Elliptic differential operators on Lipschitz domains and abstract boundary value problems

This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value problems on non-smooth domains. A key feature is the extension of the boundary maps by continuity to the duals of certain range spaces, which directly leads to a description of all self-adjoint extensions of the underlying symmetric operator with the help of abstract boundary values. In the second part of the paper a complete description is obtained of all self-adjoint realizations of the Laplacian on bounded Lipschitz domains, as well as Kre\u{\i}n type resolvent formulas and a spectral characterization in terms of energy dependent Dirichlet-to-Neumann maps. These results can be viewed as the natural generalization of recent results from Gesztesy and Mitrea for quasi-convex domains. In this connection we also characterize the maximal range spaces of the Dirichlet and Neumann trace operators on a bounded Lipschitz domain in terms of the Dirichlet-to-Neumann map. The general results from the first part of the paper are also applied to higher order elliptic operators on smooth domains, and particular attention is paid to the second order case which is illustrated with various examples.


Introduction
Spectral theory of elliptic partial differential operators has received a lot of attention in the recent past, in particular, modern techniques from abstract operator theory were applied to extension and spectral problems for symmetric and self-adjoint elliptic differential operators on bounded and unbounded domains. We refer the reader to the recent contributions [2,10,11,12,16,17,40,41,42,49] on smooth domains, [3,4,31,32,33,37,39,57,59] on non-smooth domains, and we point out the paper [34] by Gesztesy and Mitrea which has inspired parts of the present work. Many of these contributions are based on the classical works Grubb [36] and Višik [67] on the parametrization of the closed realizations of a given elliptic differential expression on a smooth domain, and other classical papers on realizations with local and non-local boundary conditions, see, e.g. [1,7,8,15,29,63] and the monograph [48] by Lions and Magenes. In [34] Gesztesy and Mitrea obtain a complete description of the selfadjoint realizations of the Laplacian on a class of bounded non-smooth, socalled quasi-convex domains. The key feature of quasi-convex domains is that the functions in the domains of the self-adjoint Dirichlet realization ∆ D and the self-adjoint Neumann realization ∆ N possess H 2 -regularity, a very convenient property which is well-known to be false for the case of Lipschitz domains; cf. [45]. Denote by τ D and τ N the Dirichlet and Neumann trace operator, respectively. Building on earlier work of Maz'ya, Mitrea and Shaposhnikova [52], see also [20,28,30], the range spaces G 0 := τ D (dom ∆ N ) and G 1 := τ N (dom ∆ D ) were characterized for quasi-convex domains in [34], and the self-adjoint realizations of the Laplacian were parametrized via tuples {X , L}, where X is a closed subspace of the anti-dual G ′ 0 or G ′ 1 and L is a self-adjoint operator from X to X ′ . This parametrization technique has its roots in [14,47] and was used in [36,67], see also [38,Chapter 13]. In [16] the connection to the notion of (ordinary) boundary triples from extension theory of symmetric operators was made explicit.
The theory of ordinary boundary triples and Weyl functions originates in the works of Kocubeȋ [46], Bruk [18], Gorbachuk and Gorbachuk [35], and Derkach and Malamud [26,27]. A boundary triple {G, Γ 0 , Γ 1 } for a symmetric operator A in a Hilbert space H consists of an auxiliary Hilbert space G and two boundary mappings Γ 0 , Γ 1 : dom A * → G which satisfy an abstract Green's identity and a maximality condition. With the help of a boundary triple the closed extensions of the underlying symmetric operator A can be parametrized in an efficient way with closed operators and subspaces Θ in the boundary space G. The concept of ordinary boundary triples was applied successfully to various problems in extension and spectral theory, in particular, in the context of ordinary differential operators, see [19] for a review and further references. However, for the Laplacian (or more general symmetric elliptic differential operators) on a domain Ω ⊂ R n , n ≥ 2, with boundary ∂Ω the natural choice Γ 0 = τ D and Γ 1 = −τ N does not lead to an ordinary boundary triple since Green's identity does not extend to the domain of the maximal operator A * . This simple observation led to a generalization of the concept of ordinary triples, the so-called quasi boundary triples, which are designed for applications to PDE problems. Here the boundary mappings Γ 0 = τ D and Γ 1 = −τ N are only defined on some suitable subset of dom A * , e.g. H 2 (Ω), and the realizations are labeled with operators and subspaces Θ in the boundary space L 2 (∂Ω) via boundary conditions of the form Θτ D f + τ N f = 0, f ∈ H 2 (Ω). One of the advantages of this approach is that the Weyl function corresponding to the quasi boundary triple {L 2 (∂Ω), τ D , −τ N } coincides (up to a minus sign) with the usual family of Dirichlet-to-Neumann maps on the boundary ∂Ω, and hence the spectral properties of a fixed self-adjoint extension can be described with the Dirichlet-to-Neumann map and the parameter Θ in the boundary condition.
The aim of the present paper is twofold. Our first objective is to further develop the abstract notion of quasi boundary triples and their Weyl functions. The main new feature is that we shall assume that the spaces G 0 = ran Γ 0 ↾ ker Γ 1 and G 1 = ran Γ 1 ↾ ker Γ 0 are reflexive Banach spaces densely embedded in the boundary space G; this assumption is natural in the context of PDE problems and related Sobolev spaces on the boundary of the domain, and is satisfied in applications to the Laplacian on Lipschitz domains and other elliptic boundary value problems treated in the second part of the present paper. In fact, this assumption is the abstract analog of the the properties of the range spaces in [34], and it is also automatically satisfied in many abstract settings, e.g. for ordinary and so-called generalized boundary triples; cf. [27] and Section 2.4 for a counterexample in the general case. Under the density assumption it then follows that the boundary maps Γ 0 and Γ 1 can be extended by continuity to surjective mappings from dom A * onto the anti-duals G ′ 1 and G ′ 0 , respectively. Then also the γ-field and the Weyl function admit continuous extensions to operators mapping in between the appropriate spaces; for the special case of generalized boundary triples and G 0 , G 1 equipped with particular topologies this was noted in the abstract setting earlier in [27,Proposition 6.3] and [25,Lemma 7.22]. Following the regularisation procedure in the PDE case we then show that a quasi boundary triple with this additional density property can be transformed into a quasi boundary triple which is the restriction of an ordinary boundary triple, and hence can be extended by continuity; a similar argument can also be found in a different abstract form in [25]. As a consequence of these considerations we obtain a complete description of all closed extensions of the underlying symmetric operator in Section 3, as well as abstract regularity results, Kreȋn type resolvent formulas and new sufficient criteria for the parameter Θ in the boundary condition to imply self-adjointness of the corresponding extension.
The second objective of this paper is to apply the abstract quasi boundary triple technique to various non-local PDE problems. In particular, in Section 4.1 we extend the characterization of the self-adjoint realizations ∆ Θ of the Laplacian on quasi convex domains to the more natural case of usual Lipschitz domains. Here the Hilbert spaces G 0 and G 1 are topologized with the help of the Dirichlet-to-Neumann map in a similar manner as in [25,27] for abstract generalized boundary triples. This also leads to a continuous extension of the Dirichlet and Neumann trace operators on a Lipschitz domain to the maximal domain of the Laplacian, and hence to a description of the Dirichlet boundary data for L 2 -solutions of −∆f = λf . For the special case of quasi convex domains and C 1,r -domains with r ∈ ( 1 2 , 1] we establish the link to the approach in [34], and recover many of the results in [34] as corollaries of the abstract methods developed in Section 2 and Section 3. In Section 4.2 we illustrate the abstract methods in the classical case of 2m-th order elliptic differential operators with smooth coefficients on smooth bounded domains, where the spaces G 0 and G 1 coincide with the usual product Sobolev trace spaces on ∂Ω. Here e.g. some classical trace extension results follow from the abstract theory developed in the first part of the paper. Finally, we pay particular attention to the second order case on bounded and unbounded domains with compact smooth boundary in Section 4.3. Here we recover various recent results on the description and the spectral properties of the self-adjoint extensions of a symmetric second order elliptic differential operator, and extend these by adding, e.g. regularity results. This section contains also some simple examples, among them selfadjoint extensions with Robin boundary conditions. One of the examples is also interesting from a more abstract point of view: It turns out that there exist self-adjoint parameters in the range of the boundary maps of a quasi boundary triple such that the corresponding extension is essentially self-adjoint, but not self-adjoint.
Acknowledgement. Till Micheler gratefully acknowledges financial support by the Studienstiftung des deutschen Volkes. The authors are indebted to Fritz Gesztesy and Marius Mitrea for very valuable comments, and to Seppo Hassi and Henk de Snoo for pointing out connections to recent abstract results. Moreover, the authors also wish to thank Vladimir Lotoreichik, Christian Kühn, and Jonathan Rohleder for many helpful discussions and remarks.

