( ∂ (cid:93) ) and Γ + u = Π + (cid:93) f + Π − (cid:93) f = f .Now, let us prove the direct inclusion and pick f ∈ ran(Γ + ). We know that thereexists u ∈ dom( ∂ (cid:93) ) such that f = Γ + u . In particular, by Lemma 27 we know thatΠ − (cid:93) f = Π − (cid:93) Γ + u ∈ H ( ∂ Ω) which concludes the proof. (cid:3)
We are now able to prove Proposition 26.
Proof of Proposition 26.
First, let us prove that the sum is direct. Let v = Φ (cid:93) ( f ) = u with Π − (cid:93) f = 0 and Π + (cid:93) Γ + u = 0. Then, taking the traces we obtain:Γ + v = Π + (cid:93) f = Π − (cid:93) Γ + u, which implies f = Γ + u = 0. Consequently, v = Φ (cid:93) ( f ) = 0.Second, let us pick v ∈ dom( ∂ (cid:93) ). There holds v = Φ (cid:93) (Π + (cid:93) Γ + v ) + v − Φ (cid:93) (Π + (cid:93) Γ + v ) . However, remark that u := v − Φ (cid:93) (Π + (cid:93) Γ + v ) ∈ dom( ∂ (cid:93) ) and satisfies Γ + u = Π − (cid:93) Γ + v ∈ H ( ∂ Ω) by Lemma 27. Hence, by Lemma 18, we obtain u ∈ H (Ω) and Γ + u ∈ ker Π + (cid:93) = ran Π − (cid:93) , which concludes the proof. (cid:3) Variational characterization of the principal eigenvalue
The aim of this section is to prove Theorem 4. In § q Ω E ) and dom( H Ω E ), where H Ω E is the unique self-adjoint operatorassociated with q Ω E via Kato’s first representation theorem. In § E ∈ [0 , + ∞ ) (cid:55)→ µ Ω ( E ). Finally, in § The quadratic form q Ω E and its associated self-adjoint operator H Ω E . For
E >
0, recall that q Ω E is defined in (2) on the domain consisting of the closureof the C ∞ (Ω) functions with respect to the norm of the quadratic form N Ω E ( u ) := (cid:113) (cid:107) ∂ ¯ z u (cid:107) L (Ω) + (cid:107) u (cid:107) L (Ω) + E (cid:107) u (cid:107) L ( ∂ Ω) . Remark that as defined, q Ω E is a closed, densely defined and bounded below quadraticform thus, by Kato’s first representation theorem (see [25, Chap. VI, Thm. 2.1]), q Ω E is associated with a unique self-adjoint operator H Ω E acting in L (Ω) satisfyingdom( H Ω E ) ⊂ dom( q Ω E ) . In this paragraph, we describe properties of the domains dom( q Ω E ) and dom( H Ω E )and start with the domain of the quadratic form q Ω E . Proposition 29.
Let
E > . The form domain dom( q Ω E ) admits the followingdirect sum decomposition dom( q Ω E ) = { u ∈ H (Ω) : Π +h Γ + u = 0 } (cid:117) H (Ω) . Moreover, dom( q Ω E ) is continuously embedded in H (Ω) . Proof . Set E = { u ∈ H (Ω) : Π +h Γ + u = 0 } (cid:117) H (Ω) and remark that the sum isdirect by the same arguments as in the proof of Proposition 26. We prove the setequality by proving both inclusions.Inclusion E ⊂ dom( q Ω E ). Let v := u + Φ h ( f ) ∈ E and take ( u n ) n ∈ N and ( f n ) n ∈ N twosequences of functions such thatfor all n ∈ N u n ∈ C ∞ (Ω) , f n ∈ C ∞ ( ∂ Ω);and when n → + ∞ there holds (cid:107) u n − u (cid:107) H (Ω) → , (cid:107) f n − f (cid:107) L ( ∂ Ω) → . By [10, Theorem 3.1.], we have v n := u n + Φ h ( f n ) ∈ C ∞ (Ω) and for E >
0, thereexists
C > q Ω E ( v − v n ) + ( E + 1) (cid:107) v − v n (cid:107) L (Ω) = (cid:107) ∂ ¯ z ( u − u n ) (cid:107) L (Ω) + E (cid:107) Γ + ( u − u n ) + Π +h ( f − f n ) (cid:107) L ( ∂ Ω) + (cid:107) ( u − u n ) + (Φ h ( f − f n )) (cid:107) L (Ω) ≤ C (cid:18) (cid:107) u − u n (cid:107) H (Ω) + (cid:107) f − f n (cid:107) L ( ∂ Ω) (cid:19) , where we have used the mapping properties of Φ h , Γ + , Π +h and the continuityof the embedding of L ( ∂ Ω) into H − ( ∂ Ω). Letting n → + ∞ , we obtain that v ∈ dom( q Ω E ) and this inclusion is proved.Inclusion dom( q Ω E ) ⊂ E . For all u ∈ C ∞ (Ω), there holds q Ω E ( u ) + ( E + 1) (cid:107) u (cid:107) L (Ω) ≥ (cid:107) u (cid:107) . In particular, the closure of C ∞ (Ω) for the norm N Ω E is included in dom( ∂ h ). Itrewrites dom( q Ω E ) ⊂ dom( ∂ h ) and by Proposition 26, any v ∈ dom( q Ω E ) writes v = u + Φ h ( f ), for some u ∈ H (Ω) with Π +h Γ + u = 0 and some f ∈ H − ( ∂ Ω)with Π − h f = 0. Now, if v n ∈ C ∞ (Ω) converges to v ∈ dom( q Ω E ) in the norm of thequadratic form, we have (cid:107) Γ + v − Γ + v n (cid:107) L ( ∂ Ω) ≤ E − q Ω E ( v − v n ) → , n → + ∞ . In particular Γ + v = Γ + u + f ∈ L ( ∂ Ω) and as Γ + u ∈ H ( ∂ Ω) we get f ∈ L ( ∂ Ω)which concludes the proof of this inclusion. Let us consider the inclusion map I := dom( q Ω E ) → H (Ω) , ( I u ) = u. By Proposition 22 for s = 0, this map is well-defined. Consider v n := u n +Φ h ( f n ) ∈ dom( q Ω E ) which converges to v in the norm of the quadratic form q Ω E and assumethat v n → w in the H (Ω)-norm. In particular, as v ∈ dom( q Ω E ), there holds v = u + Φ h ( f ) for some u ∈ H (Ω) and f ∈ H ( ∂ Ω) as in the definition of E . Inparticular, in D (cid:48) (Ω) we obtain u + Φ h ( f ) = w and as both terms belong to H (Ω), the closed graph theorem gives that I iscontinuous. (cid:3) Because of the compact embedding of H (Ω) into L (Ω), an immediate corollaryof Proposition 29 reads as follows. Corollary 30.
