INTRODUCTION. (0.1) f(a) = i f(λ)(a λ) 1 dλ,

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1 INTRODUCTION Vetenskaperna äro nyttiga därigenom att de hindra människan från att tänka. Quoted in Hjalmar Söderberg: Doktor Glas. The main purpose of this work is to set up an operational calculus for operators defined from differential and pseudodifferential boundary value problems, via a resolvent construction, and to present some applications of this to evolution problems, fractional powers, index theory, spectral theory and singular perturbation problems. A side purpose is to give a complete deduction of the properties of the calculus of pseudodifferential boundary problems with transmission, both in the first version by Boutet de Monvel and in a global version, and in versions containing a parameter running in an unbounded set. After the presentations in [BM66,69,71], which were quite sketchy in many respects, both Rempel and Schulze in [R-S82] and the author in [G-G77] (joint work with Geymonat) and [G84] have given supplementary explanations. Working out the parameter-dependent calculus forced us to rethink the whole theory and, in some respects, improve the argumentation. This resulted in the book [G86], and for the present second edition we have moreover extended the theory with or without a parameter to operators and symbols with global estimates in the space variables. This allows the treatment of operators on suitable noncompact manifolds; here we incorporate material from [G-K93] (joint work with Kokholm) and [G95]. 1. Functions of an operator. On the abstract level, there are several wellknown methods to define functions of an operator. For one thing, when A is a selfadjoint operator in a Hilbert space H, f(a) can be defined via the spectral resolution, for any measurable function f(t) on R. Another method, which works when A is not necessarily selfadjoint but has a sufficiently large resolvent set ρ(a), is to define functions f(a) by use of a Cauchy integral formula (0.1) f(a) = i f(λ)(a λ) 1 dλ, 2π C 1

2 2 INTRODUCTION when f(λ) is holomorphic on the spectrum C \ ρ(a) and the integral converges in a suitable sense; here C is a curve going around the spectrum in the positive direction. We use the latter method, and here our point is not merely to define f(a), but rather to analyze its structure in the framework of pseudodifferential boundary operators, to determine its properties in detail. In this method, the fundamental object to analyze is the resolvent R λ = (A λ) 1. As particular applications, we consider the heat operator exp( ta) and the fractional powers A z. The heat operator is a basic tool in the solution of evolution problems (0.2) u t + Au = g for t > 0, u t=0 = u 0, which we shall also discuss. 2. Pseudodifferential operators. The concept of pseudodifferential operators was invented in the 1960s as a class of operators that includes differential operators, their solution operators (or approximate solution operators, called parametrices) in the elliptic case, and certain other integral operators and integro-differential operators, especially the so-called singular integral operators. For example, the Laplace operator (0.3) = 2 x x 2 (= Dx 1 1 Dx 2 n ) n on R n is included as a differential operator, whereas the solution operator ( +1) 1 (easily defined via the Fourier transform) is a pseudodifferential operator (ps.d.o.). Also, the Newton potential solution operator (for n > 2) (0.4) Qf = c n f(x)dx x y n 2 for the equation u = f is a ps.d.o. The Laplace operator is of order 2, and ( + 1) 1 and Q are of order 2. Actually, the theory of ps.d.o.s allows operators of any real order; for example, ( +1) s (defined in L 2 (R n ) by spectral theory) is a ps.d.o. of order 2s for any s R. (Introductions to ps.d.o.s are given, e.g., in Seeley [Se69], Nirenberg [Ni73], Hörmander [H71,85], Shubin [Sh78], Treves [Tre80] and Taylor [Tay81], and we take them up in the present text in Sections 1.2 and 2.1.) An important point in the calculus is the relation between a pseudodifferential operator and its symbol p(x,ξ) (a function of (x,ξ) R n R n ) where, roughly speaking, composition of two ps.d.o.s P and P corresponds