Quasi boundary triples and their Weyl functions
The concept of boundary triples and their Weyl functions is a useful and efficient tool in extension and spectral theory of symmetric and self-adjoint operators, it originates in the works [18,46] and was further developed in [26,27,35]; cf. [19] for a review. In the recent past different generalizations of the notion of boundary triples were introduced, among them boundary relations, boundary pairs and boundary triples associated with quadratic forms, and other related concepts, see [6,23,24,25,55,56,58,59,61,62]. The concept of quasi boundary triples and their Weyl functions introduced in [10] is designed for the analysis of elliptic differential operators. It can be viewed as a slight generalization of the notions of boundary and generalized boundary triples. In this section we first recall some definitions and basic properties which can be found in [10,11]. Our main objective is to show that under an additional density condition the corresponding boundary maps can be extended by continuity and that the corresponding quasi boundary triple can be transformed (or regularized) such that it turns into an ordinary boundary triple; cf. [25,69,70] for related investigations.

Ordinary and quasi boundary triples
Let throughout this section A be a closed, densely defined, symmetric operator in a separable Hilbert space H.
In the special case T = A * a quasi boundary triple {G, Γ 0 , Γ 1 } is called ordinary boundary triple.
Let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ A * . Then the mapping Γ = (Γ 0 , Γ 1 ) ⊤ : dom T → G × G is closable with respect to the graph norm of A * and ker Γ = dom A holds; cf. [10,Proposition 2.2]. Moreover, according to [10,Theorem 2.3] (see also Proposition 2.2 below) we have T = A * if and only if ran Γ = G × G, in this case Γ = (Γ 0 , Γ 1 ) ⊤ : dom A * → G × G is onto and continuous with respect to the graph norm of A * , and the restriction A 0 = A * ↾ ker Γ 0 is automatically self-adjoint. Thus, the above definition of an ordinary boundary triple coincides with the usual one, see, e.g. [26]. We also note that a quasi boundary triple is in general not a boundary relation in the sense of [23,24], but it can be viewed as a certain transform of a boundary relation; cf. [ Then S := T ↾ ker Γ is a densely defined, closed symmetric operator in H and {G, Γ 0 , Γ 1 } is a quasi boundary triple for S * such that A = T ↾ ker Γ 0 = A 0 . Moreover, T = S * if and only if ran Γ = G × G.
Not surprisingly, suitable restrictions of ordinary boundary triples lead to quasi boundary triples.
Proof. Clearly, items (i) and (iii) in Definition 2.1 hold for the restricted For this letx ∈ G × G. Thenx ∈ ran Γ and there exists an element f ∈ dom A * such that Γf =x. Since T = A * there exists a sequence (f n ) ⊂ dom T which converges to f in the graph norm of A * . As Γ is continuous with respect to the graph norm we obtain Γ T f n = Γf n →x for n → ∞, that is, item (ii) in Definition 2.1 holds and {G, The following proposition shows that the converse of Proposition 2.3 holds under an additional continuity assumption. In particular, it implies that if a quasi boundary triple can be extended to an ordinary boundary triple then this extension is unique.
Proof. (⇒) Since Γ : dom A * → G × G is continuous with respect to the graph norm of A * the same holds for the restriction Γ T : dom T → G × G.
(⇐) Let Γ = (Γ 0 , Γ 1 ) ⊤ : dom A * → G × G be the continuous extension of Γ T with respect to the graph norm of A * . Then also the abstract Green's identity extends by continuity from dom T to dom A * , (2.2) and the range of Γ is dense in G × G. Moreover, from (2.2) it follows that the operator A * ↾ ker Γ 0 is a symmetric extension of the self-adjoint operator A 0 = T ↾ ker Γ T 0 and hence A 0 = A * ↾ ker Γ 0 . We conclude that {G, Γ 0 , Γ 1 } is a quasi boundary triple for T = A * , that is, {G, Γ 0 , Γ 1 } is an ordinary boundary triple for A * ; cf. Definition 2.1. Clearly, {G, Γ T 0 , Γ T 1 } is the restriction of this ordinary boundary triple to T . A simple and useful example of an ordinary and quasi boundary triple is provided in Lemma 2.5 below, it also implies the well-known fact that a boundary triple or quasi boundary triple exists if and only if A has equal deficiency indices n ± (A) := dim ker(A * ± i), that is, if and only if A admits self-adjoint extensions in H. Recall first that for a self-adjoint extension A 0 ⊂ T of A and η ∈ ρ(A 0 ) the domains of T and A * admit the direct sum decompositions . The orthogonal projection in H onto the defect subspace N η (A * ) will be denoted by P η .
In the next lemma a special boundary triple and quasi boundary triple are constructed. The restriction η ∈ R below is for convenience only, an example of a similar ordinary boundary triple with η ∈ C \ R can be found in, e.g. [26] or the monographs [35,64].
Lemma 2.5. Assume that the deficiency indices of A are equal and let G be a Hilbert space with dim G = n ± (A). Let A 0 be a self-adjoint extension of A in H, assume that there exists η ∈ ρ(A 0 ) ∩ R and fix a unitary operator ϕ : N η (A * ) → G. Then the following statements hold.
i.e., the abstract Green's identity holds. Moreover, Γ 0 : dom A * → G is surjective and since ran(A 0 − η) = H it follows that also Γ : dom A * → G × G is surjective. This implies that {G, Γ 0 , Γ 1 } is an ordinary boundary triple for A. It is obvious that A 0 = A * ↾ ker Γ 0 holds.