Let
E > , the operator H Ω E has compact resolvent and its spectrumconsists of a non-decreasing sequence of eigengalues denoted ( µ Ω j ( E )) j ≥ . Moreover,there holds µ Ω j ( E ) = inf F ⊂ dom ( q Ω E )dim F = j sup u ∈ F \{ } (cid:82) Ω | ∂ ¯ z u | dx − E (cid:82) Ω | u | dx + E (cid:82) ∂ Ω | u | ds (cid:82) Ω | u | dx . Remark 31.
For E = 0, the counterpart of Propostion 29, would readdom( q Ω0 ) = { u ∈ H (Ω) : Π +h Γ + u = 0 } (cid:117) A (Ω) . In particular, note that dom( q Ω0 ) can not be included in any Sobolev space H s (Ω),( s > u ∈ A (Ω), there holds q Ω0 ( u ) = 0 whichimplies that for all j ≥ µ Ω j (0) = 0. Thus 0 is an eigenvalue of H Ω0 of infinitemultiplicity which would not be possible if we had dom( q Ω0 ) ⊂ H s (Ω) because ofthe compact embedding of H s (Ω) in L (Ω). This phenomena is reminiscent of whathappens for the Dirac operator with zig-zag boundary conditions as discussed in[36].We conclude this paragraph by a description of the domain of the operator H Ω E . Proposition 32.
Let
E > , there holds: dom( H Ω E ) = { u ∈ H (Ω) : ∂ ¯ z u ∈ H (Ω) and ∂ ¯ z u + n E u = 0 on ∂ Ω } . Proof . Let E denote the set in the right-hand side of Proposition 32. The proof isperformed proving both inclusions.Inclusion dom( H Ω E ) ⊂ E . Let u ∈ dom( H Ω E ) and v ∈ C ∞ (Ω), there holds (cid:104) H Ω E u, v (cid:105) D (cid:48) (Ω) , D (Ω) = (cid:104) H Ω E u, v (cid:105) L (Ω) = q Ω E [ u, v ]= 4 (cid:104) ∂ ¯ z u, ∂ z v (cid:105) D (cid:48) (Ω) , D (Ω) − E (cid:104) u, v (cid:105) D (cid:48) (Ω) , D (Ω) = (cid:104) ( − ∆ − E ) u, v (cid:105) D (cid:48) (Ω) , D (Ω) , where q Ω E [ · , · ] denotes the sesquilinear form associated with the quadratic form q Ω E .Hence, in L (Ω), there holds H Ω E u = ( − ∆ − E ) u . Remark that if u ∈ dom( H Ω E ) then ∂ ¯ z u ∈ dom( ∂ ah ), in particular, by Green’s Formula (10), for all v ∈ C ∞ (Ω) weget: (cid:104) H Ω E u, v (cid:105) L (Ω) = − (cid:104) ∂ z ( ∂ ¯ z u ) , v (cid:105) L (Ω) − E (cid:104) u, v (cid:105) L (Ω) = 4 (cid:104) ∂ ¯ z u, ∂ ¯ z v (cid:105) L (Ω) − E (cid:104) u, v (cid:105) L (Ω) − (cid:104) n Γ + ∂ ¯ z u, Γ + v (cid:105) H − ( ∂ Ω) ,H ( ∂ Ω) = q Ω E [ u, v ] − (cid:104) n Γ + ∂ ¯ z u + Eu, Γ + v (cid:105) H − ( ∂ Ω) ,H ( ∂ Ω) . As v ∈ dom( q Ω E ) we necessarily have (cid:104) n Γ + ∂ ¯ z u + Eu, Γ + v (cid:105) H − ( ∂ Ω) ,H ( ∂ Ω) = 0. Asthis is true for all v ∈ C ∞ (Ω) we obtain2 n Γ + ∂ ¯ z u + E Γ + u = 0 , in H − ( ∂ Ω) . (18)Taking the Szeg¨o projectors in (18) we obtain(Γ + ∂ ¯ z u ) + n E Γ + u = 0 ⇐⇒ (cid:26) Π +ah (Γ + ( ∂ ¯ z u )) + E Π +ah n Γ + u = 0Π − ah (Γ + ( ∂ ¯ z u )) + E Π − ah n Γ + u = 0Nevertheless, there holdsΠ − ah = Π +h −
12 ( S h + S ah ) , Π +ah = Π − h + 12 ( S ah + S h ) . In particular, we getΠ − ah (Γ + ( ∂ ¯ z u )) = − E − ah ( n Γ + u ) = − E (cid:16) n Π +h Γ + u + [Π +h , n ]Γ + u −
12 ( S h + S ah )( n Γ + u ) (cid:17) = − E (cid:16) n Π +h Γ + u + [ S h , n ]Γ + u −
12 ( S h + S ah )( n Γ + u ) (cid:17) . It rewritesΠ +h Γ + u = − n (cid:16) E Π − ah Γ + ( ∂ ¯ z u ) + [ S h , n ]Γ + u −
12 ( S h + S ah )( n Γ + u ) (cid:17) . Remark that the right-hand side belongs to H ( ∂ Ω). This holds for the first termbecause of Lemma 27 and for the last two-terms because of Proposition 12. AsΠ − h Γ + u ∈ H ( ∂ Ω) by Lemma 27, we get Γ + u = Π +h Γ + u + Π − h Γ + u ∈ H ( ∂ Ω) thus,by Lemma 18, u ∈ H (Ω). In particular Π +ah (Γ + ( ∂ ¯ z u )) = − E Π +ah n Γ + u ∈ H ( ∂ Ω)and as Π − ah Γ + ( ∂ ¯ z u ) ∈ H ( ∂ Ω) by Lemma 27 we obtain Γ + ∂ ¯ z u = Π − ah Γ + ( ∂ ¯ z u ) +Π +ah Γ + ( ∂ ¯ z u ) ∈ H ( ∂ Ω) and by Lemma 18 we obtain ∂ ¯ z u ∈ H (Ω). It concludesthe proof of this inclusion.Inclusion E ⊂ dom( H Ω E ). Pick u ∈ E . One easily sees that ( − ∆ − E ) u ∈ L (Ω),moreover for all v ∈ dom( q Ω E ), there holds q Ω E [ u, v ] = (cid:104) ( − ∆ − E ) u, v (cid:105) L (Ω) . By definition of H Ω E it implies u ∈ dom( H Ω E ) and H Ω E u = ( − ∆ − E ) u . (cid:3) Concavity of the first min-max level.