3 INTRODUCTION 3 to multiplication of their symbols p(x,ξ) and p (x,ξ) (modulo errors of lower order), and inversion of a ps.d.o. P corresponds to inversion of the symbol (again modulo lower-order errors). The inversion can be carried out when P is elliptic, i.e., when p has an invertible highest-order part p 0 (x,ξ). Here x represents the space variable (position) and ξ is the dual variable (momentum) appearing by the Fourier transform. When x runs in a manifold Ω, (x,ξ) should be considered as a point in the cotangent space T (Ω). By the symbolic calculus, one has good control over polynomial functions of P, f(p) = k N c k P k, and over suitable rational functions f(p) in case P is elliptic. The study of general functions of P was initiated by Seeley, who analyzed the resolvent Q λ = (P λi) 1 in [Se67] and used it to prove that P z (for any z C) is a classical ps.d.o. when P is elliptic and classical. Further developments have been given in Strichartz [Str72], Dunau [Du77], Widom [Wi78 85], Taylor [Tay81], Iwasaki and Iwasaki [I-I81] and other works; the general outcome is that f(p) lies in a larger class, the more general P (or f) is taken to be. (Closely related to the question of functional calculus are the resolvent studies of various kinds of ps.d.o.s with applications in spectral theory, e.g., Robert [Ro78], Shubin [Sh78], Mohamed [Mo82 84],...) 3. Pseudodifferential boundary problems. Pseudodifferential operators are defined to act on functions (and distributions) on R n, or more generally on functions on open sets or on manifolds without boundary. One of the interesting developments of the theory lies in the study of operators on manifolds with boundary (always smooth in this book). Here one has to adjoin further classes of operators, having their most important effect near the boundary, in order to get a workable calculus. Consider for instance the boundary value problem on a bounded open set Ω R n with smooth boundary Ω: Pu =f in Ω, (0.5) Tu =ϕ at Ω, where P is the Laplace operator and T is a trace operator; for example Tu = γ 0 u u Ω (the Dirichlet case), or Tu = γ 1 u D n u Ω (the Neumann case) where D n is a normal derivative at Ω. The theory must here include both the system (0.6) A = P : C (Ω) T and its solution operator (or parametrix) (0.7) A 1 = (R K): C (Ω) C ( Ω) C (Ω) C ( Ω) C (Ω);

4 4 INTRODUCTION here R and K are the operators solving the problem (0.5) with ϕ = 0 resp. f = 0 (sometimes called the Green, resp. Poisson, operator for (0.5)). We can express R in more detail as (0.8) R = Q + + G, where Q + is the ps.d.o. (0.4) truncated to Ω (one extends f by zero on R n \ Ω, applies Q, and restricts to Ω), and G is a special term adapted to the choice of boundary condition, called a singular Green operator. Q + is also denoted Q Ω. To get a general calculus including all the terms appearing in (0.6), (0.7), (0.8), one must consider systems (0.9) A = P + + G K : T S C (Ω) N C ( Ω) M C (Ω) N, C ( Ω) M where P is a ps.d.o. on R n (satisfying additional hypotheses), P + is its truncation to Ω, G is a singular Green operator (s.g.o., acting in Ω), T is a trace operator (going from Ω to Ω), K is a Poisson operator (going from Ω to Ω) and S is a ps.d.o. acting in the manifold Ω; all the terms possibly matrix-formed (or they can act in vector bundles). Note that (0.6) is a case where N = N = 1, M = 0 and M = 1 (A is a column), and (0.7) is a case where N = N = 1, M = 1 and M = 0 (A is a row). The theory of pseudodifferential boundary problems, ps.d.b.p.s, has its origin in works of Vishik and Eskin (see, e.g., [V-E67] and [E81]), who studied row and column cases. Boutet de Monvel considered in [BM66 71] a particular class of ps.d.o.s P, namely those having the transmission property: The truncated operator P + preserves C up to the boundary. For these he defined the full systems (0.9), showing that they form an algebra : It is closed under composition and has invertible (elliptic) elements, all this being reflected in a corresponding symbolic calculus. He applied this to show an extension of the index theorem of Atiyah and Bott [A-B64] for boundary problems, using the results of Atiyah and Singer [A-S68] for the boundaryless case. Further developments have been made by, e.g., Rempel and Schulze, see in particular their book [R-S82], and the present author (in part jointly with Geymonat), see, e.g., [G-G77,79] and [G77,82,83,84]. (Generalizations of (0.9) to cases where P does not have the transmission property have been studied in [R-S82 84] and in later works of Schulze and others, e.g., [Sl91,94]. See also Melrose [M81 95].) Pseudodifferential boundary problems are of interest not just in themselves but also because they arise from manipulations with differential boundary problems e.g., reductions of multi-order systems as in [G- G77,79], [G77], [G-So87 91] (joint work with Solonnikov), or composition