Weyl functions and γ-fields of quasi boundary triples
In this subsection the notion and some properties of γ-fields and Weyl functions associated to quasi boundary triples are briefly reviewed. Furthermore, a simple but useful description of the range of the boundary mappings is given in terms of the Weyl function in Proposition 2.8.
Definition 2.6. The γ-field and the Weyl function corresponding to the quasi boundary triple {G, Γ 0 , Γ 1 } are defined by It follows that for λ ∈ ρ(A 0 ) the operator γ(λ) is continuous from G to H with dense domain dom γ(λ) = ran Γ 0 and range ran γ(λ) = N λ (T ), the function λ → γ(λ)g is holomorphic on ρ(A 0 ) for every g ∈ ran Γ 0 , and the relations hold for all λ, µ ∈ ρ(A 0 ); cf. [10, Proposition 2.6]. Note that γ(λ) * : H → G is continuous and that (ker γ(λ) * ) ⊥ = ran γ(λ) = N λ (A * ) yields the orthogonal space decomposition H = ker γ(λ) * ⊕ N λ (A * ). (2.5) For λ ∈ ρ(A 0 ) the values M (λ) of the Weyl function are operators in G with dense domain ran Γ 0 and range contained in ran Γ 1 . If, in addition, and this implies the identity We also mention that for λ, µ ∈ ρ(A 0 ) the Weyl function is connected with the γ-field via and, in particular, M (λ) is a symmetric operator in G for λ ∈ R ∩ ρ(A 0 ). It is important to note that The subspaces G 0 and G 1 of G in the next definition will play a fundamental role throughout this paper. 1 Definition 2.7. Let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ A * . Then we define the spaces G 0 := ran Γ 0 ↾ ker Γ 1 and Observe that for the spaces G 0 and G 1 in Definition 2.7 we have G 0 ×G 1 ⊂ ran Γ. Note also that the second identity in (2.4) implies Proposition 2.8. Let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ A * with A 0 = T ↾ ker Γ 0 and Weyl function M , and let G 0 and G 1 be as in Definition 2.7. Then the following assertions hold for all λ ∈ ρ(A 0 ).
(ii) We show first that ran Γ is contained in the right hand side of (2.10). Let such that Γf =x. From (2.6) and Γ 0 f = x we conclude and hencex belongs to the right hand side of (2.10). Conversely, let x ∈ ran Γ 0 and x ′ = M (λ)x + y with some y ∈ G 1 . Then there exist f 0 ∈ ker Γ 0 with Γ 1 f 0 = y and f λ ∈ N λ (T ) with Γ 0 f λ = x.
The remaining assertion in (ii) follows from the representation (2.10) and the fact that ran Γ is dense in G × G.  Im M (λ) = Im λ γ(λ) * γ(λ), λ ∈ ρ(A 0 ), holds. Hence Im M (λ) is a densely defined, invertible bounded operator in G with ran(Im M (λ)) ⊂ G 1 ; cf. (2.4). Therefore we may rewrite Proposition 2.8 (ii) in the form The continuous extension of Im M (λ) onto G is given by the closure It is important to note that for λ ∈ C \ R we have

Extensions of boundary mappings, γ-fields and Weyl functions
Let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ A * . In this section we investigate the case where the space G 1 = ran(Γ 1 ↾ ker Γ 0 ) in Definition 2.7 is dense in G. Under this assumption we show that the boundary map Γ 0 and the γ-field admit continuous extensions. If, in addition, G 0 = ran(Γ 0 ↾ ker Γ 1 ) is dense in G and A 1 = T ↾ ker Γ 1 is self-adjoint in H then also Γ 1 and the Weyl function M admit continuous extensions. We point out that in general G 1 (or G 0 ) is not dense in G, see Proposition 2.17 for a counterexample. The next proposition is a variant of [27, Proposition 6.3] (see also [25,Lemma 7.22]) for quasi boundary triples and their Weyl functions. It was proved for generalized boundary triples in [27], where the additional assumption that G 1 is dense in G is automatically satisfied; cf. (2.13) and [27,Lemma 6.1]. In the following G ′ 1 stands for the anti-dual space of G 1 .
Proposition 2.9. Let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ A * with Weyl function M , set Λ := Im M (i) and assume, in addition, that G 1 is dense in G. Then G 1 = ran Λ 1/2 and if G 1 is equipped with the norm induced by the inner product then the following assertions hold.
(i) γ(i) extends to an isometry γ(i) from G ′ 1 onto N i (A * ), (ii) Im M (i) extends to an isometry from G ′ 1 onto G 1 .
Hence ran Λ and ran Λ 1/2 are dense in G. As in the proof of [27, Proposition 6.3] we equip G := ran Λ 1/2 with the inner product Then G is a Hilbert space which is densely embedded in G and hence gives rise to a Gelfand triple G ֒→ G ֒→ G ′ , where G ′ is the completion of G equipped with the inner product (Λ 1/2 x, Λ 1/2 y) G , x, y ∈ G. As in [27,Proposition 6.3] one verifies that the mapping γ(i) admits a continuation to an isometry γ(i) from G ′ onto N i (A * ) and the mapping Im M (i) admits a continuation to an isometry Λ from G ′ onto G with Λ ⊂ Λ = γ(i) * γ(i). This implies G = ran γ(i) * = G 1 by (2.9) and assertions (i) and (ii) follow.
The next proposition contains a simple but far-reaching observation: If G 1 is dense in G and G 1 is equipped with a Hilbert or Banach space norm such that Γ 1 (A 0 −λ) −1 : H → G 1 is continuous then the boundary map Γ 0 can be extended by continuity onto dom A * . Although Proposition 2.9 provides a possible norm on G 1 it is essential for later applications to allow other norms which are a priori not connected with the Weyl function.
Proposition 2.10. Let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ A * with A 0 = T ↾ ker Γ 0 and assume, in addition, that G 1 is dense in G. Then for any norm · G 1 such that is continuous for some, and hence for all, λ ∈ ρ(A 0 ), hold the boundary mapping Γ 0 admits a unique surjective, continuous extension Moreover, the norm ||| · ||| G 1 induced by the inner product (2.14) is equivalent to any norm · G 1 on G 1 with the properties (i)-(ii).
Let ||| · ||| G 1 be the norm induced by the inner product (2.14) and let · G 1 be an arbitrary norm on Therefore we obtain . This fact together with Proposition 2.10 implies the following statement.
is continuous for some, and hence for all, λ ∈ ρ(A 1 ), hold the boundary mapping Γ 1 admits a unique surjective, continuous extension We note that in the situation of the above corollary it follows that the closure of Im(−M (i) −1 ) is an invertible bounded operator defined on G. Making use of Proposition 2.9 for the quasi boundary triple {G, −Γ 1 , Γ 0 } and setting Σ := Im(−M (i) −1 ) we then conclude that the norm ||| · ||| G 0 induced by the inner product is equivalent to any norm · G 0 on G 0 which satisfies (i)-(ii) in Corollary 2.11. The next theorem is strongly inspired by regularisation techniques used in extension theory of symmetric partial differential operators; cf. [36,67].
It will be shown that a quasi boundary triple {G, Γ 0 , Γ 1 } with the additional property that G 1 is dense in G can be transformed and extended to an ordinary boundary triple. Such a type of transform appears also in [11,16] and in a more abstract form in [25], see also [69,70]. Here we discuss only a situation which is relevant in applications, namely we assume that the spectrum of the self-adjoint operator A 0 = T ↾ ker Γ 0 does not cover the whole real line. The more general case is left to the reader; cf. Remark 2.13. Recall that for the Gelfand triple Here and in the following G 1 is equipped with some norm · G 1 such that (i) and (ii) in Proposition 2.10 hold. Recall that according to Proposition 2.9 such a norm always exists (if G 1 is dense in G) and that all such norms are equivalent by Proposition 2.10.
is an ordinary boundary triple for A * with Proof. We verify that the restriction {G, Υ T 0 , Υ T 1 }, Note first that ker Υ T 0 = ker Γ 0 holds. Thus T ↾ ker Υ T 0 coincides with the self-adjoint linear operator A 0 in H and (iii) in Definition 2.1 holds. In order to check Green's identity observe that for all f ∈ dom T the identity Here M is the Weyl function of the quasi boundary triple {G, Γ 0 , Γ 1 } and since by assumption η ∈ R ∩ ρ(A 0 ) the operator M (η) is symmetric in G; cf. (2.7). Making use of (2.15) and the fact that ·, · G 1 ×G ′ 1 is the continuous extension of the scalar product in tends tox for n → ∞. Hence (ii) in Definition 2.1 holds and it follows that are continuous with respect to the graph norm. It follows from Proposition 2.10 that this is even true for Υ 0 = ι − Γ 0 , and hence also for the restriction and together with Proposition 2.10 (ii) we conclude that Υ T 1 is also continuous with respect to the graph norm.
It remains to check that ker Remark 2.13. We note that the assumption η ∈ R in Theorem 2.12 can be dropped. respectively. Here Instead of (2.6) use the following formula when verifying Green's identity in the proof of Theorem 2.12.
With the help of the extensions Γ 0 and Γ 1 of the boundary mappings Γ 0 and Γ 1 , respectively, also the γ-field and Weyl function can be extended by continuity. Observe that by Theorem 2.12 we have 1 be the continuous extension of Γ 0 from Proposition 2.10. Then the extended γ-field γ corresponding to the quasi boundary triple {G, Γ 0 , Γ 1 } is defined by (ii) Assume that G 0 and G 1 are dense in G, that A 1 is self-adjoint in H, and let Γ 1 : dom A * → G ′ 0 be the continuous extension of Γ 1 from Corollary 2.11. Then the extended Weyl function M corresponding to the quasi boundary triple {G, We mention that the values of the extended γ-field γ are bounded linear operators from G ′ 1 to H, where G 1 is equipped with a norm such that (i) and (ii) in Proposition 2.10 hold. If also G 0 is equipped with a norm such that (i) and (ii) in Corollary 2.11 hold then the values of the extended Weyl function M are bounded linear operators from G ′ 1 to G ′ 0 . Therefore the adjoints are continuous for all λ ∈ ρ(A 0 ). Moreover we obtain the simple identity In the next two lemmas some basic, but important, facts about the extended boundary mappings, the extended γ-field and the extended Weyl function are summarized. As above it is assumed that G 1 is dense in G and that G 1 is equipped with a norm such that (i) and (ii) in Proposition 2.10 hold.
Then the following statements hold.
Proof. Let {G, Υ 0 , Υ 1 } be the ordinary boundary triple for A * from Theorem 2.12 and denote the corresponding γ-field with β. Then according to Theorem 2.12 statement (i) follows from see the text before Definition 2.14. From Proposition 2.10 we obtain that Γ 0 : 1 is bijective and continuous and this implies (ii). The identity To proof statement (iii) we only have to show that the identity Lemma 2.16. Let the assumption be as in Lemma 2.15 and assume, in addition, that G 0 is dense in G and that . Then the following statements hold for all λ ∈ ρ(A 0 ).
(v) the range of the boundary mapping Γ is given by Proof. Statement (i) follows in the same way as in Lemma 2.15 and from the fact that (2.6). In order to verify (iii) note first that according to (2.8) . Therefore the first assertion in (iv) follows from (i) and Corollary 2.11. The second assertion in (iv) is a consequence of (iii). Finally, statement (v) follows from (ii) in the same way as in the proof of Proposition 2.8 (ii).
Since ker Γ 1 = ker Γ 1 and ker Γ 0 = ker Γ 0 hold by Lemma 2.16 (i) and Lemma 2.15 (i) we conclude that the spaces G 0 and G 1 in Definition 2.7 remain the same for the extended boundary mappings, i.e., For later purposes we also note that for a quasi boundary triple {G, Γ 0 , Γ 1 } as in Lemma 2.15 and 2.16, with γ-field γ, Weyl function M , their extensions , and the corresponding ordinary boundary triple {G, Υ 0 , Υ 1 } from Theorem 2.12 with γ-field β, Weyl function M the following relations hold: In fact, the identity β(λ) = γ(λ)ι −1 − was already shown in the proof of Lemma 2.15 and the second relation in (2.17) is a direct consequence of the definition of the Weyl function M, Lemma 2.16 (ii), and the particular form of the ordinary boundary triple {G, Υ 0 , Υ 1 } in Theorem 2.12. In fact,

