Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities
aa r X i v : . [ m a t h . SP ] F e b NONLOCAL ROBIN LAPLACIANS AND SOME REMARKS ONA PAPER BY FILONOV ON EIGENVALUE INEQUALITIES
FRITZ GESZTESY AND MARIUS MITREA
Dedicated with great pleasure to Sergio Albeverio on the occasion of his 70th birthday
Abstract.
The aim of this paper is twofold: First, we characterize an es-sentially optimal class of boundary operators Θ which give rise to self-adjointLaplacians − ∆ Θ , Ω in L (Ω; d n x ) with (nonlocal and local) Robin-type bound-ary conditions on bounded Lipschitz domains Ω ⊂ R n , n ∈ N , n ≥
2. Second,we extend Friedlander’s inequalities between Neumann and Dirichlet Laplacianeigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigen-values associated with bounded Lipschitz domains Ω, following an approachintroduced by Filonov for this type of problems. Introduction
In recent years, there has been a flurry of activity in connection with 2nd-orderelliptic partial differential operators, particularly, Schr¨odinger–type operators onopen domains Ω ⊂ R n , n ∈ N , n ≥
2, with nonempty boundary ∂ Ω, under varioussmoothness assumptions (resp., lack thereof) on Ω, and associated nonlocal Robinboundary conditions. We refer, for instance, to [2], [3], [4], [10], [12], [13], [14], [15],[17], [25], [26], [28], [29], [35], [37], [40], [52], and the literature cited therein.If Ω is minimally smooth, that is, a Lipschitz domain, these Robin-type boundaryconditions are formally of the type ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) ∂ Ω + Θ( u | ∂ Ω ) = 0 (1.1)in appropriate Sobolev spaces on the boundary ∂ Ω, where ν denotes the outwardpointing normal unit vector to ∂ Ω, and Θ is an appropriate self-adjoint operator in L ( ∂ Ω; d n − ω ), with d n − ω the surface measure on ∂ Ω. The boundary conditionin (1.1) is called local and then resembles the familiar classical Robin boundarycondition for smooth domains Ω, if Θ equals the operator of multiplication M θ by an appropriate function θ on the boundary ∂ Ω (cf., e.g., [50]). Otherwise,the boundary condition (1.1) represents a generalized or nonlocal Robin boundarycondition generated by the operator Θ. The case Θ = 0 (resp., θ = 0), of course,corresponds to the case of Neumann boundary conditions on ∂ Ω. The case ofDirichlet boundary conditions on ∂ Ω, that is, the condition u | ∂ Ω = 0 (formallycorresponding to Θ = ∞ , resp., θ = ∞ ) will also play a major role in this paper. Mathematics Subject Classification.
Primary: 35P15, 47A10; Secondary: 35J25, 47A07.
Key words and phrases.
Lipschitz domains, nonlocal Robin Laplacians, spectral analysis, eigen-value inequalities.Based upon work partially supported by the US National Science Foundation under GrantNos. DMS-0400639 and FRG-0456306.
J. Diff. Eq. , 2871–2896 (2009).
Schr¨odinger operators on bounded Lipschitz domains Ω with nonlocal Robinboundary conditions of the form (1.1), have been very recently discussed in greatdetail in [25] and [26], and our treatment of nonlocal Robin Laplacians in this papernaturally builds upon these two papers.In addition to presenting a detailed approach to nonlocal Robin Laplacians onbounded Lipschitz domains, we also present an application to eigenvalue inequal-ities between the associated Robin and Dirichlet Laplacian eigenvalues, extendingFriedlander’s eigenvalue inequalities between Neumann and Dirichlet eigenvaluesfor bounded C -domains [21], employing its extension to very general bounded do-mains due to Filonov [20]. We briefly review the relevant history of these eigenvalueinequalities. We denote by0 = λ N, Ω , < λ N, Ω , ≤ · · · ≤ λ N, Ω ,j ≤ λ N, Ω ,j +1 ≤ · · · (1.2)the eigenvalues for the Neumann Laplacian − ∆ N, Ω in L (Ω; d n x ), listed accordingto their multiplicity. Similarly,0 < λ D, Ω , < λ D, Ω , ≤ · · · ≤ λ D, Ω ,j ≤ λ D, Ω ,j +1 ≤ · · · (1.3)denote the eigenvalues for the Dirichlet Laplacian − ∆ D, Ω in L (Ω; d n x ), againenumerated according to their multiplicity.Then, for any open bounded domain Ω ⊂ R n , the variational formulation of theNeumann and Dirichlet eigenvalue problem (in terms of Rayleigh quotients, cf. [11,Sect. VI.1]) immediately implies the inequalities λ N, Ω ,j ≤ λ D, Ω ,j , j ∈ N . (1.4)Moreover, P´olya [47] proved in 1952 that λ N, Ω , < λ D, Ω , , (1.5)answering a question of Kornhauser and Stakgold [36]. For a two-dimensionalbounded convex domain Ω ⊂ R , with a piecewise C -boundary ∂ Ω, Payne [46]demonstrated in 1955 that λ N, Ω ,j +2 < λ D, Ω ,j , j ∈ N . (1.6)For domains Ω with a C -boundary and ∂ Ω having a nonnegative mean curvature,Aviles [8] showed in 1986 that λ N, Ω ,j +1 < λ D, Ω ,j , j ∈ N . (1.7)This was reproved by Levine and Weinberger [39] in 1986 who also showed that λ N, Ω ,j + n < λ D, Ω ,j , j ∈ N , (1.8)for smooth bounded convex domains Ω, as well as λ N, Ω ,j + n ≤ λ D, Ω ,j , j ∈ N , (1.9)for arbitrary bounded convex domains. In addition, they also proved inequalitiesof the type λ N, Ω ,j + m < λ D, Ω ,j , j ∈ N , for all 1 ≤ m ≤ n under appropriateassumptions on ∂ Ω in [39] (see also [38]). For additional eigenvalue inequalities werefer to Friedlander [22], [23].In 1991, and most relevant to our paper, Friedlander [21] proved that actually λ N, Ω ,j +1 ≤ λ D, Ω ,j , j ∈ N , (1.10)for any bounded domain Ω with a C -boundary ∂ Ω. We also refer to Mazzeo [43]for an extension to certain smooth manifolds, and to Ashbaugh and Levine [5]
OME REMARKS ON A PAPER BY FILONOV 3 and Hsu and Wang [32] for the case of subdomains of the n -dimensional sphere S n with a smooth boundary and nonnegative mean curvature. (For intriguingconnections between these eigenvalue inequalities with the null variety of the Fouriertransform of the characteristic function of the domain Ω, we also refer to [9].)Finally, inequality (1.10) was extended to any open domain Ω with finite volume,and with the embedding H (Ω) ֒ → L (Ω; d n x ) compact, by Filonov [20] in 2004,who also proved strict inequality in (1.10), that is, λ N, Ω ,j +1 < λ D, Ω ,j , j ∈ N . (1.11)We emphasize that Filonov’s conditions on Ω are equivalent to − ∆ N, Ω , defined asthe unique self-adjoint operator associated with the Neumann sesquilinear form in L (Ω; d n x ), a N ( u, v ) = Z Ω d n x ( ∇ u )( x ) · ( ∇ v )( x ) , u, v ∈ H (Ω) , (1.12)having a purely discrete spectrum, that is, σ ess ( − ∆ N, Ω ) = ∅ (1.13)(cf. also our discussion in Lemmas 2.1, 2.2), where σ ess ( · ) abbreviates the essentialspectrum. While Friedlander used techniques based on the Dirichlet-to-Neumannmap and an appropriate trial function argument, Filonov found an elementary newproof directly based on eigenvalue counting functions (and the same trial functions).Friedlander’s result (1.10) was recently reconsidered by Arendt and Mazzeo [2],which in turn motivated our present investigation into an extension of Filonov’sresult (1.11) to nonlocal Robin Laplacians − ∆ Θ , Ω . In fact, if λ Θ , Ω , ≤ λ Θ , Ω , ≤ · · · ≤ λ Θ , Ω ,j ≤ λ Θ , Ω ,j +1 ≤ · · · , (1.14)denote the eigenvalues of the nonlocal Robin Laplacian − ∆ Θ , Ω , counting