In this paragraph we investigate thebehavior of the first min-max level µ Ω ( E ) with respect to the spectral parameter E >
0. This behavior is illustrated in Figure 3 for various domains Ω.
Proposition 33.
The map µ Ω : E ≥ (cid:55)→ µ Ω ( E ) verifies the following properties.(1) µ Ω is a continuous and concave function on R + .(2) We have µ Ω (0) = 0 and there exists E Ω (cid:63) > such that for all E ∈ (0 , E Ω (cid:63) ) there holds µ Ω ( E ) > .(3) Let < E < E , there holds µ Ω ( E ) ≤ E E µ Ω ( E ) − E ( E − E ) In particular, if µ Ω ( E ) = 0 (resp. µ Ω ( E ) = 0 ) there holds µ Ω ( E ) < (resp. µ Ω ( E ) > ). Proof . As for all u ∈ dom( q Ω E ) the function (cid:0) E ≥ (cid:55)→ q Ω E ( u ) (cid:1) is a continuous andconcave, so is (cid:0) E ≥ (cid:55)→ µ Ω ( E ) (cid:1) and Point (1) is proved.Regarding Point (2), one observes that for all u ∈ dom( q Ω E ) there holds q Ω0 ( u ) ≥ µ Ω (0) ≥
0. Now, for any f ∈ L ( ∂ Ω) we have Φ h ( f ) ∈ dom( q Ω E )and q Ω0 ( u ) = 0 because Φ h ( f ) is holomorphic in Ω. Consequently, there holds µ Ω (0) = 0.To prove the second part of Point (2), let u ∈ dom( q Ω E ) and remark that q Ω E ( u ) = (4 − E ) (cid:107) ∂ ¯ z u (cid:107) L (Ω) − E (cid:107) u (cid:107) L (Ω) + E Q ( u ) (19)where the quadratic form Q is defined as Q ( u ) = (cid:107) ∂ ¯ z u (cid:107) L (Ω) + (cid:107) u (cid:107) L ( ∂ Ω) , dom( Q ) = dom( q Ω E ) . Now, remark that Q ≥ H such that dom( H ) ⊂ dom( Q ) and its spectrum isa sequence of non-decreasing eigenvalues because dom( Q ) = dom( q Ω E ) is compactlyembedded into L (Ω). Let λ Ω1 be its smallest eigenvalue, we already know by themin-max principle that λ Ω1 ≥
0. Moreover, if λ Ω1 = 0, for an associated eigenfunction u , we obtain Q ( u ) = 0 which implies that ∂ ¯ z u = 0 hence u is holomorphic with tracein L ( ∂ Ω). Consequently, u belongs to H (Ω) and u = Φ h ( f ) for some f ∈ L ( ∂ Ω)such that Γ + u = f . However, as Q ( u ) = 0, we also obtain Γ + u = f = 0 whichyields u = 0 which is not possible because u is an eigenfunction. It implies that λ Ω1 > u ∈ dom( q Ω E ): q Ω E ( u ) ≥ (4 − E ) (cid:107) ∂ ¯ z u (cid:107) L (Ω) − E (cid:107) u (cid:107) + Eλ Ω1 (cid:107) u (cid:107) L (Ω) . In particular, if
E < q Ω E ( u ) ≥ E (cid:0) λ Ω1 − E (cid:1) (cid:107) u (cid:107) L (Ω) and the min-max principle yields µ Ω ( E ) ≥ E ( λ Ω1 − E ) . Thus, setting E Ω (cid:63) := min(4 , λ Ω1 ), for all E ∈ (0 , E Ω (cid:63) ), we have µ Ω ( E ) > u ∈ dom( q Ω E ) and 0 < E < E . There holds q Ω E ( u ) = q Ω E ( u ) − ( E − E ) (cid:90) Ω | u | dx + ( E − E ) (cid:90) ∂ Ω | u | ds. (20)Now, pick u a normalized eigenfunction of H Ω E associated with the eigenvalue µ Ω ( E ). We have q Ω E ( u ) = µ Ω ( E ) which implies (cid:90) ∂ Ω | u | ds ≤ E (cid:16) (cid:90) Ω | ∂ ¯ z u | dx + E (cid:90) ∂ Ω | u | ds (cid:17) = 1 E ( q Ω E ( u )+ E ) ≤ E + µ Ω ( E ) E . Thus, evaluating (20) with u = u we obtain q Ω E ( u ) ≤ µ Ω ( E ) − ( E − E ) + E − E E ( E + µ Ω ( E )) . The min-max principle finally gives the sought inequality µ Ω ( E ) ≤ µ Ω ( E ) − ( E − E ) + E − E E ( E + µ Ω ( E ))= E E µ Ω ( E ) − E ( E − E ) . Now, assume that µ Ω ( E ) = 0. It yields µ Ω ( E ) ≤ − E ( E − E ) < . Similarly, if µ Ω ( E ) = 0 we get0 < E ( E − E ) ≤ µ Ω ( E ) . (cid:3) Proof of the variational principle.