5 INTRODUCTION 5 of an operator like A in (0.6) with the parametrix of another, used in implicit eigenvalue problems and singular perturbation problems (see Sections ). In the present book we give a survey of the early theory in Section 1.2 and present in detail a parameter-dependent version in Chapters 2 3. The case with boundary is more complicated than the boundaryless case, not just because of the matrix form of A in (0.9) but because the symbolic calculus is no longer just simple multiplication (modulo lower-order errors). One associates two symbols with A: one is the interior symbol, namely the usual symbol p(x,ξ) of P for x Ω, ξ R n ; and the other one is the boundary symbol, which in reality is a family of operators a(x,ξ,d n ) of the form (0.9) with Ω replaced by R + and parametrized by x Ω, ξ R n 1 (more precisely by (x,ξ ) T ( Ω)). The compositions of Green operators (0.9) are now reflected in compositions of these one-dimensional boundary symbol operators, besides multiplication of interior symbols. Similarly, inversion or construction of parametrices involves not just inversion of a (possibly matrix-formed) function of (x,ξ) but also the inversion of boundary symbol operators on R +, parametrized by (x,ξ ). This kind of complexity is of course wellknown also from studies of partial differential operators prior to the systematic pseudodifferential calculus, where one successful technique was to do the following (expressed in a few words): Freeze the coefficients at a boundary point and reduce to an ordinary differential equation (with boundary conditions) by Fourier transformation in the tangential variables. Then build up the general solution from the solutions of the ordinary differential equations with respect to the normal variable at each boundary point. In the present framework, the boundary symbol operator a(x,ξ,d n ) corresponds to the Fourier transformed frozen coefficient case, and the construction of a true parametrix from the symbolic cases (or model cases ) is systematized. In this work we devote much attention to those ps.d.b.p.s that lie fairly close to differential boundary problems. For instance, one can make (0.5) nonlocal in the boundary condition only, by taking T of the form (0.10) Tu = γ 1 u + Sγ 0 u, where S is a first order ps.d.o. on Ω; or one can take (0.11) (0.12) Tu = γ 0 u + T 0u, Tu = γ 1 u + T 1 u, or where T 0 and T 1 are a kind of integral operators going from Ω to Ω, of suitable orders (fitting together with the order of γ 0 resp. γ 1 ). Here (0.10)

6 6 INTRODUCTION represents a mild form of nonlocalness that is often easily handled along with local problems (cf. [Se69]), whereas (0.11) and (0.12) are somewhat more different from differential trace operators since they contain a link between the boundary value and the interior behavior of u. Such conditions were called lateral conditions in Phillips [Ph59], and they occur as boundary renewal conditions in population theory. As for the first line in (0.5), one is not so far from differential operator problems if is replaced by, say (0.13) P = (D 4 x D 4 x n )(D 2 x D 2 x n ) 1 (where (D 2 x 1 + +D 2 x n ) 1 is a parametrix of D 2 x 1 + +D 2 x n =, e.g., Q in (0.4)). The addition of a singular Green operator G to P will give further links between the interior and the boundary; typically, G can be of the form (0.14) G = K 0 γ 0 + K 1 γ 1 + G, where G is an integral operator on Ω, and K 0 and K 1 are integral operators going from Ω to Ω (defining boundary feedback terms). We study these and more general systems {P + + G,T } under the assumption of suitable ellipticity conditions. Along with the stationary problems generalizing (0.5), we also consider time-dependent pseudodifferential evolution problems (0.15) t u + P + u + Gu =f(x,t) for x Ω, t > 0, Tu =ϕ(x,t) for x Ω, t > 0, u t=0 = u 0 (x) for x Ω, where P, G and T can be as described above. An object of importance, both in the study of (0.5) and its generalizations and in (0.15), is the realization associated with P, G and T, namely the operator (0.16) B = (P + G) T, acting like P + + G and with domain (0.17) D(B) = {u H d (Ω) Tu = 0}. Here d > 0 is the order of P (equal to 2 in the above examples); B is a closed unbounded operator in L 2 (Ω). The discussion of (0.15) can be carried out on the basis of a discussion of the operator function exp( tb),