multiplic-ity, we will prove that λ Θ , Ω ,j +1 < λ D, Ω ,j , j ∈ N , (1.15)assuming appropriate hypotheses on Θ, including, for instance,Θ ≤ h f, Θ f i / ≤ f ∈ H / ( ∂ Ω). Here, h · , · i / denotes theduality pairing between H / ( ∂ Ω) and H − / ( ∂ Ω) = (cid:0) H / ( ∂ Ω) (cid:1) ∗ . Filonov’s resultwas recently generalized to the Heisenberg Laplacian on certain three-dimensionaldomains by Hansson [31].Most recently, the relation between the eigenvalue counting functions of theDirichlet and Neumann Laplacian originally established by Friedlander in [21], wasdiscussed in an abstract setting by Safarov [49] based on sequilinear forms andan abstract version of the Dirichlet-to-Neumann map. When applied to ellipticboundary value problems, his approach avoids the use of boundary trace opera-tors and hence is not plagued by the usual regularity hypotheses on the boundary(such as Lipschitz boundaries or additional smoothness of the boundary). In par-ticular, Safarov’s approach permits the existence of an essential spectrum of theNeumann (resp., Robin) and Dirichlet Laplacians and then restricts the eigenvalueinequalities of the type (1.10) to those Dirichlet eigenvalues lying strictly beyondinf (cid:0) σ ess ( − ∆ Θ , Ω ) (cid:1) . Hence, Safarov’s results appear to be in the nature of best pos-sible in this context. In addition, as pointed out at the end in Remark 5.5, Safarov’snovel approach considerably improves upon conditions such as (1.16). F. GESZTESY AND M. MITREA
Condition (1.16) was anticipated by Filonov in the special case of local RobinLaplacians − ∆ M θ , Ω , where M θ equals the operator of multiplication by an appropri-ate real-valued function θ on the boundary ∂ Ω. The case of local Robin Laplacians − ∆ M θ , Ω associated with C ,α -domains Ω ⊂ R n , α ∈ (0 , n − h ( ξ ) ≥ θ ( ξ ), ξ ∈ ∂ Ω, h ( · ) the mean curvature on ∂ Ω,he established λ M θ , Ω ,j +1 < λ D, Ω ,j , j ∈ N . (1.17)He also proved λ M θ , Ω ,j + n < λ D, Ω ,j , j ∈ N , (1.18)under the additional assumption of convexity of Ω. (In addition, he derived inequal-ities of the type λ Θ , Ω ,j + m < λ D, Ω ,j , j ∈ N , for all 1 ≤ m ≤ n , under appropriateconditions on Ω.) Similarly, in the case of local Robin Laplacians − ∆ M θ , Ω onsmooth domains Ω ⊂ S n and ( n − h ( ξ ) ≥ θ ( ξ ), ξ ∈ ∂ Ω, Ashbaugh and Levine [5]proved λ M θ , Ω ,j +1 ≤ λ D, Ω ,j , j ∈ N , in 1997.We conclude this introduction with a brief description of the content of eachsection: Section 2 succinctly reviews the basic facts on sesquilinear forms and theirassociated self-adjoint operators. Sobolev spaces on bounded Lipschitz domainsand on their boundaries are presented in a nutshell in Section 3. Section 4 focuseson self-adjoint realizations of Laplacians with nonlocal Robin boundary conditions,and finally, Section 5 discusses the extension of Friedlander’s eigenvalue inequalitiesbetween Neumann and Dirichlet eigenvalues to that of nonlocal Robin eigenvaluesand Dirichlet eigenvalues for bounded Lipschitz domains, closely following a strat-egy of proof due to Filonov.2. Sesquilinear Forms and Associated Operators
In this section we describe a few basic facts on sesquilinear forms and linearoperators associated with them. Let H be a complex separable Hilbert space withscalar product ( · , · ) H (antilinear in the first and linear in the second argument), V a reflexive Banach space continuously and densely embedded into H . Then also H embeds continuously and densely into V ∗ . That is, V ֒ → H ֒ → V ∗ . (2.1)Here the continuous embedding H ֒ → V ∗ is accomplished via the identification H ∋ v ( · , v ) H ∈ V ∗ , (2.2)and we use the convention in this manuscript that if X denotes a Banach space, X ∗ denotes the adjoint space of continuous conjugate linear functionals on X , alsoknown as the conjugate dual of X .In particular, if the sesquilinear form V h · , · i V ∗ : V × V ∗ → C (2.3)denotes the duality pairing between V and V ∗ , then V h u, v i V ∗ = ( u, v ) H , u ∈ V , v ∈ H ֒ → V ∗ , (2.4)that is, the V , V ∗ pairing V h · , · i V ∗ is compatible with the scalar product ( · , · ) H in H .Let T ∈ B ( V , V ∗ ). Since V is reflexive, ( V ∗ ) ∗ = V , one has T : V → V ∗ , T ∗ : V → V ∗ (2.5) OME REMARKS ON A PAPER BY FILONOV 5 and V h u, T v i V ∗ = V ∗ h T ∗ u, v i ( V ∗ ) ∗ = V ∗ h T ∗ u, v i V = V h v, T ∗ u i V ∗ . (2.6) Self-adjointness of T is then defined by T = T ∗ , that is, V h u, T v i V ∗ = V ∗ h T u, v i V = V h v, T u i V ∗ , u, v ∈ V , (2.7) nonnegativity of T is defined by V h u, T u i V ∗ ≥ , u ∈ V , (2.8)and boundedness from below of T by c T ∈ R is defined by V h u, T u i V ∗ ≥ c T k u k H , u ∈ V . (2.9)(By (2.4), this is equivalent to V h u, T u i V ∗ ≥ c T V h u, u i V ∗ , u ∈ V .)Next, let the sesquilinear form a ( · , · ) : V × V → C (antilinear in the first andlinear in the second argument) be V -bounded , that is, there exists a c a > | a ( u, v ) | c a k u k V k v k V , u, v ∈ V . (2.10)Then e A defined by e A : ( V → V ∗ ,v e Av = a ( · , v ) , (2.11)satisfies e A ∈ B ( V , V ∗ ) and V (cid:10) u, e Av (cid:11) V ∗ = a ( u, v ) , u, v ∈ V . (2.12)Assuming further that a ( · , · ) is symmetric , that is, a ( u, v ) = a ( v, u ) , u, v ∈ V , (2.13)and that a is V -coercive , that is, there exists a constant C > a ( u, u ) ≥ C k u k V , u ∈ V , (2.14)respectively, then, e A : V → V ∗ is bounded, self-adjoint, and boundedly invertible. (2.15)Moreover, denoting by A the part of e A in H defined bydom( A ) = (cid:8) u ∈ V | e Au ∈ H (cid:9) ⊆ H , A = e A (cid:12)(cid:12) dom( A ) : dom( A ) → H , (2.16)then A is a (possibly unbounded) self-adjoint operator in H satisfying A ≥ C I H , (2.17)dom (cid:0) A / (cid:1) = V . (2.18)In particular, A − ∈ B ( H ) . (2.19)The facts (2.1)–(2.19) are a consequence of the Lax–Milgram theorem and thesecond representation theorem for symmetric sesquilinear forms. Details can befound, for instance, in [16, Sects. VI.3, VII.1], [18, Ch. IV], and [41].Next, consider a symmetric form b ( · , · ) : V × V → C and assume that b is bounded from below by c b ∈ R , that is, b ( u, u ) ≥ c b k u k H , u ∈ V . (2.20)Introducing the scalar product ( · , · ) V b : V ×V → C (and the associated norm k·k V b )by ( u, v ) V b = b ( u, v ) + (1 − c b )( u, v ) H , u, v ∈ V , (2.21) F. GESZTESY AND M. MITREA turns V into a pre-Hilbert space ( V ; ( · , · ) V b ), which we denote by V b . The form b is called closed in H if V b is actually complete, and hence a Hilbert space. Theform b is called closable in H if it has a closed extension. If b is closed in H , then | b ( u, v ) + (1 − c b )( u, v ) H | k u k V b k v k V b , u, v ∈ V , (2.22)and | b ( u, u ) + (1 − c b ) k u k H | = k u k V b , u ∈ V , (2.23)show that the form b ( · , · ) + (1 − c b )( · , · ) H is a symmetric, V -bounded, and V -coercive sesquilinear form. Hence, by (2.11) and (2.12), there exists a linear map e B c b : ( V b → V ∗ b ,v e B c b v = b ( · , v ) + (1 − c b )( · , v ) H , (2.24)with e B c b ∈ B ( V b , V ∗ b ) and V b (cid:10) u, e B c b v (cid:11) V ∗ b = b ( u, v ) + (1 − c b )( u, v ) H , u, v ∈ V . (2.25)Introducing the linear map e B = e B c b + ( c b − e I : V b → V ∗ b , (2.26)where e I : V b ֒ → V ∗ b denotes the continuous inclusion (embedding) map of V b into V ∗ b , one obtains a self-adjoint operator B in H by restricting e B to H ,dom( B ) = (cid:8) u ∈ V (cid:12)(cid:12) e Bu ∈ H (cid:9) ⊆ H , B = e B (cid:12)(cid:12) dom( B ) : dom( B ) → H , (2.27)satisfying the following properties: B ≥ c b I H , (2.28)dom (cid:0) | B | / (cid:1) = dom (cid:0) ( B − c b I H ) / (cid:1) = V , (2.29) b ( u, v ) = (cid:0) | B | / u, U B | B | / v (cid:1) H (2.30)= (cid:0) ( B − c b I H ) / u, ( B − c b I H ) / v (cid:1) H + c b ( u, v ) H (2.31)= V b (cid:10) u, e Bv (cid:11) V ∗ b , u, v ∈ V , (2.32) b ( u, v ) = ( u, Bv ) H , u ∈ V , v ∈ dom( B ) , (2.33)dom( B ) = { v ∈ V | there exists an f v ∈ H such that b ( w, v ) = ( w, f v ) H for all w ∈ V} , (2.34) Bu = f u , u ∈ dom( B ) , dom( B ) is dense in H and in V b . (2.35)Properties (2.34) and (2.35) uniquely determine B . Here U B in (2.31) is the partialisometry in the polar decomposition of B , that is, B = U B | B | , | B | = ( B ∗ B ) / ≥ . (2.36)The operator B is called the operator associated with the form b .The norm in the Hilbert space V ∗ b is given by k ℓ k V ∗ b = sup {| V b h u, ℓ i V ∗ b | | k u k V b } , ℓ ∈ V ∗ b , (2.37)with associated scalar product,( ℓ , ℓ ) V ∗ b = V b (cid:10)(cid:0) e B + (1 − c b ) e I (cid:1) − ℓ , ℓ (cid:11) V ∗ b , ℓ , ℓ ∈ V ∗ b . (2.38) OME REMARKS ON A PAPER BY FILONOV 7
Since (cid:13)(cid:13)(cid:0) e B + (1 − c b ) e I (cid:1) v (cid:13)(cid:13) V ∗ b = k v k V b , v ∈ V , (2.39)the Riesz representation theorem yields (cid:0) e B + (1 − c b ) e I (cid:1) ∈ B ( V b , V ∗ b ) and (cid:0) e B + (1 − c b ) e I (cid:1) : V b → V ∗ b is unitary. (2.40)In addition, V b (cid:10) u, (cid:0) e B + (1 − c b ) e I (cid:1) v (cid:11) V ∗ b = (cid:0)(cid:0) B + (1 − c b ) I H (cid:1) / u, (cid:0) B + (1 − c b ) I H (cid:1) / v (cid:1) H = ( u, v ) V b , u, v ∈ V b . (2.41)In particular, (cid:13)(cid:13) ( B + (1 − c b ) I H ) / u (cid:13)(cid:13) H = k u k V b , u ∈ V b , (2.42)and hence( B +(1 − c b ) I H ) / ∈ B ( V b , H ) and ( B +(1 − c b ) I H ) / : V b → H is unitary. (2.43)The facts (2.20)–(2.43) comprise the second representation theorem of sesquilinearforms (cf. [18, Sect. IV.2], [19, Sects. 1.2–1.5], and [34, Sect. VI.2.6]).A special but important case of nonnegative closed forms is obtained as fol-lows: Let H j , j = 1 ,
2, be complex separable Hilbert spaces, and T : dom( T ) →H , dom( T ) ⊆ H , a densely defined operator. Consider the nonnegative form a T : dom( T ) × dom( T ) → C defined by a T ( u, v ) = ( T u, T v ) H , u, v ∈ dom( T ) . (2.44)Then the form a T is closed (resp., closable) in H if and only if T is. If T isclosed, the unique nonnegative self-adjoint operator associated with a T in H , whoseexistence is guaranteed by the second representation theorem for forms, then equals T ∗ T ≥
0. In particular, one obtains in addition to (2.44), a T ( u, v ) = ( | T | u, | T | v ) H , u, v ∈ dom( T ) = dom( | T | ) . (2.45)Moreover, since b ( u, v ) + (1 − c b )( u, v ) H = (cid:0) ( B + (1 − c b ) I H ) / u, ( B + (1 − c b ) I H ) / v (cid:1) H ,u, v ∈ dom( b ) = dom (cid:0) | B | / (cid:1) = V , (2.46)and ( B + (1 − c b ) I H ) / is self-adjoint (and hence closed) in H , a symmetric, V -bounded, and V -coercive form is densely defined in H × H and closed in H (a factwe will be using in the proof of Theorem 4.5). We refer to [34, Sect. VI.2.4] and[53, Sect. 5.5] for details.Next we recall that if a j are sesquilinear forms defined on dom( a j ), j = 1 , a + a ) : ( (dom( a ) ∩ dom( a )) × (dom( a ) ∩ dom( a )) → C , ( u, v ) ( a + a )( u, v ) = a ( u, v ) + a ( u, v ) (2.47)is bounded from below and closed (cf., e.g., [34, Sect. VI.1.6]).Finally, we also recall the following perturbation theoretic fact: Suppose a is asesquilinear form defined on V × V , bounded from below and closed, and let b be asymmetric sesquilinear form bounded with respect to a with bound less than one,that is, dom( b ) ⊇ V × V , and that there exist 0 α < β > | b ( u, u ) | α | a ( u, u ) | + β k u k H , u ∈ V . (2.48) F. GESZTESY AND M. MITREA
Then ( a + b ) : ( V × V → C , ( u, v ) ( a + b )( u, v ) = a ( u, v ) + b ( u, v ) (2.49)defines a sesquilinear form that is bounded from below and closed (cf., e.g., [34,Sect. VI.1.6]). In the special case where α can be chosen arbitrarily small, the form b is called infinitesimally form bounded with respect to a .Finally we turn to a brief discussion of operators with purely discrete spectra.We denote by S the cardinality of the set S . Lemma 2.1.
Let V , H be as in (2.1) , (2.2) . Assume that the inclusion ι V : V ֒ → H is compact, and that the sesquilinear form a ( · , · ) : V × V → C is symmetric, V -bounded, and suppose that there exists κ > with the property that a κ ( u, v ) := a ( u, v ) + κ ( u, v ) H , u, v ∈ V , (2.50) is V -coercive. Then the operator A associated with a ( · , · ) is self-adjoint and boundedfrom below. In addition, A has purely discrete spectrum σ ess ( A ) = ∅ , (2.51) and hence σ ( A ) = { λ j ( A ) } j ∈ N , with λ j ( A ) → ∞ as j → ∞ , − κ < λ ( A ) ≤ λ ( A ) ≤ · · · ≤ λ j ( A ) ≤ λ j +1 ( A ) ≤ · · · . (2.52) Here, the eigenvalues λ j ( A ) of A are listed according to their multiplicity. Moreover,the following min-max principle holds: λ j ( A ) = min Lj subspace of V dim( L j )= j (cid:16) max = u ∈ L j R a [ u ] (cid:17) , j ∈ N , (2.53) where R a [ u ] denotes the Rayleigh quotient R a [ u ] := a ( u, u ) k u k H , = u ∈ V . (2.54) As a consequence, if N A is the eigenvalue counting function of A , that is, N A ( λ ) := { j ∈ N | λ j ( A ) ≤ λ } , λ ∈ R , (2.55) then for each λ ∈ R one has N A ( λ ) = max (cid:8) dim( L ) ∈ N (cid:12)(cid:12) L a subspace of V with a ( u, u ) ≤ λ k u k H , u ∈ L (cid:9) . (2.56) Proof.
Analogous claims for the operator B associated with the V -coercive form a κ ( · , · ) are well-known (cf., e.g., [16, Sect. VI.3.2.5, Ch. VII]). Then the corre-sponding claims for A follow from these, after observing that B = A + κI H , R a κ [ u ] = R a [ u ] + κ , λ j ( B ) = λ j ( A ) + κ , and N B ( λ ) = N A ( λ − κ ), λ ∈ R . (cid:3) A closely related result is provided by the following elementary observations: Let c ∈ R and B ≥ cI H be a self-adjoint operator in H , and introduce the sesquilinearform b in H associated with B via b ( u, v ) = (cid:0) ( B − cI H ) / u, ( B − cI H ) / v (cid:1) H + c ( u, v ) H ,u, v ∈ dom( b ) = dom (cid:0) | B | / (cid:1) . (2.57)Given B and b , one introduces the Hilbert space H b ⊆ H by H b = (cid:0) dom (cid:0) | B | / (cid:1) , ( · , · ) H b (cid:1) , OME REMARKS ON A PAPER BY FILONOV 9 ( u, v ) H b = b ( u, v ) + (1 − c )( u, v ) H (2.58)= (cid:0) ( B − cI H ) / u, ( B − cI H ) / v (cid:1) H + ( u, v ) H = (cid:0) ( B + (1 − c ) I H ) / u, ( B + (1 − c ) I H ) / v (cid:1) H . Of course, H b plays a role analogous to V b in (2.21). As in (2.43) one then observesthat ( B + (1 − c ) I H ) / : H b → H is unitary. (2.59) Lemma 2.2.