In our way to prove Theorem 4 we willneed the following two propositions.
Proposition 34.
Let
E > be such that µ Ω ( E ) = 0 then E ∈ Sp dis ( D Ω ) . Proof . Let
E > µ Ω ( E ) = 0 and consider a normalized associatedeigenfunction v ∈ dom( H Ω E ). Set u = ( u , u ) (cid:62) = ( v, − E ∂ ¯ z v ) (cid:62) , by Proposition 32, u ∈ H (Ω , C ) and as v ∈ dom( H Ω E ), in H ( ∂ Ω) there holdsΓ + ( ∂ ¯ z v ) + n E + v = 0 ⇐⇒ − E − iΓ + ( ∂ ¯ z v ) = i n Γ + u ⇐⇒ Γ + u = i n Γ + u . Hence, ( u , u ) (cid:62) ∈ dom( D Ω ) and there holds D Ω ( u , u ) (cid:62) = (cid:18) − ∂ z − ∂ ¯ z (cid:19) ( u , u ) (cid:62) = ( − ∂ z u , − ∂ ¯ z u ) (cid:62) = ( − E ∆ u, Eu ) (cid:62) = E ( u , u ) (cid:62) . Hence, E ∈ Sp dis ( D Ω ) and it concludes the proof of Proposition 34. (cid:3) Proposition 35.
Let E ∈ Sp dis ( D Ω ) ∩ R ∗ + then µ Ω ( E ) ≤ . Proof . Let E ∈ Sp dis ( D Ω ) ∩ R ∗ + and pick u = ( u , u ) (cid:62) ∈ dom( D Ω ) a normalizedeigenfunction of D Ω associated with E . We have (cid:26) D Ω u = Eu in Ω ,u = i n u on ∂ Ω . In particular, we have − ∂ ¯ z u = Eu and ∂ ¯ z u ∈ H (Ω). It yields Eu = − ∂ z u = − E ∂ z ∂ ¯ z u . Taking the scalar product with respect to u on both side of the previous equationwe get E (cid:90) Ω | u | dx = − (cid:90) Ω ( ∂ z ∂ ¯ z u ) u dx = 4 (cid:90) Ω | ∂ ¯ z u | dx − (cid:90) ∂ Ω n ( ∂ ¯ z u ) u ds. (21)Now, remark that on ∂ Ω, we have − E ∂ ¯ z u = u = i n u which implies that on ∂ Ω 2 n ∂ ¯ z u + Eu = 0 . Hence, (21) becomes E (cid:90) Ω | u | = 4 (cid:90) Ω | ∂ ¯ z u | dx + E (cid:90) ∂ Ω | u | ds which reads q Ω E ( u ) = 0 thus, the min-max principle gives µ Ω ( E ) ≤ (cid:3) Now, we have all the tools to prove Theorem 4. The proof is performed provingeach implication. Proof of Theorem 4.
By Proposition 35, we have µ Ω ( E (Ω)) ≤
0. Assume that µ Ω ( E (Ω)) <
0, by Proposition 33 we know that there exists 0 < E < E (Ω)such that µ Ω ( E ) = 0 which, by Proposition 34, implies E ∈ Sp dis ( D Ω ). It isnot possible because, by definition of E (Ω), E ≥ E (Ω) consequently, we obtain µ Ω ( E (Ω)) = 0.Let E > µ Ω ( E ) = 0. By Proposition 34, E ∈ Sp dis ( D Ω ) andnecessarily E ≥ E (Ω). If E > E (Ω), by Proposition 33, we obtain µ Ω ( E (Ω)) > µ Ω ( E (Ω)) ≤ E = E (Ω). (cid:3) Geometric upper bounds on the spectral gap
The goal of this section is to prove Theorem 3 and this is discussed in § § A simple upper bound.
An immediate consequence of Theorem 4 reads asfollows.
Proposition 36.
Let Ω ⊂ R be C ∞ and simply connected. There holds E (Ω) ≤ | ∂ Ω || Ω | . There is no reason for the above upper bound to be attained among Euclideandomains. However, the bound brings into play simple geometric quantities: theperimeter and the area of Ω.
Proof . Let
E > u ≡ u ∈ dom( q Ω E ), bythe min-max principle we obtain µ Ω ( E ) ≤ q Ω E ( u ) (cid:107) u (cid:107) L (Ω) = E (cid:16) | ∂ Ω || Ω | − E (cid:17) . So in E crit := | ∂ Ω || Ω | we get µ Ω ( E crit ) ≤ E (Ω) ≤ E crit = | ∂ Ω || Ω | . (cid:3) A sharp upper bound.
It turns out Theorem 3 is a consequence of thefollowing result.
Theorem 37.
Let Ω ⊂ R be a C ∞ simply connected domain. There holds E (Ω) ≤ | ∂ Ω | + (cid:112) | ∂ Ω | + 8 πE ( D )( E ( D ) − πr i + | Ω | )2( πr i + | Ω | ) with equality if and only if Ω is a disk. Now, we have all the tools to prove Theorem 3.
Proof of Theorem 3.
Using that πr i ≤ | Ω | and the isoperimetric inequality we ob-tain 4 π r i ≤ π | Ω | ≤ | ∂ Ω | . It gives | ∂ Ω | + 8 πE ( D )( E ( D ) − πr i + | Ω | ) ≤ | ∂ Ω | (2 E ( D ) − . Note that in the above inequalities, we have equality if and only if Ω is a disk andcombining this bound with the one of Theorem 37 we get Theorem 3. (cid:3) In the rest of this section we focus on proving Theorem 37 and assume, withoutloss of generality, the following.(i) 0 ∈ Ω is such that r i = max x ∈ ∂ Ω | x | ,(ii) f : D → Ω is a conformal map such that f (0) = 0 and we write f ( z ) = (cid:88) n ≥ c n z n , where ( c n ) n ≥ is a sequence of complex numbers.Before going through the proof of Theorem 37, we gather in the following para-graph some known properties linking the geometry of Ω with the conformal map f .6.2.1. Preliminaries.