7 INTRODUCTION 7 which we call the heat operator in view of the resemblance with the case where P = and G = 0. Problems like (0.15) can arise as models of concrete problems, for example, in control theory, as considered in Nambu [Nm79], Triggiani [Trg79], Pedersen [Pe91]. They also enter as technical tools in studies of differential boundary problems, e.g., as a result of factorization and other reductions. For example, in boundary problems for the nonstationary Stokes equation, the condition div u = 0 can be eliminated by a reduction to a parabolic problem of the form (0.15) [G-So87,91]. 4. Functional calculus. In the questions of functional calculus of pseudodifferential boundary problems, there are several possible lines to follow. Clearly, one can consider operators A as in (0.9) in the square matrix case, where N = N and M = M so that one can define f(a) as an operator acting in C (Ω) M C ( Ω) N (and in suitable Sobolev spaces). Neither (0.6) nor (0.7) belongs to this case. But they do define the unbounded realization B = P T acting in L 2 (Ω) (and its bounded inverse R); it is of great interest to study functions of B, e.g., in view of the applications to (0.15). The theory we present in this work permits a study, both of square matrix formed systems A as in (0.9) and of realizations B as in (0.16); we focus particularly on the latter case, which we find most interesting. The basic step in the calculus is a study of the resolvent R λ = (B λi) 1. This study is imbedded in a calculus of parameter-dependent pseudodifferential boundary problems. In the differential operator case, the parameter µ = λ 1/d is easily absorbed as an extra cotangent variable because of the polynomial nature of the symbols. In the ps.d.o. case, however, the parameter gives severe extra trouble. This is felt already in the case without boundary: The strictly homogeneous principal symbol p h (x,ξ) of a ps.d.o. P of order d 1 is continuous at ξ = 0 with locally bounded derivatives in ξ up to order d only; the next derivatives are unbounded at ξ = 0 unless P is a differential operator. The ps.d.o. calculus handles easily such an irregularity in a compact neighborhood of ξ = 0 in R n, but when an extra cotangent variable µ R is adjoined, the irregularity extends to a noncompact neighborhood of the full µ-axis in R n+1. (One does not just get (anisotropic) ps.d.o.s in one more variable, as in Fabes and Rivière [F-R67], Lascar [Ls77], Shubin [Sh78], [R-S82, 4.3.6], [S-S94], cf. Section 1.5.) Now when boundary conditions are included, the strictly homogeneous principal boundary symbol operator a h (x,ξ,d n ) is generally irregular at ξ = 0, and the addition of an extra cotangent variable µ gives similar trouble as for p(x, ξ), or even moreso, because here we are treating operator families depending on the parameter (x,ξ ), not just functions. To clarify these phenomena we introduce the regularity number ν for the system {P + + G,T },