Let H , B , b , and H b be as in (2.57) – (2.59) . Then B has purelydiscrete spectrum, that is, σ ess ( B ) = ∅ , if and only if H b ֒ → H compactly.Proof. Denoting by J H b = I H | H b the inclusion map from H b into H , one infers that H ( B +(1 − c ) I H ) − / −−−−−−−−−−−−→ H b J H b ֒ → H . (2.60)Thus, one concludes that H b ֒ → H compactly ⇐⇒ J H b ∈ B ∞ ( H b , H ) ⇐⇒ (cid:2) J H b ( B + (1 − c ) I H ) − / (cid:3) ( B + (1 − c ) I H ) / ∈ B ∞ ( H b , H ) ⇐⇒ J H b ( B + (1 − c ) I H ) − / ∈ B ∞ ( H ) ⇐⇒ ( B + (1 − c ) I H ) − / ∈ B ∞ ( H ) ⇐⇒ ( B − zI H ) − ∈ B ∞ ( H ) , z ∈ C \ σ ( B ) ⇐⇒ σ ess ( B ) = ∅ , (2.61)since ( B + (1 − c b ) I H ) / : H b → H is unitary by (2.59). (cid:3) Throughout this paper we are employing the following notation: The Banachspaces of bounded and compact linear operators on a Hilbert space H are denotedby B ( H ) and B ∞ ( H ), respectively. The analogous notation B ( X , X ), B ∞ ( X , X ),etc., will be used for bounded and compact operators between two Banach spaces X and X . Moreover, X ֒ → X denotes the continuous embedding of the Banachspace X into the Banach space X .3. Sobolev Spaces in Lipschitz Domains
The goal of this section is to introduce the relevant material pertaining to Sobolevspaces H s (Ω) and H r ( ∂ Ω) corresponding to subdomains Ω of R n , n ∈ N , and discussvarious trace results.We start by recalling some basic facts in connection with Sobolev spaces corre-sponding to open subsets Ω ⊂ R n , n ∈ N . For an arbitrary m ∈ N ∪ { } , we followthe customary way of defining L -Sobolev spaces of order ± m in Ω as H m (Ω) := (cid:8) u ∈ L (Ω; d n x ) (cid:12)(cid:12) ∂ α u ∈ L (Ω; d n x ) for 0 ≤ | α | ≤ m (cid:9) , (3.1) H − m (Ω) := (cid:26) u ∈ D ′ (Ω) (cid:12)(cid:12)(cid:12)(cid:12) u = X | α |≤ m ∂ α u α , with u α ∈ L (Ω; d n x ) , ≤ | α | ≤ m (cid:27) , (3.2)equipped with natural norms (cf., e.g., [1, Ch. 3], [42, Ch. 1]). Here D ′ (Ω) denotesthe usual set of distributions on Ω ⊆ R n . Then one sets H m (Ω) := the closure of C ∞ (Ω) in H m (Ω) , m ∈ N ∪ { } . (3.3) As is well-known, all three spaces above are Banach, reflexive and, in addition, (cid:0) H m (Ω) (cid:1) ∗ = H − m (Ω) . (3.4)Again, see, for instance, [1, Ch. 3], [42, Sect. 1.1.14]. Throughout this paper, weagree to use the adjoint (rather than the dual) space X ∗ of a Banach space X .One recalls that an open, nonempty, bounded set Ω ⊂ R n is called a boundedLipschitz domain if the following property holds: There exists an open covering {O j } ≤ j ≤ N of the boundary ∂ Ω of Ω such that for every j ∈ { , ..., N } , O j ∩ Ωcoincides with the portion of O j lying in the over-graph of a Lipschitz function ϕ j : R n − → R (considered in a new system of coordinates obtained from the originalone via a rigid motion). The number max {k∇ ϕ j k L ∞ ( R n − ; d n − x ′ ) | ≤ j ≤ N } issaid to represent the Lipschitz character of Ω.The classical theorem of Rademacher of almost everywhere differentiability ofLipschitz functions ensures that, for any Lipschitz domain Ω, the surface measure d n − ω is well-defined on ∂ Ω and that there exists an outward pointing normalvector ν at almost every point of ∂ Ω.In the remainder of this paper we shall make the following assumption:
Hypothesis 3.1.
Let n ∈ N , n ≥ , and assume that Ω ⊂ R n is a bounded Lipschitzdomain. As regards L -based Sobolev spaces of fractional order s ∈ R , in a bounded Lipschitz domain Ω ⊂ R n we set H s ( R n ) := (cid:26) U ∈ S ′ ( R n ) (cid:12)(cid:12)(cid:12)(cid:12) k U k H s ( R n ) = Z R n d n ξ (cid:12)(cid:12) b U ( ξ ) (cid:12)(cid:12) (cid:0) | ξ | s (cid:1) < ∞ (cid:27) , (3.5) H s (Ω) := (cid:8) u ∈ D ′ (Ω) (cid:12)(cid:12) u = U | Ω for some U ∈ H s ( R n ) (cid:9) . (3.6)Here S ′ ( R n ) is the space of tempered distributions on R n , and b U denotes theFourier transform of U ∈ S ′ ( R n ). These definitions are consistent with (3.1)–(3.2).Moreover, so is H s (Ω) := (cid:8) u ∈ H s ( R n ) (cid:12)(cid:12) supp( u ) ⊆ Ω (cid:9) , s ∈ R , (3.7)equipped with the natural norm induced by H s ( R n ), in relation to (3.3). One alsohas (cid:0) H s (Ω) (cid:1) ∗ = H − s (Ω) , s ∈ R (3.8)(cf., e.g., [33]). For a bounded Lipschitz domain Ω ⊂ R n it is known that (cid:0) H s (Ω) (cid:1) ∗ = H − s (Ω) , − / < s < / . (3.9)See [51] for this and other related properties.To discuss Sobolev spaces on the boundary of a Lipschitz domain, consider firstthe case when Ω ⊂ R n is the domain lying above the graph of a Lipschitz function ϕ : R n − → R . In this setting, we define the Sobolev space H s ( ∂ Ω) for 0 ≤ s ≤ f ∈ L ( ∂ Ω; d n − ω ) with the property that f ( x ′ , ϕ ( x ′ )),as a function of x ′ ∈ R n − , belongs to H s ( R n − ). This definition is easily adaptedto the case when Ω is a Lipschitz domain whose boundary is compact, by using asmooth partition of unity. Finally, for − ≤ s ≤
0, we set H s ( ∂ Ω) = (cid:0) H − s ( ∂ Ω) (cid:1) ∗ , − s . (3.10)From the above characterization of H s ( ∂ Ω) it follows that any property of Sobolevspaces (of order s ∈ [ − , OME REMARKS ON A PAPER BY FILONOV 11 multiplication by smooth, compactly supported functions as well as compositionsby bi-Lipschitz diffeomorphisms, readily extends to the setting of H s ( ∂ Ω) (vialocalization and pull-back). As a concrete example, for each Lipschitz domain Ωwith compact boundary, one has H s ( ∂ Ω) ֒ → L ( ∂ Ω; d n − ω ) compactly if 0 < s ≤ . (3.11)For additional background information in this context we refer, for instance, to [6],[7], [18, Chs. V, VI], [30, Ch. 1], [44, Ch. 3], [54, Sect. I.4.2].Assuming Hypothesis 3.1, we introduce the boundary trace operator γ D (theDirichlet trace) by γ D : C (Ω) → C ( ∂ Ω) , γ D u = u | ∂ Ω . (3.12)Then there exists a bounded linear operator γ D γ D : H s (Ω) → H s − (1 / ( ∂ Ω) ֒ → L ( ∂ Ω; d n − ω ) , / < s < / ,γ D : H / (Ω) → H − ε ( ∂ Ω) ֒ → L ( ∂ Ω; d n − ω ) , ε ∈ (0 ,
1) (3.13)(cf., e.g., [44, Theorem 3.38]), whose action is compatible with that of γ D . Thatis, the two Dirichlet trace operators coincide on the intersection of their domains.Moreover, we recall that γ D : H s (Ω) → H s − (1 / ( ∂ Ω) is onto for 1 / < s < / . (3.14)Next, retaining Hypothesis 3.1, we introduce the operator γ N (the strong Neu-mann trace) by γ N = ν · γ D ∇ : H s +1 (Ω) → L ( ∂ Ω; d n − ω ) , / < s < / , (3.15)where ν denotes the outward pointing normal unit vector to ∂ Ω. It follows from(3.13) that γ N is also a bounded operator. We seek to extend the action of theNeumann trace operator (3.15) to other (related) settings. To set the stage, assumeHypothesis 3.1 and observe that the inclusion ι : H s (Ω) ֒ → (cid:0) H r (Ω) (cid:1) ∗ , s > − / , r > / , (3.16)is well-defined and bounded. We then introduce the weak Neumann trace operator e γ N : (cid:8) u ∈ H s +1 / (Ω) (cid:12)(cid:12) ∆ u ∈ H s (Ω) (cid:9) → H s − ( ∂ Ω) , s ∈ (0 , , s > − / , (3.17)as follows: Given u ∈ H s +1 / (Ω) with ∆ u ∈ H s (Ω) for some s ∈ (0 ,
1) and s > − /
2, we set (with ι as in (3.16) for r := 3 / − s > / h φ, e γ N u i − s = H / − s (Ω) h∇ Φ , ∇ u i ( H / − s (Ω)) ∗ + H / − s (Ω) h Φ , ι (∆ u ) i ( H / − s (Ω)) ∗ , (3.18)for all φ ∈ H − s ( ∂ Ω) and Φ ∈ H / − s (Ω) such that γ D Φ = φ . We note that thefirst pairing on the right-hand side of (3.18) is meaningful since (cid:0) H / − s (Ω) (cid:1) ∗ = H s − / (Ω) , s ∈ (0 , , (3.19)and that the definition (3.18) is independent of the particular extension Φ of φ ,and that e γ N is a bounded extension of the Neumann trace operator γ N defined in(3.15). Laplace Operators with Nonlocal Robin-TypeBoundary Conditions
In this section we primarily focus on various properties of general Laplacians − ∆ Θ , Ω in L (Ω; d n x ) including Dirichlet, − ∆ D, Ω , and Neumann, − ∆ N, Ω , Lapla-cians, nonlocal Robin-type Laplacians, and Laplacians corresponding to classicalRobin boundary conditions associated with bounded Lipschitz domains Ω ⊂ R n .For simplicity of notation we will denote the identity operators in L (Ω; d n x )and L ( ∂ Ω; d n − ω ) by I Ω and I ∂ Ω , respectively. Also, in the sequel, the sesquilinearform h · , · i s = H s ( ∂ Ω) h · , · i H − s ( ∂ Ω) : H s ( ∂ Ω) × H − s ( ∂ Ω) → C , s ∈ [0 , , (4.1)(antilinear in the first, linear in the second factor), will denote the duality pairingbetween H s ( ∂ Ω) and H − s ( ∂ Ω) = (cid:0) H s ( ∂ Ω) (cid:1) ∗ , s ∈ [0 , , (4.2)such that h f, g i s = Z ∂ Ω d n − ω ( ξ ) f ( ξ ) g ( ξ ) ,f ∈ H s ( ∂ Ω) , g ∈ L ( ∂ Ω; d n − ω ) ֒ → H − s ( ∂ Ω) , s ∈ [0 , , (4.3)where, as before, d n − ω stands for the surface measure on ∂ Ω.We also recall the notational conventions summarized at the end of Section 2.