The next proposition can be found in [32, § f . Proposition 38 (Area formula) . There holds | Ω | = π (cid:88) n ≥ n | c n | . The second proposition is a consequence of the Schwarz lemma (see Koebe’sestimate in [22, Chap. I, Thm. 4.3]). It gives a relation between the first coefficient c of the conformal map f and the inradius r i . Proposition 39 (Koebe’s estimate) . There holds | f (cid:48) (0) | = | c | ≥ r i . Finally, the last geometric relation between the conformal map f and the ge-ometry of Ω we need to prove Theorem 37 is that the perimeter | ∂ Ω | of Ω can beexpressed as | ∂ Ω | = (cid:90) π | f (cid:48) ( e iθ ) | dθ. (22)(22) is a simple consequence of the fact that f | S is a parametrization of ∂ Ω.6.2.2.
Proof of the upper bound on the spectral gap.
To prove Theorem 37, weconstruct an adequate test function for q Ω E transplanting the eigenfunction of theunit disk D in the domain Ω thanks to the conformal map f . We obtain an upperbound on µ Ω ( E ) which is a second order polynomial in the spectral parameter E >
E >
Proof of Theorem 37.
Let us go through all the steps of the proof.Step 1. Let us denote by J (resp. J ) the Bessel function of the first kind oforder 0 (resp. of order 1). For x ∈ D , consider u ( x ) = J (cid:0) E ( D ) | x | (cid:1) ∈ H ( D ) ⊂ dom( q Ω E ( D ) ). As explained in Remark 2 u ( x ) = ( u ( x ) , i x + ix | x | J (cid:0) E ( D ) | x | (cid:1) ) (cid:62) isan eigenfunction of D D associated with E ( D ). Theorem 4 implies0 = q D E ( D ) ( u ) = 2 πE ( D ) (cid:90) J (cid:0) E ( D ) r (cid:1) rdr − πE ( D ) (cid:90) J (cid:0) E ( D ) r (cid:1) rdr + 2 πE ( D ) J (cid:0) E ( D ) (cid:1) . (23) Step 2. For x = ( x , x ) ∈ Ω, consider v ( x , x ) = u ( f − ( x + i x )) ∈ H (Ω) ⊂ dom( q Ω E ). By the min-max principle, there holds µ Ω ( E ) ≤ q Ω E ( v ) (cid:107) v (cid:107) L (Ω) = (cid:107)∇ v (cid:107) L (Ω) + E (cid:107) v (cid:107) L ( ∂ Ω) (cid:107) v (cid:107) L (Ω) − E , (24)where we have used that v is real valued to ensure that (cid:107)∇ v (cid:107) L (Ω) = 4 (cid:107) ∂ ¯ z v (cid:107) L (Ω) .Step 3. Now, as f is a conformal map, we know that (cid:107)∇ v (cid:107) L (Ω) = (cid:107)∇ u (cid:107) L ( D ) = 2 πE ( D ) (cid:90) J (cid:0) E ( D ) r (cid:1) rdr. (25)Using (22), we obtain (cid:107) v (cid:107) L ( ∂ Ω) = (cid:90) π | v ( f ( e i θ )) | | f (cid:48) ( e i θ ) | dθ = J (cid:0) E ( D ) (cid:1) | ∂ Ω | . (26)Finally, the last integral reads (cid:107) v (cid:107) L (Ω) = (cid:90) (cid:90) π | u ( r ) | | f (cid:48) ( re i θ ) | rdrdθ = (cid:90) | u ( r ) | (cid:16) (cid:90) π (cid:12)(cid:12)(cid:12) (cid:88) n ≥ nc n r n − e i( n − θ (cid:12)(cid:12)(cid:12) dθ (cid:17) rdr =2 π (cid:88) n ≥ n | c n | M n , where for n ≥ , M n := n (cid:90) J (cid:0) E ( D ) r (cid:1) r n − dr, (27)where we have used Parseval identity.Step 4. Taking into account (25),(26) and (27), (24) becomes µ Ω ( E ) ≤ πE ( D ) (cid:90) J (cid:0) E ( D ) r (cid:1) rdr π (cid:88) n ≥ n | c n | M n − E + E J (cid:0) E ( D ) (cid:1) | ∂ Ω | π (cid:88) n ≥ n | c n | M n . (28)Let us find a lower bound on the sequence ( M n ) n ≥ . Using first an integrationby parts we find M n = 12 J (cid:0) E ( D ) (cid:1) + E ( D )2 (cid:90) J (cid:0) E ( D ) r (cid:1) J (cid:0) E ( D ) r (cid:1) r n dr. In particular, for n = 1 it gives M = (cid:90) J (cid:0) E ( D ) r (cid:1) rdr = J (cid:0) E ( D ) (cid:1) (29)= E ( D ) (cid:90) J (cid:0) E ( D ) r (cid:1) J ( E ( D ) r ) r dr. Now, for n ≥
1, one notices that h := (cid:16) r (cid:55)→ ( J J ) (cid:0) E ( D ) r (cid:1) r (cid:17) and h := (cid:16) r (cid:55)→ r n − (cid:17) are non-decreasing functions on [0 ,
1] and by Chebyschev’s inequality fornon-decreasing functions, we obtain M n ≥ M + 12 M (cid:90) r n − dr = n n − M . In particular, we have2 π (cid:88) n ≥ n | c n | M n ≥ J (cid:0) E ( D ) (cid:1) (cid:16) π | c | + 2 π (cid:88) n ≥ n n − | c n | (cid:17) ≥ J (cid:0) E ( D ) (cid:1) (cid:16) π | c | + π (cid:88) n ≥ n | c n | (cid:17) = J (cid:0) E ( D ) (cid:1) ( π | c | + | Ω | ) ≥ J (cid:0) E ( D ) (cid:1) ( π | r i | + | Ω | ) , (30)where we have used Proposition 38 and Proposition 39. Remark that in the firsttwo inequalities above we have equality if and only if c n = 0 for all n ≥
