The theory of selfadjoint extensions of symmetric operators, and more generally the theory of extensions of dual pairs, was implemented some years ago for boundary value problems for elliptic operators on smooth bounded domains. Recently, the questions have been taken up again for nonsmooth domains, with results first on -domains for symmetric or smooth second-order operators, and next on quasi-convex Lipschitz domains for the selfadjoint realizations of the Laplacian. In the present work we show that pseudodifferential methods can be used to obtain a full characterization, including Kreĭn resolvent formulas, of the realizations of nonselfadjoint second-order operators on domains; more precisely, we treat domains with -smoothness and operators with -coefficients, for suitable and . The advantage of the pseudodifferential boundary operator calculus is that the operators are represented by a principal part and a lower-order remainder, leading to regularity results; in particular we analyze resolvents, Poisson solution operators and Dirichlet-to-Neumann operators in this way, also in Sobolev spaces of negative order. Some unbounded domains are allowed.
Key words: Extension theory; Krein resolvent formula; elliptic
boundary value problems; pseudodifferential boundary operators; symbol
smoothing; M-functions; nonsmooth domains; nonsmooth coefficients
MSC (2000): 35J25, 35P05, 35S15, 46E35, 47A10, 47A20, 47G30
The systematic theory of selfadjoint extensions of a symmetric operator in a Hilbert space , or more generally, adjoint pairs of extensions of a given dual pair of operators in , has its origin in fundamental works of Kreĭn , Vishik  and Birman . There have been several lines of development since then. For one thing, there are the early works of Grubb – completing and extending the theories and giving an implementation for results for boundary value problems for elliptic PDEs. Another line has been the development by, among others, Kochubei , Gorbachuk–Gorbachuk , Derkach–Malamud , Malamud–Mogilevskii , where the tendency has been to incorporate the problems into studies of relations (generalizing operators), with applications to (operator valued) ODEs; keywords in this development are boundary triples, Weyl-Titchmarsh -functions. More recently this has been applied to PDEs (e.g., Amrein-Pearson , Behrndt and coauthors [11, 12, 13, 14, 16], Brown-Marletta-Naboko-Wood , Kopachevskiĭ-Kreĭn , Malamud , Ryzhov ). Further references are given in Brown-Grubb-Wood , where a connection between the two lines of development is worked out.
One of the interesting aims is to establish Kreĭn resolvent formulas, linking the resolvent of a general operator with the resolvent of a fixed reference operator by expressing the difference in terms of operators connected to boundary conditions, encoding spectral information.
In the applications to elliptic PDEs, Kreĭn-type resolvent formulas are by now well-established in the case of operators with smooth coefficients on smooth domains, but there remain challenging questions about the validity in nonsmooth cases, and their applications.
One difficulty in implementing the extension theory in nonsmooth cases lies in the fact that one needs mapping properties of direct and inverse operators not only in the most usual Sobolev spaces, but also in spaces of low order, even of negative order over the boundary. Another difficulty is to arrive at a theory where ellipticity considerations are still applicable, in the way that the operators are defined from principal symbols plus lower-order error terms. This is important for regularity questions, as well as for questions of spectral estimates.
Gesztesy and Mitrea have addressed the extension problem for the Laplacian on Lipschitz domains, showing Kreĭn-type resolvent formulas in [24, 25, 26] involving Robin problems under the hypothesis that the boundary is of Hölder class . More recently, they have described the selfadjoint realizations of the Laplacian in  (based on the abstract theory of ), under a more general hypothesis of quasi-convexity, which includes convex domains and necessitates nonstandard boundary value spaces. Posilicano and Raimondi gave in  an analysis of selfadjoint realizations of second-order problems on -domains. Grubb treated nonselfadjoint realizations on -domains in , including Neumann-type boundary conditions
with a differential operator of order 1, where the other mentioned works mainly treat cases (1.1) with of order or nonlocal. ( can be considered as a pilot project for the present paper.) It should also be mentioned that Behrndt and Micheler  recently have shown how a parametrization of the selfadjoint realizations of the Laplace operator on a Lipschitz domain can be obtained by use of the theory of quasi-boundary triples due to Behrndt et al. (cf. e.g. ). Compared with our results less regularity of the boundary is needed in the analysis. But the results are restricted to the Laplacian, while in the following we work with general second order elliptic operators. Moreover, in order to deal with Lipschitz boundary, where the usual results on elliptic regularity might fail, Behrndt and Micheler work with suitable more abstractly defined function spaces. In the case that the boundary is of class for some , their function spaces coincide with the classical ones, which we use in the following, cf. [15, Theorem 4.10] and the discussion below.