Hypothesis 4.1.
Assume Hypothesis , suppose that δ > is a given number,and assume that Θ ∈ B (cid:0) H / ( ∂ Ω) , H − / ( ∂ Ω) (cid:1) is a self-adjoint operator which canbe written as Θ = Θ + Θ + Θ , (4.4) where Θ j , j = 1 , , , have the following properties: There exists a closed sesquilin-ear form a Θ in L ( ∂ Ω; d n − ω ) , with domain H / ( ∂ Ω) × H / ( ∂ Ω) , bounded frombelow by c Θ ∈ R ( hence, a Θ is symmetric ) such that if Θ > c Θ I ∂ Ω denotes theself-adjoint operator in L ( ∂ Ω; d n − ω ) uniquely associated with a Θ ( cf. (2.27)) ,then Θ = e Θ , the extension of Θ to an operator in B (cid:0) H / ( ∂ Ω) , H − / ( ∂ Ω) (cid:1) ( asdiscussed in (2.26) and (2.32)) . In addition, Θ ∈ B ∞ (cid:0) H / ( ∂ Ω) , H − / ( ∂ Ω) (cid:1) , (4.5) whereas Θ ∈ B (cid:0) H / ( ∂ Ω) , H − / ( ∂ Ω) (cid:1) satisfies k Θ k B ( H / ( ∂ Ω) ,H − / ( ∂ Ω)) < δ. (4.6)We recall the following useful result.
Lemma 4.2.
Assume Hypothesis . Then for every ε > there exists a β ( ε ) > with β ( ε ) = ε ↓ O (1 /ε )) such that k γ D u k L ( ∂ Ω; d n − ω ) ε k∇ u k L (Ω; d n x ) n + β ( ε ) k u k L (Ω; d n x ) , u ∈ H (Ω) . (4.7)A proof from which it is possible to read off how the constant β ( ε ) depends onthe Lipschitz character of Ω appears in [25]. Below we discuss a general abstractscheme which yields results of this type, albeit with a less descriptive constant β ( ε ).The lemma below is inspired by [2, Lemma 2.3]: OME REMARKS ON A PAPER BY FILONOV 13
Lemma 4.3.
Let V be a reflexive Banach space, W a Banach space, assume that K ∈ B ∞ ( V , V ∗ ) , and that T ∈ B ( V , W ) is one-to-one. Then for every ε > thereexists C ε > such that (cid:12)(cid:12) V h u, Ku i V ∗ (cid:12)(cid:12) ≤ ε k u k V + C ε k T u k W , u ∈ V . (4.8) Proof.
Seeking a contradiction, assume that there exist ε > u j ∈ V , k u j k V = 1, j ∈ N , for which (cid:12)(cid:12) V h u j , Ku j i V ∗ (cid:12)(cid:12) ≥ ε + j k T u j k W , j ∈ N . (4.9)Furthermore, since V is reflexive, there is no loss of generality in assuming thatthere exists u ∈ V such that u j → u as j → ∞ , weakly in V (cf., e.g., [45,Theorem 1.13.5]). In addition, since T (and hence T ∗ ) is bounded, one concludesthat T u j → T u as j → ∞ weakly in W , as is clear from W h T u j , Λ i W ∗ = V h u j , T ∗ Λ i V ∗ −→ j →∞ V h u, T ∗ Λ i V ∗ = W h T u, Λ i W ∗ , Λ ∈ W ∗ . (4.10)Moreover, since K is compact, we may choose a subsequence of { u j } j ∈ N (still de-noted by { u j } j ∈ N ) such that Ku j → Ku as j → ∞ , strongly in V ∗ . This, in turn,yields that V h u j , Ku j i V ∗ → V h u, Ku i V ∗ as j → ∞ . (4.11)Together with k T u j k W ≤ j − (cid:12)(cid:12) V h u j , Ku j i V ∗ (cid:12)(cid:12) , j ∈ N , (4.12)this also shows that T u j → j → ∞ , in W . Hence, T u = 0 in W whichforces u = 0, since T is one-to-one. Given these facts, we note that, on the onehand, we have V h u j , Ku j i V ∗ → j → ∞ by (4.11), while on the other hand (cid:12)(cid:12)(cid:12) V h u j , Ku j i V ∗ (cid:12)(cid:12)(cid:12) ≥ ε for every j ∈ N by (4.9). This contradiction concludes theproof. (cid:3) Parenthetically, we note that Lemma 4.2 (with a less precise description of theconstant β ( ε )) follows from Lemma 4.3 by taking V := H (Ω) , W := L (Ω , d n x ) , (4.13)and, with γ D ∈ B ∞ (cid:0) H (Ω) , L ( ∂ Ω; d n − ω ) (cid:1) denoting the Dirichlet trace, K := γ ∗ D γ D ∈ B ∞ (cid:0) H (Ω) , (cid:0) H (Ω) (cid:1) ∗ (cid:1) , T := ι : H (Ω) ֒ → L (Ω , d n x ) , (4.14)the inclusion operator. Lemma 4.4.
Assume Hypothesis , where the number δ > is taken to besufficiently small relative to the Lipschitz character of Ω . Consider the sesquilinearform a Θ ( · , · ) defined on H (Ω) × H (Ω) by a Θ ( u, v ) := Z Ω d n x ( ∇ u )( x ) · ( ∇ v )( x ) + (cid:10) γ D u, Θ γ D v (cid:11) / , u, v ∈ H (Ω) . (4.15) Then there exists κ > with the property that the form a Θ ,κ ( u, v ) := a Θ ( u, v ) + κ ( u, v ) L (Ω; d n x ) , u, v ∈ H (Ω) , (4.16) is H (Ω) -coercive.As a consequence, the form (4.15) is symmetric, H (Ω) -bounded, bounded frombelow, and closed in L (Ω; d n x ) . Proof.