A counterexample
In this supplementary subsection we show that the assumption G ⊥ 1 = {0}, which is essential for Proposition 2.9, Proposition 2.10, Corollary 2.11 and Theorem 2.12, is not satisfied automatically. For this we construct a quasi boundary triple {H , Υ 0 , Υ 1 } with the property G ⊥ 1 = {0} as a transform of the quasi boundary triple in Lemma 2.5 (ii).
, and let H be an auxiliary Hilbert space. Choose a densely defined, bounded operator γ : and let M be an (unbounded) self-adjoint operator in H defined on dom γ.
is a quasi boundary triple for T ⊂ A * such that A 0 = T ↾ ker Υ 0 , In particular, if M (·) is the Weyl function corresponding to the quasi bound- for all f, g ∈ dom T , and hence the abstract Green's identity holds. Observe that Next it will be shown that the range of Υ : Here we have used in the last step that ran Γ T for all x ∈ dom γ and all y ∈ ran γ * . We note that if for all x ∈ dom M . As M is self-adjoint we conclude z ′ ∈ dom M = dom γ and from ker γ = {0} we find z ′ = 0. Assume now that z ′ ∈ ker γ = (ran γ * ) ⊥ . Then there exists y ∈ ran γ * such that (−z ′ , y) Since

Extensions of symmetric operators
The main objective of this section is to parameterize the extensions of a symmetric operator A with the help of a quasi boundary triple {G, Γ 0 , Γ 1 } for T ⊂ A * . In contrast to ordinary boundary triples there is no immediate direct connection between the properties of the extensions and the properties of the corresponding parameters ϑ in G × G, as, e.g. selfadjointness. The key idea in Theorem 3.3 and Theorem 3.4 is to mimic a regularization procedure which is used in the investigation of elliptic differential operators and goes back to [36,67], see also [11,16,25,34,49,56,57]. This also leads to an abstract complete description of the extensions A ϑ ⊂ A * via the extended boundary mappings Γ 0 and Γ 1 in Theorem 3.7. The general results are illustrated with various examples and sufficient conditions on the parameters to imply self-adjointness, as well as a variant of Kreȋn's formula is discussed.

Parameterization of extensions with quasi boundary triples
Let in the following A be a closed, densely defined, symmetric operator in the Hilbert space H with equal, in general, infinite deficiency indices.
In the first theorem in this subsection we recall one of the key features of ordinary boundary triples {G, Γ 0 , Γ 1 } for A * : A complete description and parameterization of the extensions A Θ of A given by and their properties in terms of linear relations Θ in the boundary space G, see, e.g. [26,27,35].
establishes a bijective correspondence between the set of closed linear relations Θ in G and the set of closed extensions A Θ ⊂ A * of A. Furthermore, and the operator A Θ is symmetric (self-adjoint, (maximal) dissipative, (maximal) accumulative) in H if and only if the closed linear relation Θ is symmetric (self-adjoint, (maximal) dissipative, (maximal) accumulative, respectively) in G.
Not surprisingly Theorem 3.1 does not hold for quasi boundary triples {G, Γ 0 , Γ 1 }, see, e.g. [10,Proposition 4.11] for a counterexample. In particular, ϑ = {0} × G 1 ⊂ ran Γ (see Definition 2.7 and Proposition 2.8 (ii)) is symmetric and not self-adjoint in G but the corresponding extension A ϑ of A in (3.1) coincides with the self-adjoint operator A 0 = T ↾ ker Γ 0 in H. Note that for a quasi boundary triple {G, Γ 0 , Γ 1 } the range of the boundary map Γ = (Γ 0 , Γ 1 ) ⊤ is only dense in G × G, so that for a linear relation ϑ in G only the part ϑ ∩ ran Γ can be "detected" by the boundary maps. However, even for a self-adjoint linear relation ϑ ⊂ ran Γ the corresponding extension A ϑ of A in (3.1) is in general not self-adjoint, see Example 4.24. Nevertheless, the following weaker statement is a direct consequence of the abstract Green's identity (2.1); cf. [10,Proposition 2.4].
establishes a bijective correspondence between the set of symmetric linear relations ϑ ⊂ ran Γ in G and the set of symmetric extensions A ϑ ⊂ T of A in H.
In the next theorem we make use of a different type of parameterization to characterize the restrictions of T with the help of a quasi boundary triple. The idea of the proof is to relate the given quasi boundary triple {G, Γ 0 , Γ 1 } to the quasi boundary triple in Lemma 2.5 (ii) and to transform the parameters accordingly. We also point out that in contrast to most of the results in Section 2.3 here it is not assumed that the space G 1 = ran(Γ 1 ↾ ker Γ 0 ) is dense in G.