Our aim in this paper is to set up a construction of general extensions and resolvents that works in Sobolev spaces when the regularity of is in a scale of function spaces larger than , the coefficients of the elliptic operator in another larger scale, yet allowing the use of pseudodifferential calculi that can take ellipticity of boundary conditions into account and give precise information on the principal parts of the operators. We here choose to work with operators having coefficients in scales of Sobolev spaces and their generalizations to Besov and Bessel-potential spaces, since this allows rather precise multiplication properties, and convenient trace mapping results; then Hölder space properties can be read off using the well-known embedding theorems. The resulting hypothesis on is that it can be parametrized by functions in the Besov space for some . We note that this assumption is equivalent to , where is the regularity number of the Besov space , which is the scaling exponent of the highest order parts of the norms under dilations of functions. It is the most relevant number for Sobolev embeddings, estimates of nonlinearities and applications to nonlinear partial differential equations. We note that (locally) is inbetween and for and any . But the regularity number of is the same as the one of and can be much smaller than .
The theory of pseudodifferential boundary value problems (originating in Boutet de Monvel  and further developed e.g. in the book of Grubb ; introductory material is given in ) is well-established for operators with -coefficients on domains. It has been extended to nonsmooth cases by Abels , along the lines of the extension of pseudodifferential operators on open sets in Kumano-Go and Nagase , Marshall , Taylor , . These results have been applied to studies of the Stokes operator in Abels  and Abels and Terasawa , which in particular imply optimal regularity results for the instationary Stokes system, cf. Abels . For applications to quasi-linear differential equations and free boundary value problems non-smooth coefficients are essential, cf. e.g. Abels  and Abels and Terasawa . The present paper builds on  and ideas of  and develops additional material.
Our final results will be formulated for operators acting between Sobolev spaces, but along the way we also need -based variants with for the operator- and domain-coefficients. Here the integral exponent will be called when we describe the domain and its boundary , and when we describe the given partial differential operators and boundary operators and their rules of calculus. There is then an optimal choice of how to link and , together with the dimension and the smoothness parameters of the spaces where the operators act; this is expressed in Assumption 2.18.
The results in the paper have been applied in Grubb  to show spectral asymptotic estimates for the boundary term in Kreĭn formulas established here.
We originally intended to include -order operators with , but the coefficients in Green’s formula needed an extra, lengthy development of symbol classes that made us postpone this to a future publication.
Plan of the paper. In Section 2, we recall the facts on Besov and Bessel-potential function spaces that we shall need, define the domains with boundary in these smoothness classes, and establish a useful diffeomorphism property. Nonsmooth pseudodifferential operators are recalled, with mapping- and composition-properties, and Green’s formula for second-order nonsmooth elliptic operators on appropriate nonsmooth domains is established. The Appendix gives further information on pseudodifferential boundary operators (dbo’s) with nonsmooth coefficients, extending some results of  to classes. Section 3 recalls the abstract extension theory of , , . In Section 4 we use the dbo calculus to construct the resolvent and Poisson solution operator for the Dirichlet problem in the nonsmooth situation, by localization and parameter-dependent estimates. The construction shows that the principal part of the resolvent belongs to the class of non-smooth pseudodifferential boundary operators, which is essential for the subsequent analysis. Section 5 gives an extension of Green’s formula to low-order spaces, and provides an analysis of and the associated Dirichlet-to-Neumann operator , needed for the interpretation of the abstract theory. In particular it is shown that the operators coincide with operators of the pseudodifferential calculi up to lower order operators, which is one of the central results of the paper. Finally, the interpretation is worked out in Section 6, leading to a full validity of the characterization of the closed realizations of in terms of boundary conditions, and including Kreĭn-type resolvent formulas for all closed realizations . Section 7 gives a special analysis of the Neumann-type boundary conditions (1.1) entering in the theory, showing in particular that regularity of solutions holds when is elliptic.
2 Basics on function spaces and operators on nonsmooth domains
2.1 Function spaces on nonsmooth domains
For convenience we here recall the definitions and properties of function spaces that will be used throughout this paper. Proofs can be found e.g. in Triebel  and Bergh and Löfström . All spaces are Banach spaces, some -based spaces are also Hilbert spaces.
The usual multi-index notation for differential operators with , , and , , will be employed.
For the spaces defined over , the Fourier transform is used to define operators such as (also called ), for suitable functions . In particular, with , stands for . denotes the Schwartz space of smooth, rapidly decreasing functions and its dual space, the space of tempered distributions.
Function spaces. The Bessel potential space in of order is defined for by
normed by . For , a non-negative integer, equals the space of -functions with derivatives up to order in , also denoted . In the case , we omit the lower index and simply write instead of . We denote the sesquilinear duality pairing of with by (linear in , conjugate linear in ).