We shall show that κ > k u k H (Ω) ≤ Z Ω d n x | ( ∇ u )( x ) | + κ Z Ω d n x | u ( x ) | + (cid:10) γ D u, Θ j γ D v (cid:11) / ,u ∈ H (Ω) , j = 1 , , , (4.17)where Θ j , j = 1 , ,
3, are as introduced in Hypothesis 4.1. Summing up these threeinequalities then proves that the form (4.16) is indeed H (Ω)-coercive. To this end,we assume first j = 1 and recall that there exists c Θ ∈ R such that (cid:10) γ D u, Θ γ D u (cid:11) / ≥ c Θ k γ D u k L ( ∂ Ω; d n − ω ) , u ∈ H (Ω) . (4.18)Thus, in this case, it suffices to show thatmax {− c Θ , } k γ D u k L ( ∂ Ω; d n − ω ) + 16 k u k H (Ω) ≤ Z Ω d n x | ( ∇ u )( x ) | + κ Z Ω d n x | u ( x ) | , u ∈ H (Ω) , (4.19)or, equivalently, thatmax {− c Θ , } k γ D u k L ( ∂ Ω; d n − ω ) ≤ Z Ω d n x | ( ∇ u )( x ) | + 2 κ − Z Ω d n x | u ( x ) | , u ∈ H (Ω) , (4.20)with the usual convention, k u k H (Ω) = k∇ u k L dnx ) n + k u k L dnx ) , u ∈ H (Ω) . (4.21)The fact that there exists κ > j = 2 ,
3, estimate (4.17) is implied by (cid:12)(cid:12)(cid:10) γ D u, Θ j γ D u (cid:11) / (cid:12)(cid:12) ≤ Z Ω d n x | ( ∇ u )( x ) | + 2 κ − Z Ω d n x | u ( x ) | , u ∈ H (Ω) , (4.22)or, equivalently, by (cid:12)(cid:12)(cid:10) γ D u, Θ j γ D u (cid:11) / (cid:12)(cid:12) ≤ k u k H (Ω) + κ − k u k L (Ω; d n x ) , u ∈ H (Ω) . (4.23)When j = 2, in which case Θ ∈ B ∞ (cid:0) H (Ω) , (cid:0) H (Ω) (cid:1) ∗ (cid:1) , we invoke Lemma 4.3with V , W as in (4.13) and, with γ D ∈ B (cid:0) H (Ω) , H / ( ∂ Ω) (cid:1) denoting the Dirichlettrace, K := γ ∗ D Θ γ D ∈ B ∞ (cid:0) H (Ω) , (cid:0) H (Ω) (cid:1) ∗ (cid:1) , T := ι : H (Ω) ֒ → L (Ω , d n x ) , (4.24)the inclusion operator. Then, with ε = 1 / κ := 3 C / + 1, estimate (4.8)yields (4.23) for j = 2.Finally, consider (4.23) in the case where j = 3 and note that by hypothesis, (cid:12)(cid:12)(cid:10) γ D u, Θ γ D u (cid:11) / (cid:12)(cid:12) ≤ k Θ k B ( H / ( ∂ Ω) ,H − / ( ∂ Ω)) k γ D u k H / ( ∂ Ω) ≤ δ k γ D k B ( H (Ω) ,H / ( ∂ Ω)) k u k H (Ω) , u ∈ H (Ω) . (4.25)Thus (4.23) also holds for j = 3 if0 < δ ≤ k γ D k − B ( H (Ω) ,H / ( ∂ Ω)) and κ > . (4.26)This completes the justification of (4.17), and hence finishes the proof. (cid:3) OME REMARKS ON A PAPER BY FILONOV 15
Next, we turn to a discussion of nonlocal Robin Laplacians in bounded Lipschitzsubdomains of R n . Concretely, we describe a family of self-adjoint Laplace operators − ∆ Θ , Ω in L (Ω; d n x ) indexed by the boundary operator Θ. We will refer to − ∆ Θ , Ω as the nonlocal Robin Laplacian. Theorem 4.5.
Assume Hypothesis , where the number δ > is taken to besufficiently small relative to the Lipschitz character of Ω . Then the nonlocal RobinLaplacian, − ∆ Θ , Ω , defined by − ∆ Θ , Ω = − ∆ , (4.27)dom( − ∆ Θ , Ω ) = (cid:8) u ∈ H (Ω) (cid:12)(cid:12) ∆ u ∈ L (Ω; d n x ) , (cid:0)e γ N + Θ γ D (cid:1) u = 0 in H − / ( ∂ Ω) (cid:9) is self-adjoint and bounded from below in L (Ω; d n x ) . Moreover, dom (cid:0) | − ∆ Θ , Ω | / (cid:1) = H (Ω) , (4.28) and − ∆ Θ , Ω , has purely discrete spectrum bounded from below, in particular, σ ess ( − ∆ Θ , Ω ) = ∅ . (4.29) Finally, − ∆ Θ , Ω is the operator uniquely associated with the sesquilinear form a Θ in Lemma .Proof. Denote by a − ∆ Θ , Ω ( · , · ) the sesquilinear form introduced in (4.15). FromLemma 4.4, we know that a − ∆ Θ , Ω is symmetric, H (Ω)-bounded, bounded frombelow, as well as densely defined and closed in L (Ω; d n x ) × L (Ω; d n x ). Thus, ifas in (2.34), we now introduce the operator − ∆ Θ , Ω in L (Ω; d n x ) bydom( − ∆ Θ , Ω ) = (cid:26) v ∈ H (Ω) (cid:12)(cid:12)(cid:12)(cid:12) there exists a w v ∈ L (Ω; d n x ) such that Z Ω d n x ∇ w ∇ v + (cid:10) γ D w, Θ γ D v (cid:11) / = Z Ω d n x ww v for all w ∈ H (Ω) (cid:27) , − ∆ Θ , Ω u = w u , u ∈ dom( − ∆ Θ , Ω ) , (4.30)it follows from (2.20)–(2.43) (cf., in particular (2.27)) that − ∆ Θ , Ω is self-adjointand bounded from below in L (Ω; d n x ) and that (4.28) holds. Next we recall that H (Ω) = (cid:8) u ∈ H (Ω) (cid:12)(cid:12) γ D u = 0 on ∂ Ω (cid:9) . (4.31)Taking v ∈ C ∞ (Ω) ֒ → H (Ω) ֒ → H (Ω), one concludes Z Ω d n x vw u = − Z Ω d n x v ∆ u for all v ∈ C ∞ (Ω), and hence w u = − ∆ u in D ′ (Ω) , (4.32)with D ′ (Ω) = C ∞ (Ω) ′ the space of distributions on Ω. Going further, suppose that u ∈ dom( − ∆ Θ , Ω ) and v ∈ H (Ω). We recall that γ D : H (Ω) → H / ( ∂ Ω) andcompute Z Ω d n x ∇ v ∇ u = − Z Ω d n x v ∆ u + h γ D v, e γ N u i / = Z Ω d n x vw u + (cid:10) γ D v, (cid:0)e γ N + Θ γ D (cid:1) u (cid:11) / − (cid:10) γ D v, Θ γ D u (cid:11) / = Z Ω d n x ∇ v ∇ u + (cid:10) γ D v, (cid:0)e γ N + Θ γ D (cid:1) u (cid:11) / , (4.33) where we used the second line in (4.30). Hence, (cid:10) γ D v, (cid:0)e γ N + Θ γ D (cid:1) u (cid:11) / = 0 . (4.34)Since v ∈ H (Ω) is arbitrary, and the map γ D : H (Ω) → H / ( ∂ Ω) is actuallyonto, one concludes that (cid:0)e γ N + Θ γ D (cid:1) u = 0 in H − / ( ∂ Ω) . (4.35)Thus,dom( − ∆ Θ , Ω ) ⊆ (cid:8) v ∈ H (Ω) (cid:12)(cid:12) ∆ v ∈ L (Ω; d n x ) , (cid:0)e γ N + Θ γ D (cid:1) v = 0 in H − / ( ∂ Ω) (cid:9) . (4.36)Next, assume that u ∈ (cid:8) v ∈ H (Ω) (cid:12)(cid:12) ∆ v ∈ L (Ω; d n x ) , (cid:0)e γ N + Θ γ D (cid:1) v = 0 (cid:9) , w ∈ H (Ω), and let w u = − ∆ u ∈ L (Ω; d n x ). Then, Z Ω d n x ww u = − Z Ω d n x w div( ∇ u )= Z Ω d n x ∇ w ∇ u − h γ D w, e γ N u i / = Z Ω d n x ∇ w ∇ u + (cid:10) γ D w, Θ γ D u (cid:11) / . (4.37)Thus, applying (4.30), one concludes that u ∈ dom( − ∆ Θ , Ω ) and hencedom( − ∆ Θ , Ω ) ⊇ (cid:8) v ∈ H (Ω) (cid:12)(cid:12) ∆ v ∈ L (Ω; d n x ) , (cid:0)e γ N + e Θ γ D (cid:1) v = 0 in H − / ( ∂ Ω) (cid:9) . (4.38)Finally, the last claim in the statement of Theorem 4.5 follows from the fact that H (Ω) embeds compactly into L (Ω; d n x ) (cf., e.g., [18, Theorem V.4.17]); seeLemma 2.1. (cid:3) In the special case Θ = 0, that is, in the case of the Neumann Laplacian, we willalso use the notation − ∆ N, Ω := − ∆ , Ω . (4.39)The case of the Dirichlet Laplacian − ∆ D, Ω associated with Ω formally correspondsto Θ = ∞ and so we isolate it in the next result (cf. also [24], [27]): Theorem 4.6.