8 8 INTRODUCTION defined, roughly speaking, as the highest order of derivatives bounded at ξ = 0, possessed by the strictly homogeneous symbols (the boundedness should hold in suitable symbol norms). It is necessary for the calculus to introduce also noninteger and negative regularity numbers. Keeping account of the regularity numbers is all-important for the strength of the calculus. Then the parametrix construction works best for systems with strictly positive regularity in the principal part, for this assures a good principal parametrix symbol as well as a decrease of the lower-order terms with respect to the parameter. The parameter-elliptic systems occurring in our resolvent study have regularity in the interval [ 1 2,d]. It is perhaps surprising that the three conditions (0.10), (0.11) and (0.12) give different regularities ν for the problem (0.5), namely ν = 1 for (0.10), ν = 1 2 for (0.11), and ν = 3 2 for (0.12). What was most surprising to the author was that the Dirichlet-type condition (0.11) with T 0 = 0 has a very low regularity 1 2, barely missing 0. A large part of the book, namely Chapters 2 and 3, is devoted to the systematic parameter-dependent calculus and its application to the resolvent study. The development of the calculus required a very thorough analysis of the Boutet de Monvel calculus; for instance, there are several systems of symbol seminorms that are equivalent in the nonparametrized case, but which have quite different qualities when the parameter is included (see the discussions of (2.2.82), (2.2.85), (2.4.24), (2.4.26)). This investigation led to improvements also in the nonparametrized case; see [G84]. The operator-theoretic approach in [G77] (applied to strongly elliptic ps.d.o.s resulting from reductions of matrix-formed differential operators) gave results on the principal symbol level, whereas the present calculus treats the full asymptotic expansions. It was first written up in the prepublication [G79], and the results were announced in two short notes [G81,81 ] (and in [G80]); [G86] gave full details with improvements. Also, Rempel and Schulze have taken up the resolvent studies and operational calculus for their operator class, giving principal estimates in [R-S83,84]. Elements of parameter-dependent calculi of ps.d.b.p.s moreover enter in Eskin [E81] and Frank and Wendt [F-W82 85], [We83]. Cordes [Co79] treats the functional calculus from a more abstract point of view, using Banach algebra techniques in a framework of L 2 estimates. 5. Outline. Some prerequisites for our presentation are collected in the Appendix at the end of the book, which the reader is invited to consult for the notation. Chapter 1 starts with some examples and then gives a survey of the previously known theory of parameter-independent ps.d.b.p.s on compact manifolds. The original contributions begin with our study of Green s formula (Section 1.3) and realizations (Section 1.4) of ps.d.b.p.s. Here the

9 INTRODUCTION 9 realizations defined by normal boundary conditions (which we introduce as a natural generalization of the concept for differential operators) are especially interesting. The class of these realizations has the good property of being closed under composition and passage to adjoints (in a precise sense when ellipticity holds). Moreover, the normality is (essentially) necessary for the type of ellipticity with a parameter that our resolvent construction requires. The parameter-ellipticity and parabolicity concepts are explained in Section 1.5, where we also briefly discuss the so-called regularity number ν. In order to provide concrete examples, we hereafter treat some special cases, in Section 1.6 the selfadjoint realizations, and in Section 1.7 the semibounded and coercive realizations. In particular, we show how the Friedrichs extension, alias the Dirichlet realization, fits in. Chapter 1 is only marginally changed since the first edition; e.g., the information on the transmission property is made more precise by use of [G-H90] (joint work with Hörmander). Chapters 2 and 3 give the details of the parameter-dependent calculus and resolvent construction, on which the rest of the book depends. These chapters have in the present edition been completely reworked (using [G-K93] and [G95]) in order to include x-uniformly estimated symbol classes, which allow the study of operators on suitable noncompact manifolds (called admissible here, defined in Section A.5). We go through the full program of building up the calculus: Chapter 2 gives the definition of (uniformly estimated) parameter-dependent ps.d.o.s and their regularity in Section 2.1, the appropriate version of the transmission condition and its background (sometimes called the Wiener-Hopf calculus) in Section 2.2, the definition of boundary symbol classes in Section 2.3, the associated operators and kernels (in particular the negligible ones) and their behavior under coordinate changes in Section 2.4, mapping properties in Section 2.5, and composition rules in Sections An advantage of the x-uniform calculus, developed from the techniques for ps.d.o.s on R n given in [H85, Sect. 18.1], is that the composition rules have a more exact form; one does not need a lot of different types of negligible operators as in the first edition of this book. Moreover, the case of fully x n -dependent symbols, which needed a particular effort in [G86], is more manageable here. The choice of symbol classes is aimed towards a good parametrix construction. This is carried out in Chapter 3, in particular for the resolvent of (0.16) in Section 3.3. A reader who wants a complete explanation of the nonparametrized theory can actually find it in Chapters 2 3 by disregarding the effects of the parameter. Finally, Chapter 4 gives some applications. The presentation here follows the first edition closely, extending, however, the results to noncompact