To describe the regularity, both of domains and of operator-coefficients, we shall also need Besov spaces , where . These are defined by , where
Here, , , is a partition of unity on such that and if , chosen such that for all , .
The parameter indicates the smoothness of the functions. The second parameter is called the integration exponent. The third parameter is called the summation exponent; it measures smoothness on a finer scale than , which can be seen by the following simple relations:
where , , and are arbitrary. (The sign indicates continuous embedding.) The embeddings follow directly from the definition and the fact that if . Here, is the space of sequences such that in the case and if , provided with the hereby defined norm.
We recall that for and , equals the Sobolev-Slobodetskiĭ space , whereas for , it is that equals . (In the following, the - and -notation will be used for clarity; these scales of spaces have the best interpolation properties.) In the case , all three spaces coincide, for general :
The spaces , also denoted when (Hölder-Zygmund spaces), play a special role. For , can be identified with the Hölder space , defined for , and , and also denoted when . For , there are sharp inclusions
here, is the usual space of bounded continuous functions with bounded continuous derivatives up to order .
At this point, let us recall some interpolation results: Denoting the real and complex interpolation functors by and , respectively, we have that if with , , and , , then
If additionally for some and , then
(cf. [17, Theorem 6.4.5]).
General embedding properties. For any , we have the following embeddings between Besov spaces and Bessel potential spaces:
cf. e.g. [17, Theorem 6.4.4].
There are the following Sobolev embeddings for Bessel potential spaces:
provided that , . In particular, for .
For the Besov spaces a Sobolev-type embedding is given by
for any . In particular, combining this with (2.1), we get
when , . We also note that
if and ; this can be found in [57, Section 2.8.1, equation (17)].
Function spaces over subsets of . The Bessel potential and Besov spaces are defined on a domain with -boundary (see Definition 2.4 below) simply by restriction:
for and . Here is defined by for all , embedded in by extension by zero. The spaces are equipped with the quotient norms, e.g.,
In particular, is for and equal to the usual Sobolev space of -functions with derivatives up to order in . We recall that there is an extension operator which is a bounded linear operator , for all , , and satisfies for all . This holds when is merely a Lipschitz domain, cf. e.g. Stein [54, Chapter VI, Section 3.2] and trivially carries over to for . Moreover, in view of the fact that is a retract of , one has that all interpolation and Sobolev embedding results for are inherited by the spaces on .
We shall also need the spaces
Traces. Next, let us recall the well-known trace theorems: The trace map from to , defined on smooth functions with bounded support, extends by continuity to continuous maps for , , ,
All of these maps are surjective and have continuous right inverses.
Vector-valued Besov and Bessel potential spaces. In the following let be a Banach space. Then , , is defined as the space of strongly measurable functions with
and is the space of all strongly measurable and essentially bounded functions. Similarly, , , denotes the -valued variant of .
Furthermore, let be the space of smooth rapidly decreasing functions and let denote the space of tempered -valued distributions. Then the -valued variants of the Bessel potential and Besov spaces of order are defined as
where . Here, is defined by
We will also make use of the Banach space
with the supremum norm.
The properties of the function spaces discussed above for the scalar case, carry over to the vector-valued case. For details, we refer to e.g. [10, 38] (for the Bochner integral and its properties) and to  (for vector-valued function spaces).
In the following we will use some special anisotropic Sobolev spaces.
Let or , and let , with coordinates . For , , set
One has for that
Here, the trace mapping is surjective from to . Namely, when , then is in with (where is the semigroup generated by ). extends to a function .
First of all,
one can apply [17, Corollary 3.12.3] with , , a result from Lions’ trace method of real interpolation, to obtain
For the last assertion, let , ; here . Then
for all , , as explained in [17, Theorem 6.2.3]. Now when is given, let for . Since
we have by [22, Corollary 3.5.6, Theorem 3.4.2] that , so . Here, we use that , as noted above in (2.17). is extended to a function in by a standard “reflection” in , as explained e.g. in [45, Th. I 2.2]. ∎
By use of this lemma we derive the following product estimate, which is essential for the low boundary regularity that we shall allow:
Let and be as in Definition 2.1. Then for every and such that there is some such that
for all , . Moreover, if and , then
uniformly with respect to .
First of all
Since and , where , we have . For the other terms where , we note that
One has in general
provided that and . This estimate easily follows from the Sobolev-type embedding theorems: If , then from (2.11) we have and the statement is trivial. If , then (2.13) implies with , and implies by (2.8) that with . If , one can choose some such that and apply the preceding case. The estimate is also a consequence of Hanouzet [37, Théorème 3].
Finally, if , then we have that