2. Similarly,in the last equality, we have equality if and only if | c | = r i . In particuliar there isequality in the above inequalities if and only if f ( z ) = c z and Ω is a disk centeredin 0 of radius r i .Combining (23) and (30) in (28), we obtain µ Ω ( E ) ≤ − E + 2 πE ( D ) (cid:82) J (cid:0) E ( D ) r (cid:1) rdr + J (cid:0) E ( D ) (cid:1) (cid:16) E | ∂ Ω | − πE ( D ) (cid:17) J (cid:0) E ( D ) (cid:1) ( πr i + | Ω | ) . Using (29), we obtain µ Ω ( E ) ≤ − E + 2 πE ( D ) + (cid:0) E | ∂ Ω | − πE ( D ) (cid:1) πr i + | Ω | = (cid:0) πE ( D ) − ( πr i + | Ω | ) E (cid:1) + (cid:0) E | ∂ Ω | − πE ( D ) (cid:1) πr i + | Ω | = P ( E ) πr i + | Ω | , P ( E ) := − E ( πr i + | Ω | ) + E | ∂ Ω | + 2 πE ( D ) (cid:0) E ( D ) − (cid:1) . Step 5. Remark that by (1), there holds E ( D ) − ≥ √ − >
0. In particular,the discriminant of P satisfies δ ( P ) := | ∂ Ω | + 8 πE ( D ) (cid:0) E ( D ) − (cid:1) ( πr i + | Ω | ) > . Thus, P has two real roots and as P (0) >
0, the only positive root is E crit := | ∂ Ω | + (cid:113) | ∂ Ω | + 8 πE ( D ) (cid:0) E ( D ) − (cid:1) ( πr i + | Ω | )2( πr i + | Ω | ) . One obtains µ Ω ( E crit ) ≤ P ( E crit ) πr i + | Ω | = 0 and by Proposition 33 and Theorem 4 we get E ( D ) ≤ E crit which is precisely Theorem 37. (cid:3) About the Faber-Krahn conjecture
In this section we discuss how the variational formulation established in Theorem4 can be used to investigate Conjecture 1. § H Ω E introduced in § § A new conjecture.
Let us introduce a new Faber-Krahn type conjecture for µ Ω ( E ), the first eigenvalue of H Ω E . Conjecture 40.
Let Ω ⊂ R be C ∞ and simply connected. For all E >
0, thereholds µ Ω ( E ) ≥ π | Ω | µ D (cid:16)(cid:114) | Ω | π E (cid:17) . Moreover, there is equality in the above inequality if and only if Ω is a disk.It turns out Conjecture 40 is equivalent to Conjecture 1 and this is what weprove in the rest of this paragraph.
Proof . First, remark that a simple scaling argument gives, for all
E >
0, that (cid:114) π | Ω | E ( D ) = E ( ρ D ) , µ ρ D ( E ) = π | Ω | µ D (cid:16)(cid:114) | Ω | π E (cid:17) where ρ := (cid:114) | Ω | π . Second, assume that Conjecture 1 holds true. If Ω is a disk, there holds µ Ω ( E ) = µ ρ D ( E ) so now, we assume that Ω is not a disk. Let us prove that for all E > µ Ω ( E ) > µ ρ D ( E ) . Let us reason by reduction ad absurdum and assume there exists E (cid:63) > µ Ω ( E (cid:63) ) ≤ µ ρ D ( E (cid:63) ).Case E (cid:63) < E ( ρ D ). By hypothesis and Proposition 33, there holds µ Ω ( E (cid:63) ) ≤ µ ρ D ( E (cid:63) ) ≤ E ( ρ D ) E (cid:63) µ ρ D (cid:0) E ( ρ D ) (cid:1) − E ( ρ D )( E ( ρ D ) − E (cid:63) )= − E ( ρ D )( E ( ρ D ) − E (cid:63) ) < . In particular, µ Ω ( E (cid:63) ) < E (cid:63) > E (Ω). However, if Conjecture 1holds true we obtain E (cid:63) > E (Ω) > E ( ρ D ) which contradicts our hypothesis.Case E ( ρ D ) ≤ E (cid:63) ≤ E (Ω). By hypothesis and Proposition 33, there holds0 ≤ µ Ω ( E (cid:63) ) ≤ µ ρ D ( E (cid:63) ) ≤ , which contradicts our hypothesis because we obtain E (cid:63) = E (Ω) = E ( ρ D ) but wehave assumed that Ω is not a disk thus, this equality can not hold if Conjecture 1holds true.Case E (cid:63) > E (Ω). By hypothesis and Proposition 33, there holds0 = µ Ω (cid:0) E (Ω) (cid:1) ≤ E (cid:63) E (Ω) µ Ω ( E (cid:63) ) − E (cid:63) (cid:0) E (cid:63) − E (Ω) (cid:1) ≤ E (cid:63) E (Ω) µ ρ D ( E (cid:63) ) − E (cid:63) (cid:0) E (cid:63) − E (Ω) (cid:1) . In particular, we obtain µ ρ D ( E (cid:63) ) ≥ E (Ω) (cid:0) E (cid:63) − E (Ω) (cid:1) >
0. Hence, E (cid:63) < E ( ρ D )which contradicts Conjecture 1.Consequently, we have proved that if Conjecture 1 holds true so does Conjecture40.Finally, let us assume that Conjecture 40 holds true. If Ω is a disk, we obtain thatfor all E > µ Ω ( E ) = µ ρ D ( E ). In particular, in E = E (Ω) we get µ ρ D (cid:0) E (Ω) (cid:1) = 0and E ( ρ D ) = E (Ω).When Ω is not a disk, for all E > µ ρ D ( E ) < µ Ω ( E ). In E = E (Ω)we obtain µ ρ D (cid:0) E (Ω) (cid:1) < E ( ρ D ) < E (Ω) whichis precisely Conjecture 1. (cid:3) Link with the Bossel-Daners inequality.
The first eigenvalue of the RobinLaplacian with positive parameter
E > λ ΩRob ( E ), isgiven by the variational characterization λ ΩRob ( E ) := inf u ∈ C ∞ (Ω) \{ } (cid:107)∇ u (cid:107) L (Ω) + E (cid:82) ∂ Ω | u | ds (cid:107) u (cid:107) L (Ω) and the Bossel-Daners inequality states that λ ΩRob ( E ) ≥ π | Ω | λ D Rob (cid:16)(cid:114) | Ω | π E (cid:17) , (31)with equality if and only if Ω is a disk. Note that the structure of (31) is similar tothat of Conjecture 40 and it turns out they are intimately connected. This is thepurpose of the following proposition. Proposition 41.