Proof. Let Θ be a linear relation in
and by (2.6) this can be rewritten as Denote the orthogonal projection in H onto N η (A * ) by P η . Making use of (2.4) and (2.5) we find According to Proposition 2.3 and Lemma 2.5 the quasi boundary triple {G, f → ϕf η , f → ϕP η (A 0 −η)f 0 } is the restriction of the ordinary boundary triple {G, f → ϕf η , f → ϕP η (A 0 − η)f 0 } for A * . Now the statement is a consequence of Theorem 3.1. In fact, if e.g. Θ is self-adjoint in G with dom Θ ⊂ ran(ϕ ↾ N η (T )), then by Theorem 3.1 the operator is a self-adjoint restriction of A * in H. As dom Θ ⊂ ran(ϕ ↾ N η (T )) we conclude that the domain of the operator in (3.4) is contained in dom T . Hence by (3.3) the operator in (3.4) can be written as and A ϑ is a self-adjoint operator in H. Conversely, by Theorem 3.1 for any self-adjoint extension A ϑ of A which is contained in T there exists a selfadjoint relation Θ in G such that A ϑ can be written in the form (3.4), where N η (A * ) can be replaced by N η (T ). Therefore dom Θ ⊂ ran(ϕ ↾ N η (T )) and together with (3.3) we conclude that A ϑ can be written in the form (3.5).
The next theorem is of similar flavor as Theorem 3.3 but more explicit and relevant for elliptic boundary value problems; cf. Section 4. Under the additional assumption that the space G 1 = ran(Γ 1 ↾ ker Γ 0 ) in Definition 2.7 is dense in G a more natural parameterization of the extensions is found. Here we will again make use of the Gelfand triple G 1 ֒→ G ֒→ G ′ 1 and the corresponding isometric isomorphisms ι + and ι − in (2.15). We also note that after suitable modifications the assumption η ∈ R can be dropped, see Remark 2.13.
Proof. Let Θ be a linear relation in G and decompose f ∈ dom T in the form According to Theorem 2.12 the triple {G, f → ι − Γ 0 f, f → ι + Γ 1 f 0 } is an ordinary boundary triple for A * . Now the statement follows from Theorem 3.1 in the same form as in the proof of Theorem 3.3.
Corollary 3.5. Let the assumptions be as in Theorem 3.4 and let ϑ be a linear relation in G. Then the extension A ϑ of A in H given by is closed (symmetric, self-adjoint, (maximal) dissipative, (maximal) accumulative) in H if and only if the linear relation Proof. (⇒) Assume that A ϑ in (3.8) is a closed (symmetric, self-adjoint, (maximal) dissipative, (maximal) accumulative) operator in H. According to Theorem 3.4 there exists a closed (symmetric, self-adjoint, (maximal) dissipative, (maximal) accumulative, respectively) linear relation Θ in G with dom Θ ⊂ ran ι − Γ 0 and From ι −1 + Θι − ⊂ ran Γ 0 × G 1 and Proposition 2.8 (ii) we conclude θ ⊂ ran Γ. Furthermore, we have θ = ϑ∩ran Γ, (see the text below Lemma 3.2). Solving equation (3.9) leads to the identity We recall that a symmetric linear relation Θ in G with ran Θ = G is self-adjoint in G with 0 ∈ ρ(Θ). This together with Corollary 3.5 yields the following example.
Example 3.6. Let the assumptions be as in Corollary 3.5 and let ϑ be a symmetric linear relation in G such that ran(ϑ − M (η)) = G 1 . Then In the next result the assumptions on the quasi boundary triple are strengthened further such that both boundary maps Γ 0 and Γ 1 extend by continuity to dom A * . In that case one obtains a description of all extensions A ϑ ⊂ A * which is very similar to the parameterization in Theorem 3.4. The additional abstract regularity result will turn out to be useful when considering the regularity of solutions of elliptic boundary value problems in Section 4. Theorem 3.7. Let the assumptions be as in Theorem 3.4 and assume, in addition, that A 1 = T ↾ ker Γ 1 is self-adjoint in H, η ∈ ρ(A 0 ) ∩ ρ(A 1 ) ∩ R, and that G 0 dense in G. Let M be the extension of the Weyl function M from Definition 2.14 (ii). Then the mapping establishes a bijective correspondence between all closed (symmetric, selfadjoint, (maximal) dissipative, (maximal) accumulative) linear relations Θ in G and all closed (symmetric, self-adjoint, (maximal) dissipative, (maximal) accumulative, respectively) extensions A ϑ ⊂ A * of A in H. Moreover, the following abstract regularity result holds: If Θ is a linear relation in G and S is an operator in H such that T ⊂ S ⊂ A * then Proof. The proof of the first part is very similar to the proof of Theorem 3.4 and will not be repeated here. We show the abstract regularity result. Let Θ and S be as in the theorem and assume that dom Θ is contained in the range of the map ι − Γ 0 ↾ dom S. Let be the corresponding extension and let f ∈ dom A ϑ . As Γf ∈ ι −1 The next corollary is a counterpart of Corollary 3.5 and can be proved in the same way using Lemma 2.16 (v) instead of Proposition 2.8 (ii).
A simple application of Theorem 3.7 is discussed in the next example.
Example 3.9. Set Θ = 0 in Theorem 3.7. Then ϑ = M (η) and it follows that . This implies that A ϑ = A ∔ N η (A * ), which coincides with the Kreȋn-von Neumann extension if A is uniformly positive and η = 0.

Sufficient conditions for self-adjointness and a variant of Kreȋn's formula
In this subsection we provide different sufficient conditions for the parameter ϑ in G × G such that the corresponding extension in Theorem 3.4 becomes self-adjoint in H; cf. [10, Theorem 4.8], [12, Theorem 3.11] and, e.g. Example 3.6. In Proposition 3.10 below we will make use standard perturbation results as the Kato-Rellich theorem, thus we will restrict ourselves to operators ϑ instead of relations. Recall also the following notions from perturbation theory: If M is a linear operator acting between two Banach spaces then a sequence ( Proposition 3.10. Let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ A * with A j = T ↾ ker Γ j , j = 0, 1, and Weyl function M , and assume that A 1 is self-adjoint in H and that there exists η ∈ ρ(A 0 ) ∩ ρ(A 1 ) ∩ R. Furthermore, suppose that G 0 and G 1 are dense in G and equip G 0 and G 1 with norms which satisfy (i)-(ii) in Corollary 2.11 and Proposition 2.10, respectively. If ϑ is a symmetric operator in G such that