Assume Hypothesis . Then the Dirichlet Laplacian, − ∆ D, Ω ,defined by − ∆ D, Ω = − ∆ , dom( − ∆ D, Ω ) = (cid:8) u ∈ H (Ω) (cid:12)(cid:12) ∆ u ∈ L (Ω; d n x ) , γ D u = 0 in H / ( ∂ Ω) (cid:9) (4.40)= (cid:8) u ∈ H (Ω) (cid:12)(cid:12) ∆ u ∈ L (Ω; d n x ) (cid:9) , is self-adjoint and strictly positive in L (Ω; d n x ) . Moreover, dom (cid:0) ( − ∆ D, Ω ) / (cid:1) = H (Ω) . (4.41)Since Ω is open and bounded, it is well-known that − ∆ D, Ω has purely discretespectrum contained in (0 , ∞ ), in particular, σ ess ( − ∆ D, Ω ) = ∅ . (4.42)This follows from (4.41) since H (Ω) embeds compactly into L (Ω; d n x ); the latterfact holds for arbitrary open, bounded sets Ω ⊂ R n (see, e.g., [18, Theorem V.4.18]). OME REMARKS ON A PAPER BY FILONOV 17 Eigenvalue Inequalities
Assume Hypothesis 4.1 and denote by λ Θ , Ω , ≤ λ Θ , Ω , ≤ · · · ≤ λ Θ , Ω ,j ≤ λ Θ , Ω ,j +1 ≤ · · · (5.1)the eigenvalues for the Robin Laplacian − ∆ Θ , Ω in L (Ω; d n x ), listed according totheir multiplicity. Similarly, we let0 < λ D, Ω , < λ D, Ω , ≤ · · · ≤ λ D, Ω ,j ≤ λ D, Ω ,j +1 ≤ · · · (5.2)be the eigenvalues for the Dirichlet Laplacian − ∆ D, Ω in L (Ω; d n x ), again enumer-ated according to their multiplicity. Theorem 5.1.
Assume Hypothesis , where the number δ > is taken to besufficiently small relative to the Lipschitz character of Ω and, in addition, supposethat (cid:10) γ D ( e ix · η ) , Θ γ D ( e ix · η ) (cid:11) / ≤ for all η ∈ R n . (5.3) Then λ Θ , Ω ,j +1 < λ D, Ω ,j , j ∈ N . (5.4) Proof.
One can follow Filonov [20] closely. The main reason we present Filonov’selegant argument is to ensure that this continues to hold in the case when a non-local Robin boundary condition is considered (in lieu of the Neumann boundarycondition). Recalling the eigenvalue counting functions for the Dirichlet and RobinLaplacians, one sets for each λ ∈ R , N D ( λ ) := { σ ( − ∆ D, Ω ) ∩ ( −∞ , λ ] } , N Θ ( λ ) := { σ ( − ∆ Θ , Ω ) ∩ ( −∞ , λ ] } . (5.5)Then Lemmas 2.1 and 4.4 ensure that for each λ ∈ R one has N D ( λ ) = max (cid:26) dim( L ) ∈ N (cid:12)(cid:12)(cid:12)(cid:12) L a subspace of H (Ω) such that Z Ω d n x | ( ∇ u )( x ) | ≤ λ k u k L (Ω; d n x ) for all u ∈ L (cid:27) , (5.6)and N Θ ( λ ) = max (cid:26) dim( L ) ∈ N (cid:12)(cid:12)(cid:12)(cid:12) L a subspace of H (Ω) with the property that Z Ω d n x | ( ∇ u )( x ) | + h γ D u, Θ γ D u i / ≤ λ k u k L (Ω; d n x ) for all u ∈ L (cid:27) . (5.7)Next, observe that for any λ ∈ C , H (Ω) ∩ ker( − ∆ Θ , Ω − λ I Ω ) = { } . (5.8)Indeed, if u ∈ H (Ω) ∩ ker( − ∆ Θ , Ω − λ I Ω ), then u ∈ H (Ω) satisfies ( − ∆ − λ ) u = 0in Ω and γ D u = e γ N u = 0. It follows that the extension by zero of u to the entire R n belongs to H ( R n ), is compactly supported, and is annihilated by − ∆ − λ .Hence, this function vanishes identically, by unique continuation (see, e.g., [48, p.239–244]).To continue, we fix λ > U λ of H (Ω) such that dim( U λ ) = N D ( λ ) and Z Ω d n x | ( ∇ u )( x ) | ≤ λ Z Ω d n x | u ( x ) | , u ∈ U λ . (5.9) Then the sum U λ ˙+ ker( − ∆ Θ , Ω − λ I Ω ) is direct, by (5.8). Since the functions (cid:8) e ix · η (cid:12)(cid:12) η ∈ R n , | η | = √ λ (cid:9) are linearly independent, it follows that there existsa vector η ∈ R n with | η | = √ λ and such that e ix · η does not belong to thefinite-dimensional space U λ ˙+ ker( − ∆ Θ , Ω − λ I Ω ). Assuming that this is the case,introduce W λ := U λ ˙+ ker( − ∆ Θ , Ω − λ I Ω ) ˙+ (cid:8) ce ix · η (cid:12)(cid:12) c ∈ C (cid:9) , (5.10)so that W λ is a finite-dimensional subspace of H (Ω). Let w = u + v + ce ix · η bean arbitrary vector in W λ , where u ∈ U λ , v ∈ ker( − ∆ Θ , Ω − λ I Ω ), and c ∈ C . Wethen write Z Ω d n x | ( ∇ w )( x ) | + h γ D w, Θ γ D w i / = Z Ω d n x |∇ ( u + v + ce ix · η ) | + h γ D ( v + ce ix · η ) , Θ γ D ( v + ce ix · η ) i / = Z Ω d n x (cid:0) |∇ u | + |∇ v | + | cη | (cid:1) + 2Re (cid:18) Z Ω d n x (cid:2) ∇ v · ∇ ( u + ce ix · η ) + ∇ ( ce ix · η ) · ∇ u (cid:3)(cid:19) + h γ D ( v + ce ix · η ) , Θ γ D ( v + ce ix · η ) i / =: I + I + I . (5.11)An integration by parts shows that Z Ω d n x |∇ v | = − Z Ω d n x v ∆ v + h γ D v, e γ N v i / = λ Z Ω d n x | v | − h γ D v, Θ γ D v i / (5.12)where the last equality holds thanks to − ∆ v = λ v and e γ N v = − Θ γ D v . We nowmake use of this, (5.9), the fact that | η | = λ , in order to estimate I ≤ λ Z Ω d n x (cid:2) | u | + | v | + | c | (cid:3) − h γ D v, Θ γ D v i / . (5.13)Similarly, I = − (cid:18) Z Ω d n x [∆ v ( u + ce ix · η ) + ∆( ce ix · η ) u ] (cid:19) + 2Re (cid:0) h γ D ( ce ix · η ) , e γ N v i / (cid:1) = 2 λ Re (cid:18) Z Ω d n x [ v ( u + ce ix · η ) + ce ix · η u ] (cid:19) − (cid:0) h γ D ( ce ix · η ) , Θ γ D v i / (cid:1) . (5.14)Thus, altogether, Z Ω d n x | ( ∇ w )( x ) | + h γ D w, Θ γ D w i / ≤ λ Z Ω d n x | w ( x ) | + | c | h γ D ( e ix · η ) , Θ γ D ( e ix · η ) i / . (5.15)Upon recalling (5.3), this yields Z Ω d n x | ( ∇ w )( x ) | + h γ D w, Θ γ D w i / ≤ λ Z Ω d n x | w ( x ) | , w ∈ W λ . (5.16) OME REMARKS ON A PAPER BY FILONOV 19
Consequently, N Θ ( λ ) ≥ dim( W λ ) = dim( U λ ) + dim(ker( − ∆ Θ , Ω − λ I Ω )) + 1= N D ( λ ) + dim(ker( − ∆ Θ , Ω − λ I Ω )) + 1 . (5.17)Specializing this to the case when λ = λ D, Ω ,j then yields { σ ( − ∆ Θ , Ω ) ∩ ( −∞ , λ D, Ω ,j ) } = N Θ ( λ D, Ω ,j ) − dim(ker( − ∆ Θ , Ω − λ D, Ω ,j I Ω )) ≥ N D ( λ D, Ω ,j ) + 1 ≥ j + 1 . (5.18)Now, the fact that { σ ( − ∆ Θ , Ω ) ∩ ( −∞ , λ D, Ω ,j ) } ≥ j + 1 is reinterpreted as (5.4). (cid:3) We briefly pause to describe a class of examples satisfying the hypotheses ofTheorem 5.1:
Example 5.2.
Consider the special case s = 1 / T ∈ B ( L ( ∂ Ω; d n − ω ))satisfying T ≤ T with the compact embedding operator J H / ( ∂ Ω) : H / ( ∂ Ω) → L ( ∂ Ω; d n − ω ) (5.19)yields a boundary operator Θ = T J H / ( ∂ Ω) ∈ B ∞ (cid:0) H / ( ∂ Ω) , L ( ∂ Ω) (cid:1) and henceΘ ∈ B ∞ (cid:0) H / ( ∂ Ω) , H − / ( ∂ Ω) (cid:1) is of the type Θ in Hypothesis 4.1.We note that condition (5.3) in Theorem 5.1 can be further refined and we willreturn to this issue in our final Remark 5.5.The case treated in [20] is that of a local Robin boundary condition. That is, itwas assumed that Θ is the operator of multiplication M θ by a function θ defined on ∂ Ω (which satisfies appropriate conditions). To better understand the way in whichthis scenario relates to the more general case treated here, we state and prove thefollowing result:
Lemma 5.3.