Conjecture 1 implies the Bossel-Daners inequality (31) . Proof . As Conjecture 1 is equivalent to Conjecture 40 as discussed in § E >
0, if u ∈ dom( H D E ) is a normalized eigenfunction associated with µ D ( E ) then u can bepicked real-valued. Hence, we get µ D ( E ) = inf v ∈ C ∞ ( D , R ) (cid:107)∇ v (cid:107) L ( D ) − E (cid:107) v (cid:107) L ( D ) + (cid:82) ∂ D | v | ds (cid:107) v (cid:107) L ( D ) = λ D Rob ( E ) − E . (32)Now, we remark that for any domain Ω there holds λ ΩRob ( E ) − E = inf v ∈ C ∞ (Ω , R ) \{ } (cid:107)∇ v (cid:107) L (Ω) − E (cid:107) v (cid:107) L (Ω) + E (cid:82) ∂ Ω | v | ds (cid:107) v (cid:107) L (Ω) = inf v ∈ C ∞ (Ω , R ) \{ } (cid:107) ∂ ¯ z v (cid:107) L (Ω) − E (cid:107) v (cid:107) L (Ω) + E (cid:82) ∂ Ω | v | ds (cid:107) v (cid:107) L (Ω) ≥ inf v ∈ dom( q Ω E )) \{ } (cid:107) ∂ ¯ z v (cid:107) L (Ω) − E (cid:107) v (cid:107) L (Ω) + E (cid:82) ∂ Ω | v | ds (cid:107) v (cid:107) L (Ω) = µ Ω ( E ) . (33)Hence, using (32) and (33), we get λ ΩRob ( E ) − E ≥ µ Ω ( E ) ≥ π | Ω | µ D (cid:16)(cid:114) | Ω | π E (cid:17) = π | Ω | λ D Rob (cid:16)(cid:114) | Ω | π E (cid:17) − E . If Ω is a disk, all the above inequalities are equalities. Else, we obtain λ ΩRob ( E ) > π | Ω | λ D Rob (cid:16)(cid:114) | Ω | π E (cid:17) , which is precisely the Bossel-Daners inequality (31). (cid:3) Numerics
The goal of this section is to illustrate numerically some theoretical results dis-cussed in the previous sections and to support the validity of Conjecture (1).In § § Numerical Methods.
In this paragraph we present a brief description of thenumerical methods that we use in this work.We have implemented two different numerical approaches, respectively to calcu-late the eigenvalues of the Dirac operator with infinite mass boundary conditions,directly from the formulation of the eigenvalue problem and to solve the mini-mization problem associated with the non-linear variational characterization (3),defining µ Ω ( E ).The eigenvalues of the Dirac operator with infinite mass boundary conditionsare calculated using a numerical method based on Radial Basis Functions (RBF)(see eg . [24, 21]). We have chosen a set of RBF centers y , ..., y N ∈ R , for some N ∈ N , which are generated by a node repel algorithm (see [2] for details). Theeigenfunction u = ( u , u ) (cid:62) is defined in H (Ω , C ) and we use the notation u = v + iw and u = v + iw , where v , w and v , w are the real and imaginaryparts of u and u , respectively. The RBF numerical approximation for each ofthese functions is defined by v ( x ) = (cid:80) Nj =1 α (1) j φ j ( x ) , w ( x ) = (cid:80) Nj =1 β (1) j φ j ( x ) ,v ( x ) = (cid:80) Nj =1 α (2) j φ j ( x ) , w ( x ) = (cid:80) Nj =1 β (2) j φ j ( x ) , (34)where φ j ( x ) = φ ( | x − y j | ), for some function φ : R +0 → R . Several RBF functionscan be considered (eg. [21, 2]), but in this work we consider the multiquadric one φ ( r ) = (cid:112) (cid:15)r ) , for some (cid:15) > − ∂v ∂x + ∂w ∂x + i (cid:16) − ∂v ∂x − ∂w ∂x (cid:17) = E ( v + i w ) in Ω ∂w ∂x + ∂v ∂x + i (cid:16) − ∂v ∂x + ∂w ∂x (cid:17) = E ( v + i w ) in Ω( v + i w ) = i( n + i n )( v + i w ) on ∂ Ωand splitting in real and imaginary parts we have − ∂v ∂x + ∂w ∂x = Ev in Ω − ∂v ∂x − ∂w ∂x = Ew in Ω ∂w ∂x + ∂v ∂x = Ev in Ω − ∂v ∂x + ∂w ∂x = Ew in Ω v = − n w − n v on ∂ Ω w = n v − n w on ∂ Ω (35)These equations are imposed at a discrete set of interior and boundary points.We consider M ∂ Ω ∈ N points p , ..., p M ∂ Ω uniformly distributed on ∂ Ω and M Ω ∈ N points q , ..., q M Ω located at a grid defined on Ω. Then, we calculate the matrices M Ω = φ ( q ) · · · φ N ( q )... . . . ... φ ( q M Ω ) · · · φ N ( q M Ω ) , M Ω1 = ∂ φ ( q ) · · · ∂ φ N ( q )... . . . ... ∂ φ ( q M Ω ) · · · ∂ φ N ( q M Ω ) , M Ω2 = ∂ φ ( q ) · · · ∂ φ N ( q )... . . . ... ∂ φ ( q M Ω ) · · · ∂ φ N ( q M Ω ) , M ∂ Ω = φ ( p ) · · · φ N ( p )... . . . ... φ ( p M ∂ Ω ) · · · φ N ( p M ∂ Ω ) . and M ∂ Ω1 = n ( p ) φ ( p ) · · · n ( p ) φ N ( p )... . . . ... n ( p M ∂ Ω ) φ ( p M ∂ Ω ) · · · n ( p M ∂ Ω ) φ N ( p M ∂ Ω ) , M ∂ Ω2 = n ( p ) φ ( p ) · · · n ( p ) φ N ( p )... . . . ... n ( p M ∂ Ω ) φ ( p M ∂ Ω ) · · · n ( p M ∂ Ω ) φ N ( p M ∂ Ω ) Taking into account the definitions of the RBF linear combinations (34), thenumerical approximations for the eigenvalues are the values E for which we havenonzero solutions of the overdetermined system of linear equations = − M Ω2 M Ω1 − M Ω1 − M Ω2 M Ω2 M Ω1 − M Ω1 M Ω2 ∂ Ω2 M ∂ Ω1 M ∂ Ω − M ∂ Ω1 M ∂ Ω2 ∂ Ω − E M Ω Ω Ω
00 0 0 M Ω . α (1) β (1) α (2) β (2) . (36)The numerical solution of the minimization problem associated to the non-linearvariational characterization is obtained directly from (3), defining the function F ( α (1)1 , ..., α (1) N , β (1)1 , ..., β (1) N ) = 4 (cid:82) Ω | ∂ ¯ z u | dx − E (cid:82) Ω | u | dx + E (cid:82) ∂ Ω | u | ds (cid:82) Ω | u | dx that we minimize by a gradient type method. We refer to [2] for details about thenumerical quadratures to approximate the boundary and volume integrals in thedefinition of F .8.2. Numerical Results.