10)
and one of the followings conditions (i)-(iii) hold, (i) ϑ regarded as an operator from G 0 to G 1 is compact, (ii) ϑ regarded as an operator from G 0 to G 1 is relatively compact with respect to M (η) regarded as an operator from G 0 to G 1 , (iii) there exist c 1 > 0 and c 2 ∈ [0, 1) such that Proof. Note first that condition (i) is a special case of condition (ii). Hence it suffices to prove the proposition under conditions (ii) or (iii). By (3.10) the restriction θ := ϑ ↾ G 0 maps into G 1 and the corresponding extensions of A in H satisfy A θ ⊂ A ϑ . We show below that (ii) or (iii) imply the self-adjointness of A θ and hence, as A ϑ is symmetric by Lemma 3.2, the self-adjointness of A ϑ . By Corollary 3.5 the operator A θ = T ↾ {f ∈ dom T : Γf ∈ θ} is selfadjoint in H if and only if Θ = ι + (θ − M (η))ι −1 − is self-adjoint in G. Since ϑ is assumed to be a symmetric operator the same holds for θ, ι + θι −1 − and Θ. From Lemma 2.16 (iv) we obtain that M := M (η) ↾ G 0 is an isomorphism onto G 1 . Thus the symmetric operator −ι + Mι −1 − defined on ι − G 0 is surjective and hence self-adjoint in G. Therefore can be regarded as an additive symmetric perturbation of the self-adjoint operator −ι + Mι −1 − , and the assertion of the proposition holds if we show that Θ is self-adjoint in G.
Assume first that condition (ii) holds, that is, θ is relatively compact with respect to M, and hence also with respect to −M. Making use of the fact that ι + : G 1 → G and ι − : G ′ 1 → G are isometric isomorphisms it is not difficult to verify that ι + θι −1 − is relatively compact with respect to −ι + Mι −1 − in G. Hence by well known perturbation results the operator Θ in (3.11) is self-adjoint in G, see, e.g. [68, Theorem 9.14]. Suppose now that (iii) holds and set ξ = ι − x for x ∈ G 0 . Then shows that the symmetric operator ι + θι −1 − is ι + Mι −1 − -bounded with a relative bound c 2 < 1. Hence the Kato-Rellich theorem [60, Theorem X.12] implies that Θ in (3.11) is a self-adjoint operator in G.
The next proposition is of the same flavor as Proposition 3.10. It can be proved similarly with the help of a variant of the Kato-Rellich theorem due to Wüst; cf. [60, Theorem X.14] and [72]. Proposition 3.11. Let the assumptions be as in Proposition 3.10 and assume that there exists c > 0 such that Then A ϑ = T ↾ {f ∈ dom T : Γf ∈ ϑ} is essentially self-adjoint in H.
Example 3.12. Let ϑ be a symmetric operator in G with G 0 ⊂ dom ϑ, such that ϑ is continuous from (G 0 , · G ′ 1 ) to G 1 . Then condition (iii) in Proposition 3.10 is satisfied with c 2 = 0 and hence the extension A ϑ of A is self-adjoint in H. Now consider ϑ := M (η) ↾ G 0 as an operator from G 0 to G 1 . Then Proposition 3.11 implies that A ϑ is essentially self-adjoint in H. In fact, as in Example 3.9 one verifies A ϑ = A ∔ N η (T ), which is a proper restriction of A ϑ = A ∔ N η (A * ) from Example 3.9.
For completeness we provide a version of Kreȋn's formula for quasi boundary triples in Corollary 3.14 which can be viewed as a direct consequence of Kreȋn's formula for the ordinary boundary triple in Theorem 2.12. A similar type of resolvent formula can also be found in [25,Theorem 7.26] for generalized boundary triples. For the convenience of the reader we first recall Kreȋn's formula for ordinary boundary triples, see, e.g. [26]. For the definition of the point, continuous and residual spectrum of a closed linear relation we refer the reader to the appendix. Theorem 3.13. Let {G, Γ 0 , Γ 1 } be an ordinary boundary triple for A * with γ-field γ and Weyl function M and A 0 = A * ↾ ker Γ 0 , let Θ be a closed linear relation in G and let A Θ be the corresponding closed extension in Theorem 3.1. Then for all λ ∈ ρ(A 0 ) the following assertions (i)-(iv) hold. Corollary 3.14. Let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ A * with γ-field γ, Weyl function M , A j = T ↾ ker Γ j , j = 0, 1, such that A 1 is selfadjoint in H, there exists η ∈ ρ(A 0 )∩ R and G 0 , G 1 are dense in G. Moreover let ϑ ⊂ G ′ 1 × G ′ 0 be a linear relation in ran Γ such that the extension is closed in H. Then for all λ ∈ ρ(A 0 ) the following assertions (i)-(iv) hold.

Applications to elliptic boundary value problems
In this section the abstract theory from Section 2 and Section 3 is applied to elliptic differential operators. In Section 4.1 we first study the Laplacian on bounded Lipschitz-, quasi-convex and C 1,r -domains with r ∈ ( 1 2 , 1]. Then we investigate 2 mth order elliptic differential operators on bounded smooth domains in Section 4.2 and second order elliptic differential operators on domains with compact boundary in Section 4.3. Throughout this section let Ω ⊂ R n , n ≥ 2, be a domain with boundary ∂Ω (which is at least Lipschitz). In Section 4.1 and Section 4.2 the domain Ω is assumed to be bounded, in Section 4.3 the domain Ω may be unbounded as well but its boundary ∂Ω is assumed to be compact. We denote by H s (Ω) the Sobolev spaces of order s ∈ R on Ω and by H s (∂Ω) the Sobolev spaces on ∂Ω of order s (with at least s ∈ [−1, 1] in the Lipschitz case). By H s 0 (Ω) we denote the closure of C ∞ 0 (Ω) in H s (Ω), s ≥ 0, and with C ∞ (Ω) the functions in C ∞ 0 (R n ) restricted to Ω; see, e.g. [53, Chapter 3].