Assume Hypothesis and suppose that
Θ = M θ , the operatorof multiplication with a measurable function θ : ∂ Ω → R . Suppose that θ ∈ L p ( ∂ Ω; d n − ω ) , where p = n − if n > , and p ∈ (1 , ∞ ] if n = 2 . (5.20) Then Θ ∈ B ∞ (cid:0) H / ( ∂ Ω) , H − / ( ∂ Ω) (cid:1) (5.21) is a self-adjoint operator which satisfies k Θ k B ( H / ( ∂ Ω) ,H − / ( ∂ Ω)) ≤ C k θ k L p ( ∂ Ω; d n − ω ) , (5.22) where C = C (Ω , n, p ) > is a finite constant.Proof. Standard embedding results for Sobolev spaces (which continue to hold inthe case when the ambient space is the boundary of a bounded Lipschitz domain)yield that H / ( ∂ Ω) ֒ → L q ( ∂ Ω; d n − ω ) , where q := ( n − n − if n > , any number in (1 , ∞ ) if n = 2 . (5.23) Since the above embedding is continuous with dense range, via duality we alsoobtain that L q ( ∂ Ω; d n − ω ) ֒ → H − / ( ∂ Ω) , where q := ( n − n if n > , any number in (1 , ∞ ) if n = 2 . (5.24)Together, (5.23) and (5.24) yield that B (cid:0) L q ( ∂ Ω; d n − ω ) , L q ( ∂ Ω; d n − ω ) (cid:1) ֒ → B (cid:0) H / ( ∂ Ω) , H − / ( ∂ Ω) (cid:1) , (5.25)continuously. With p as in the statement of the lemma, H¨older’s inequality yieldsthat M θ ∈ B (cid:0) L q ( ∂ Ω; d n − ω ) , L q ( ∂ Ω; d n − ω ) (cid:1) (5.26)and k M θ k B ( L q ( ∂ Ω; d n − ω ) ,L q ( ∂ Ω; d n − ω )) ≤ C k θ k L p ( ∂ Ω; d n − ω ) , (5.27)for some finite constant C = C ( ∂ Ω , p, q , q ) >
0, granted that1 p + 1 q ≤ q . (5.28)Inequality (5.28) then holds with equality when n > p ∈ (1 , ∞ ), q , q can always be chosen as in (5.23) and (5.24) when n = 2 so that (5.28)continues to hold in this case as well. In summary, the above reasoning shows thatΘ = M θ ∈ B (cid:0) H / ( ∂ Ω) , H − / ( ∂ Ω) (cid:1) and the estimate (5.22) holds. Let us alsopoint out that Θ is a self-adjoint operator, since θ is real-valued.It remains to establish (5.21), that is, to show that Θ is also a compact operator.To this end, fix p > p and let θ j ∈ L p ( ∂ Ω; d n − ω ), j ∈ N , be a sequence ofreal-valued functions with the property that θ j → θ in L p ( ∂ Ω; d n − ω ) as j → ∞ .Set Θ j := M θ j , j ∈ N . From what we proved above, it follows thatΘ j → Θ in B (cid:0) H / ( ∂ Ω) , H − / ( ∂ Ω) (cid:1) as j → ∞ , (5.29)and there exists r ∈ (1 / ,
1) with the property thatΘ j ∈ B (cid:0) H r ( ∂ Ω) , H − / ( ∂ Ω) (cid:1) , j ∈ N . (5.30)Since the embedding H r ( ∂ Ω) ֒ → H / ( ∂ Ω) is compact, one concludes thatΘ j ∈ B ∞ (cid:0) H / ( ∂ Ω) , H − / ( ∂ Ω) (cid:1) , j ∈ N . (5.31)Thus, (5.21) follows from (5.31) and (5.29). (cid:3) We end by including a special case of Theorem 5.1 which is of independentinterest. In particular, this links our conditions on Θ with Filonov’s condition Z ∂ Ω d n − ω ( ξ ) θ ( ξ ) ≤ M θ . Corollary 5.4.
Assume Hypothesis , where the number δ > is taken to besufficiently small relative to the Lipschitz character of Ω and, in addition, supposethat Θ ≤ in the sense that h f, Θ f i / ≤ for every f ∈ H / ( ∂ Ω) . Then (5.4) holds.In particular, assuming Hypothesis and Θ = M θ , with θ ∈ L p ( ∂ Ω; d n − ω ) ,where p is as in (5.20) , is a function satisfying (5.32) , then (5.4) holds. OME REMARKS ON A PAPER BY FILONOV 21
Proof.
The first part is directly implied by Theorem 5.1. The second part is aconsequence of Lemma 5.3 and the conclusion in the first part of Corollary 5.4,since (5.33) reduces precisely to (5.32) for Θ = M θ . (cid:3) Remark 5.5.
After submitting our manuscript to the preprint archives we re-ceived a preprint version of Safarov’s paper [49] in which an abstract approach toeigenvalue counting functions and Dirichlet-to-Neumann maps was developed. Hismethods permit a considerable improvement of condition (5.3) as described in thefollowing: First, we note that in order to obtain the particular inequality λ Θ , Ω ,j +1 < λ D, Ω ,j for some fixed j ∈ N , (5.34)the proof of Theorem 5.1 uses condition (5.3) for only one value η j ∈ R n with | η j | = λ D, Ω ,j . Unfortunately, we have no manner to determine which η j to chooseon the sphere | η | = λ / D, Ω ,j such that e ix · η j does not belong to the finite-dimensionalspace U λ D, Ω ,j ˙+ ker( − ∆ Θ , Ω − λ D, Ω ,j I Ω ).On the other hand, applying Remark 1.11 (3) of Safarov [49] (and using that σ ess ( − ∆ Θ , Ω ) = ∅ ), to prove that the slightly weaker inequality λ Θ , Ω ,j +1 ≤ λ D, Ω ,j for some fixed j ∈ N , (5.35)holds, it suffices to find just one element u j ∈ H (Ω) \ H (Ω) satisfying∆ u j ∈ L (Ω; d n x ) , − ∆ u j = λ D, Ω ,j u j , (5.36)and a Θ ( u, v ) − λ D, Ω ,j k u j k L (Ω; d n x ) ≤ . (5.37)Since one can choose u j ( x ) = e ix · η j for any η j ∈ R n with | η j | = λ / D, Ω ,j , as long as(5.3) holds for η = η j , this proves that (5.35) holds whenever (cid:10) γ D ( e ix · η j ) , Θ γ D ( e ix · η j ) (cid:11) / ≤ η j ∈ R n with | η j | = λ / D, Ω ,j .Going further, and applying Remark 1.11 (4) of Safarov [49] (see also the proof ofCorollary 1.13 in [49]), one obtains strict inequality in (5.35) if there exist two ele-ments u j, , u j, ∈ H (Ω) \ H (Ω) satisfying (5.36) and (5.37) and lin.span { u j, , u j, } does not contain an element satisfying the boundary condition in − ∆ Θ , Ω . Butthe latter follows from (5.8). The two elements u j, , u j, can again be chosen as u j,k ( x ) = e ix · η j,k for any η j,k ∈ R n with | η j,k | = λ / D, Ω ,j , k = 1 ,
2, as long as (5.3)holds for η = η j, and η j, . Summing up, λ Θ , Ω ,j +1 < λ D, Ω ,j for some fixed j ∈ N , (5.39)holds whenever (cid:10) γ D ( e ix · η j,k ) , Θ γ D ( e ix · η j,k ) (cid:11) / ≤ η j,k ∈ R n with | η j,k | = λ / D, Ω ,j , k = 1 , a priori knowledge of λ D, Ω ,j , one canfinesse this dependence as follows: For instance, (5.35) holds for all j ∈ N wheneverthe set of η satisfying inequality (5.38) intersects every sphere in R n centered atthe origin. Similarly, if for some λ > (cid:10) γ D ( e ix · η ) , Θ γ D ( e ix · η ) (cid:11) / < η ∈ R n with | η | = λ , (5.41) then by continuity of (5.41) with respect to η (using the boundedness propertyΘ ∈ B (cid:0) H / ( ∂ Ω) , H − / ( ∂ Ω) (cid:1) ), one infers that (5.39) holds for all eigenvaluessufficiently close to λ , etc. Acknowledgments.
We are indebted to Mark Ashbaugh for helpful discussionsand very valuable hints with regard to the literature and especially to Yuri Safarovfor pointing out the validity of Remark 5.5 to us.
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Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
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