We start by testing our numerical algorithm for thecalculation of the eigenvalues of the Dirac operator with infinite mass boundaryconditions in the case of the unit disk, for which we know that the principal eigen-value E ( D ) is the smallest non-negative solution of the equation J ( µ ) = J ( µ )and we have E ( D ) = 1 . ... In Table 1 we show the absolute errors ofthe numerical approximations for the principal eigenvalue E ( D ), for several choicesof (cid:15) and N and show that the numerical method can be highly accurate, even witha moderate value of N . N=242 N=323 N=402 (cid:15) = 5 4 . × − . × − . × − (cid:15) = 10 1 . × − . × − . × − (cid:15) = 15 4 . × − . × − . × − Table 1.
Absolute errors of the numerical approximations for theprincipal eigenvalue λ ( D ), for several choices of (cid:15) and N .We have computed the principal eigenvalue for 2500 domains (with smoothboundary) randomly generated satisfying | Ω | = π . The corresponding eigenval-ues are plotted in Figure 1, as a function of the perimeter. We observe that theprincipal eigenvalue is minimized for the domain which also minimizes the perime-ter. By the classical isoperimetric inequality it is well know that for fixed area,the perimeter is minimized by the ball. Thus, these numerical results suggest thatthe Faber-Krahn type inequality stated in Conjecture 1 shall hold for the Diracoperator with infinite mass boundary conditions.Next, we present some numerical results for the minimization problem associatedto the non-linear variational characterization (3). Figure 2 shows three domains(denoted by Ω , Ω and Ω ) verifying | Ω i | = π, ( i = 1 , ,
3) to illustrate the Figure 1.
Plot of the principal eigenvalue for 2500 domains (withsmooth boundary) randomly generated satisfying | Ω | = π , as afunction of the perimeter.numerical results that we gathered. In Figure 3 we plot µ Ω i ( E ) , i = 1 , , µ D ( E ). We verify that for all E >
0, we have µ Ω i ( E ) ≥ µ D ( E ) , i = 1 , , Figure 2.
Plots of domains Ω , Ω and Ω .Finally, Figure 4 shows the absolute value (left plots) and argument (right plots)of a (normalized) eigenfunction associated to the principal eigenvalue of the domainsΩ i , i = 1 , ,
3. Remark that the point of maximal modulus seems to be localizedat the incenter of Ω i which is in line with our choice of test function in the proof ofTheorem (3). However, there is absolutely no reason for the associated eigenfunc-tion to be real-valued and this has two consequences. First, Theorem 3 could beimproved if one considers an adequate test function in the domain of the operatorand not only in the form domain as we do. Second, Conjecture (1) can not bereduced to the Bossel-Daners inequality because, contrary to the Robin eigenvalueproblem, there is a priori no reason for an eigenfunction to have a non-constantargument as illustrated in Figure 4. Figure 3.
Plots of µ Ω i , i = 1 , ,
3, together with the curve µ D asa function of the spectral parameter E > Figure 4.
Plots of the absolute value (left plots) and argument(right plots) of the eigenfunction associated to the principal eigen-value of Ω i , i = 1 , , Acknowledgments
The work of R. D. Benguria has been partially supported by FONDECYT (Chile)project 116-0856.R. D. Benguria, V. Lotoreichik and T. Ourmi`eres-Bonafos are very grateful tothe American Institute of Mathematics (AIM) for supporting their participation tothe AIM workshop
Shape optimization with surface interactions in 2019, where thisproject was initiated.T. Ourmi`eres-Bonafos thanks Nicolas Raymond for pointing out that the projec-tors introduced in Definition 19 are named after the famous mathematician G´aborSzeg¨o.
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J. Ration. Mech.Anal. , 354–356, 1954. (P. R. S. Antunes) Secc¸˜ao de Matem´atica, Departamento de Ciˆencias e Tecnologia,Universidade Aberta, Pal´acio Ceia, 1269-001 Lisbon, Portugal, and Grupo de F´ısicaMatema´atica, Faculdade de Ciˆencias, Universidade de Lisboa, Campo Grande, Edif´ıcioC6, P-1749-016 Lisboa, Portugal E-mail address : [email protected] URL : http://webpages.ciencias.ulisboa.pt/~prantunes/ (R. D. Benguria) Instituto de F´ısica, Pontificia Universidad Cat´olica de Chile, Avda.Vicu˜na Mackenna 4860, Santiago, Chile. E-mail address : [email protected] URL : (V. Lotoreichik) Department of Theoretical Physics, Nuclear Physics Institute,Czech Academy of Sciences, 25068 ˇReˇz, Czech Republic. E-mail address : [email protected] URL : http://gemma.ujf.cas.cz/~lotoreichik/ (T. Ourmi`eres-Bonafos) Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Mar-seille, France E-mail address : [email protected] URL ::