A description of all self-adjoint extensions of the Laplacian on bounded Lipschitz domains
In this subsection we give a complete description of the self-adjoint extensions of the Laplacian −∆ = − n j=1 ∂ 2 j on a bounded Lipschitz domain Ω in terms of linear operators and relations Θ in L 2 (∂Ω) with the help of Theorem 3.7. This description extends the one by Gesztesy and Mitrea in [34], where the class of so-called quasi-convex domains was treated; cf. [34,Definition 8.9]. In addition we introduce Hilbert spaces G 0 and G 1 such that the Dirichletand Neumann trace operator admit continuous and surjective extensions from the maximal domain of the Laplacian onto the anti-dual spaces G ′ 1 and G ′ 0 respectively.
Let Ω ⊂ R n , n ≥ 2, be a bounded Lipschitz domain. For s ≥ 0 we define the Hilbert spaces . We mention that A is a closed, densely defined, symmetric operator in L 2 (Ω) with equal infinite deficiency indices. Let n = (n 1 , n 2 , . . . , n n ) ⊤ be the unit vector field pointing out of Ω, which exists almost everywhere, see, e.g. [53,71]. The Dirichlet and Neumann trace operator τ D and τ N defined by  Let Ω be a bounded Lipschitz domain, let T be as in (4.2) and let Then {L 2 (∂Ω), Γ 0 , Γ 1 } is a quasi boundary triple for T ⊂ A * = ∆ max such that the minimal realization A = ∆ min coincides with T ↾ ker Γ and the following statements hold.
(i) The Dirichlet realization ∆ D and Neumann realization ∆ N correspond to ker Γ 0 and ker Γ 1 , respectively, and both operators are self-adjoint in L 2 (Ω).
(ii) The spaces where f ∈ L 2 (Ω) is the unique solution of the boundary value problem   [44,45]. In particular, item (iii) in Definition 2.1 is valid and assertion (i) of the theorem holds.
The fact that ran Γ is dense in L 2 (∂Ω) × L 2 (∂Ω) will follow below when we verify assertion (ii) of the theorem. For the moment we note that item (ii) in Definition 2.1 holds.
Most of the assertions in (iii) and (iv) are immediate consequences of the definition of the γ-field and the Weyl function corresponding to the quasi boundary triple  Let Ω be a bounded Lipschitz domain. Then the following statements hold.
(i) The Dirichlet trace operator τ D and Neumann trace operator τ N can be extended by continuity to surjective mappings such that ker τ D = ker τ D = dom ∆ D and ker τ N = ker τ N = dom ∆ N .
(ii) For all λ ∈ ρ(∆ D ) the values of the γ-field γ from Theorem 4.1 admit continuous extensions is the unique solution of (4.4). In particular, the space G ′ 1 is maximal in the sense that whenever f ∈ L 2 (Ω) is a solution of the Dirichlet problem (4.4) then the boundary value ϕ belongs to G ′ 1 .
(iii) For all λ ∈ ρ(∆ D ) the values M (λ) of the Weyl function M from Theorem 4.1 admit continuous extensions where f = γ(λ)ϕ is the unique solution of (4.4).
establishes a bijective correspondence between all closed (symmetric, selfadjoint, (maximal) dissipative, (maximal) accumulative) linear relations Θ in L 2 (∂Ω) and all closed (symmetric, self-adjoint, (maximal) dissipative, (maximal) accumulative, respectively) extensions ∆ ϑ ⊂ A * = ∆ max of A = ∆ min in L 2 (Ω). Moreover, the following regularity result holds: If ∆ s is an exten- We note that the abstract propositions from Section 3.2 can be applied to the quasi boundary triple {L 2 (∂Ω), Γ 0 , Γ 1 }, see also Section 4.3. We leave the formulations to the reader and state only a version of Kreȋn's formula as in Corollary 3.14.
In the following we slightly improve Lemma 3.2 by using the fact that ker τ N = ker τ N = dom ∆ N . Lemma 4.6. Let Ω be a bounded Lipschitz domain and let ϑ be a linear relation in L 2 (∂Ω). Then Proof. For f ∈ dom ∆ ϑ we have ϑ τ D f = − τ N f ∈ L 2 (∂Ω) as ϑ is assumed to be a linear relation in L 2 (∂Ω). By (4.1) there exists g ∈ H 3/2 ∆ (Ω) such that τ N g = τ N f and hence where {L 2 (∂Ω), Γ 0 , Γ 1 } is the quasi boundary triple from Theorem 4.1. Then by Lemma 3.2 ∆ ϑ is symmetric in L 2 (Ω) if and only if ϑ is symmetric L 2 (∂Ω).
In the end of this subsection we establish the link to [34] and briefly discuss two more special cases of bounded Lipschitz domains: so-called quasiconvex domains in Theorem 4.9 and C 1,r -domains with r ∈ ( 1 2 , 1] in Theorem 4.10.
For the definition of quasi-convex domains we refer to [34,Definition 8.9]. We mention that all convex domains, all almost-convex domains, all domains that satisfy a local exterior ball condition, as well as all C 1,r -domains with r ∈ ( 1 2 , 1] are quasi-convex, for more details on almost-convex domains see [54].
The key feature of a quasi-convex domain is that the Dirichlet-and Neumann Laplacian have H 2 -regularity, For the next theorem we recall the definition of the tangential gradient operator ..,n from [34, (6.1)]. Here ∂ τ j,k := n j ∂ k − n k ∂ j , j, k ∈ {1, . . . , n}, are the firstorder tangential differential operators acting continuously from H 1 (∂Ω) to L 2 (∂Ω).

Theorem 4.9.
Let Ω be a quasi-convex domain. Then the following statements hold.
(i) The spaces G 0 and G 1 in Theorem 4.1 are given by and for the norms · G 0 and · G 1 induced by the inner products in (4.5) the following equivalences hold: (ii) The Dirichlet trace operator τ D and Neumann trace operator τ N admit continuous, surjective extensions to Proof.
We note that Theorem 4.9 is essentially the same as [34, Theorems 6.4 and 6.10], and also implies [34,Corollaries 10.3 and 10.7]. Theorem 4.9 together with Corollary 4.4 yields results of similar form as in [34,Sections 14 and 15]; the Kreȋn type resolvent formulas in [34,Section 16] can also be viewed as consequences of Corollary 4.5.

Elliptic differential operators of order 2m on bounded smooth domains
In this subsection we illustrate some of the abstract results from Section 2 and Section 3 for elliptic differential operators of order 2m on a bounded smooth domain. The description of the selfadjoint realizations in this case can already be found in Grubb [36], other extension properties obtained below can be found in the monograph of Lions and Magenes [48]. We also refer the reader to the classical contributions [7,8,15,29,36,48,63] for more details on the notation and references, and to, e.g. [16,42,49] for some recent connected publications.
Let Ω ⊂ R n , n ≥ 2, be a bounded domain with C ∞ -boundary ∂Ω. Let A and T be the realizations of the 2m-th order, properly elliptic, formally self-adjoint differential expression with norms induced by the inner products given by We note that H s L (Ω) = H s (Ω) with equivalent norms if s ≥ 2m and that C ∞ (Ω) is dense in H s L (Ω) for s ≥ 0. The minimal and the maximal realization of the differential expression L are given by respectively. We mention that A is a closed, densely defined, symmetric operator in L 2 (Ω) with equal infinite deficiency indices.
In the next theorem a quasi boundary triple for the elliptic differential operator T is defined. Here we make use of normal systems D = {D j } m−1 j=0 and N = {N j } m−1 j=0 of boundary differential operators, Theorem 4.11. Let D be a normal system of boundary differential operators as in (4.14). Then there exists a normal system of boundary differential operators N of the form (4.15) of order µ j = 2m − m j − 1, such that is a quasi boundary triple for T ⊂ A * . The minimal realization A = L min coincides with T ↾ ker Γ and the following statements hold.
(i) The Dirichlet realization L D and Neumann realization L N correspond to ker Γ 0 and ker Γ 1 , respectively, and L D is self-adjoint in L 2 (Ω).
Proof. First we remark that C ∞ (Ω), and hence H 2m (Ω), is dense in H 0 L (Ω). This implies T = A * . According to [48, Chapter 2.1] for a given normal system D of boundary differential operators as in (4.14) there exists a system a normal system N of boundary differential operators of the form (4.15) of order µ j = 2m − m j − 1 such that {D, N } is a Dirichlet system of order 2 m, which acts as a mapping from H 2m (Ω) onto (4.18) The kernel of this map is H 2m 0 (Ω) and Green's formula holds for all f, g ∈ H 2m (Ω); cf. [  such that ker D = ker D = dom L D .
In the next corollary we assume, in addition, that L N = T ↾ ker Γ 1 is self-adjoint. Corollary 4.13. Let {L 2 (∂Ω) m , Γ 0 , Γ 1 } be the quasi boundary triple for T ⊂ A * from Theorem 4.11 with γ-field γ and Weyl function M . Assume that the realization L N of L is self-adjoint in L 2 (Ω). Then the following statements hold.
(i) The mapping Γ 1 = N admits a continuous extension to a surjective mapping such that ker N = ker N = dom L N .
By applying Theorem 2.12 to the quasi boundary triple {L 2 (∂Ω) m , Γ 0 , Γ 1 } from Theorem 4.11 we get the ordinary boundary triple introduced by Grubb in [36], see also [16,38] and [49,Proposition 3.5,5.1]. Recall that there exist isometric isomorphisms Then {L 2 (∂Ω) m , Υ 0 , Υ 1 } is an ordinary boundary triple for A * = L max with A * ↾ ker Υ 0 = L D and As an example of the consequences of the abstract results from Section 2 and Section 3 we state only a version of Kreȋn's formula for the case of 2m-th order elliptic differential operators. We leave it to the reader to formulate the other corollaries from the general results, e.g. the description of the closed (symmetric, self-adjoint, (maximal) dissipative, (maximal) accumulative, respectively) extensions L ϑ ⊂ L max of L min in L 2 (Ω), regularity results or sufficient criteria for self-adjointness, see also Section 4.3 for the second order case. Corollary 4.16. Let {L 2 (∂Ω) m , Γ 0 , Γ 1 } be the quasi boundary triple from Theorem 4.11, and let γ(λ) and M (λ), λ ∈ ρ(L D ), be the extended γ-field and Weyl function, respectively. Assume that L N is self-adjoint, that is a linear relation in ran( D, N ) and that the corresponding extension is closed in L 2 (Ω). Then for all λ ∈ ρ(L D ) the following assertions (i)-(iv) hold: holds for all λ ∈ ρ(L ϑ ) ∩ ρ(L D ).

Second order elliptic differential operators on smooth domains with compact boundary
In this section we pay particular attention to a special second order case which appears in the literature in different contexts, see, e.g., [9,11,12,13,39,40,41].
Let Ω ⊂ R n , n ≥ 2, be a bounded or unbounded domain with a compact C ∞ -smooth boundary ∂Ω and consider the second order differential expression on Ω given by ∂ j a jk ∂ k + a with coefficients a jk ∈ C ∞ (Ω) such that a jk (x) = a kj (x) for all x ∈ Ω and j, k ∈ {1, . . . , n}, and a ∈ L ∞ (Ω) real. In the case that Ω is unbounded we also assume that the first partial derivatives of the functions a jk are bounded in Ω. Furthermore, for some c > 0 the ellipticity condition n j, k=1 a jk (x)ξ j ξ k ≥ c n k=1 ξ 2 k is assumed to hold for all ξ ∈ R n and x ∈ Ω. As in Section 4.2 we define the Hilbert spaces H s L (Ω) and inner products via (4.12) and (4.13), respectively. The minimal and maximal realization of the differential expression L are and we set T := L ↾ H 2 (Ω). The minimal operator A is a closed, densely defined, symmetric operator in L 2 (Ω) with equal infinite deficiency indices. The Dirichlet and Neumann trace operator are defined by and extended by continuity to a surjective mapping (τ D , τ N ) ⊤ : H 2 (Ω) → H 3/2 (∂Ω) × H 1/2 (∂Ω); cf. [48]. Here n = (n 1 , n 2 , . . . , n n ) ⊤ denotes the unit vector field pointing out of Ω. The next theorem is a variant of Theorem 4.1 and Theorem 4.11 with D = τ D and N = −τ N ; cf. [11,12]. We do not repeat the proof here and refer only to [15,Theorem 5] and [8, Theorem 7.1] for the self-adjointness of L D and L N , respectively. As in the previous theorems the spaces G 0 and G 1 from Definition 2.7 turn out to be dense in L 2 (∂Ω), the γ-field coincides with a family of Poisson operators and the values of the Weyl function are (up to a minus sign) Dirichlet-to-Neumann maps. Then {L 2 (∂Ω), Γ 0 , Γ 1 } is a quasi boundary triple for T ⊂ A * = L max such that the minimal realization A = L min coincides with T ↾ ker Γ and the following statements hold.
(i) The Dirichlet realization L D and Neumann realization L N correspond to ker Γ 0 and ker Γ 1 , respectively, and both operators are self-adjoint in L 2 (Ω).
Let {L 2 (∂Ω), Γ 0 , Γ 1 } be the quasi boundary triple from Theorem 4.17. In the same way as in (4.19) and (4.20) we obtain that (τ D , τ N ) ⊤ admits a continuous extension to a mapping where for all s ∈ L (Ω). We note that the latter space coincides with the first order Beals space B 1 L (Ω), see [8].
Then {L 2 (∂Ω), Υ 0 , Υ 1 } is an ordinary boundary triple for A * = L max with A * ↾ ker Υ 0 = L D and As in Section 4.1 we apply Theorem 3.7 to the quasi boundary triple from Theorem 4.17. The regularity statement can be proven in the same way as in Theorem 4.10.
Next we state a version of Lemma 4.6 for the realizations of the second order elliptic differential expression in L .  The next corollary is a consequence of Proposition 3.10 and Proposition 3.11. In item (i) we obtain an additional regularity statement. Then the following statements hold.
(ii) follows in the same way as (i) from Proposition 3.11 and Lemma 4.21.
In the next example we consider a one parameter family L ϑα of extensions of L min which correspond to ϑ α = α M (η). It turns out that for α = 1 the extensions are self-adjoint and for α = 1 essentially self-adjoint.
For α ≤ −1 and α > 1 we make use of Corollary 3.5. For this we set and note that the operators Θ α are self-adjoint in L 2 (∂Ω). Hence Corollary 3.5 yields that for α ≤ −1 and α > 1 the extensions L ϑα are self-adjoint in L 2 (Ω).
The following example is related to the case α = 1 in the above example. It contains an observation which can also be interpreted from a slightly more abstract point of view. Namely, Example 4.24 shows that there exists a quasi boundary triple {G, Γ 0 , Γ 1 } for T ⊂ A * and a self-adjoint relation ϑ in G with ϑ ⊂ ran Γ such that the extension A ϑ := T ↾ {f ∈ dom T : Γf ∈ ϑ} is not self-adjoint in H; cf. Section 3.1. The values of the corresponding Weyl function M 3/2 are mappings from H 1 (∂Ω) to L 2 (∂Ω). For η ∈ R ∩ ρ(L D ) ∩ ρ(L N ) set ϑ := M 3/2 (η) with dom ϑ = H 1 (∂Ω). Then ϑ is a bijective symmetric operator in L 2 (∂Ω) and hence self-adjoint. As in Example 3.9 one verifies that the corresponding extension L ϑ is given by and that L ϑ = L min ∔ N η (A * ) = A * ↾ ker Υ 0 holds; here Υ 0 is the boundary mapping from Corollary 4.19. Therefore L ϑ is a proper restriction of the self-adjoint extension L ϑ and it follows, in particular, that L ϑ is essentially self-adjoint, but not self-adjoint in L 2 (Ω).
The next example is a variant of Example 3.12; cf. Proposition 3.10 (iii).
Proposition 3.10 together with well known compact embedding properties of Sobolev spaces yield some simple sufficient conditions for self-adjoint realizations of L .
write dom Θ := x ∈ G : x x ′ ∈ Θ for some x ′ ∈ G , ran Θ := x ′ ∈ G : x x ′ ∈ Θ for some x ∈ G , ker Θ := x ∈ G : for the domain, range, kernel and multivalued part of Θ, respectively. Note that each linear operator Θ in G is a linear relation if we identify the operator with its graph, and that a linear relation Θ is (the graph of) an operator if and only if the multivalued part of Θ is trivial, that is, mul Θ = {0}.
For a linear relation Θ in G the adjoint relation Θ * is defined by Θ * := y y ′ : (x ′ , y) = (x, y ′ ) for all It follows that the adjoint relation Θ * is closed in G and that Θ * * = Θ. Observe that mul Θ * = (dom Θ) ⊥ and that, in particular, Θ * is an operator if and only if Θ is densely defined. This also implies that for a densely defined operator Θ the above definition of the adjoint coincides with the usual one for (unbounded) operators. A linear relation Θ in G is said to be symmetric if Θ ⊂ Θ * and self-adjoint if Θ = Θ * . We say that Θ is dissipative (accumulative) if Im(x ′ , x) ≥ 0 (Im(x ′ , x) ≤ 0, respectively) holds for all (x, x ′ ) ⊤ ∈ Θ, and Θ is said to be maximal dissipative (maximal accumulative) if Θ is dissipative (accumulative) and does not admit proper dissipative (accumulative, respectively) extensions in G.
Finally we note that a selfadjoint (maximal dissipative, maximal accumulative) relation Θ in G can always be decomposed into the direct sum of a selfadjoint (maximal dissipative, maximal accumulative, respectively) operator in the Hilbert space dom Θ and a purely multivalued relation in the Hilbert space mul Θ. This also shows that the spectral theory of selfadjoint (maximal dissipative, maximal accumulative) operators in Hilbert spaces extends in a natural form to selfadjoint (maximal dissipative, maximal accumulative, respectively) relations in Hilbert spaces.