10 10 INTRODUCTION manifolds whenever relevant. First we consider (0.15). For one thing, the solvability can be discussed as in classical works on parabolic differential operators, on the basis of the estimates of the resolvent obtained in the preceding chapter. This is done rather briefly in Section 4.1. Second, we discuss in much detail the heat operator exp( tb) defined by (0.1), in particular its kernel properties. Here the heat operator exp( tp) for the boundaryless case is also discussed, for comparison and for the sake of completeness. However, the largest efforts are devoted to the term (0.18) W(t) = exp( tb) exp( tp) +, which is of a singular Green operator type. An interesting consequence of the study of exp( tb) is the trace formula in the case where Ω is compact: (0.19) Tr exp( tb) = c n (B)t n/d + c 1 n (B)t (1 n)/d + + c ν (B)t ν /d + O(t (ν 1 4 )/d ) for t 0+, where ν is the regularity of {P + + G,T } and ν is the largest integer in [0,ν[. For exp( tp), one has a more complete asymptotic expansion of the trace, in terms of powers t (j n)/d with j N and logarithmic expressions t k+1 log t with k N; cf. Duistermaat and Guillemin [D-G75] and Widom [Wi78,80]. We show how that expansion follows from our resolvent calculus in Section 4.2. Now the regularity number ν in (0.19) is always d when {P + +G,T } is genuinely pseudodifferential, so (0.19) does not get near the first logarithmic term t log t, which one would expect to have. For exp( tp) and exp( tp) +, one can overcome this by composition of the resolvent with high powers of P, which improves the regularity; but for exp( tb), the composition with high powers of B does not improve the regularity. This is discussed at the end of Section 3.3 and in Section 4.2. (A full expansion of Tr exp( tb) with logarithmic terms has recently been obtained in Grubb and Seeley [G-Se95] for the special case of Atiyah-Patodi-Singer problems; this is not included here.) Formula (0.19) is accurate enough (just accurate enough, for Dirichlettype trace operators (0.11)!) to lead to a new index formula (Section 4.3) for general normal elliptic realizations B of elliptic ps.d.o. boundary problems: (0.20) index B = c 0 (B B) c 0 (B B) (previously known for differential boundary problems; cf. Atiyah, Bott and Patodi [A-B-P73], Greiner [Grei67]). It involves slightly fewer terms from the symbols of T and G than the formula of Rempel [Re80] for general elliptic ps.d.o. problems.

11 INTRODUCTION 11 Section 4.4 discusses another operator function, namely the complex powers B z, again defined by use of (0.1). Here (0.21) B z = (P z ) + + G (z) where the ps.d.o. P z has been well studied (cf. [Se67]), so that the main interest lies in the study of G (z). It usually is not a singular Green operator but a generalized kind, the symbol satisfying certain but not all of the usual estimates. The symbol is analyzed for Re z < 0, and we furthermore show that the kernel (integrated in the normal variable) extends to a meromorphic function of z in the region {z C Rez < (ν 1 4 )/d}, with poles in the set (0.22) {z = (j n)/d j = 1,2,...,n + ν,j = n}. In the case of a compact manifold this also holds for the trace, which is welldefined for Re z < n/d. (The obstacles to extension above Re z = (ν 1 4 )/d are the same as those for getting more terms in (0.19).) For some cases where B is selfadjoint positive, we characterize the domains of fractional powers B θ, θ ]0,1[, as in Grisvard [Gri67], [Se71] for differential operators. Section 4.5 is devoted to spectral theory in the case where Ω is compact. Asymptotic eigenvalue estimates with remainder estimates were obtained for ps.d.b.p.s already in [G78,78,83] (independently of the present resolvent calculus), and we begin by recalling these and some corollaries, e.g., the estimate (0.23) N(t;B) = C(p 0,Ω)t n/d + O(t (n θ)/d ) for t, which holds for general B = (P + G) T with θ < 1 2, and with θ < 1 in some special cases. Here N(t; B) denotes the number of characteristic values s k (B) (eigenvalues of B = (B B) 1 2 ) in the interval [0,t]. (It should be recalled here that the finer estimates of Hörmander [H68], Demay [D77], Seeley [Se78], Ivrii [Iv80 84], Métivier [Me83], Vasiliev [Va83,84],..., with θ = 1 in the remainder term, or with a more precise asymptotic estimate replacing the remainder term, are essentially concerned with either boundary value problems for differential operators or with ps.d.o.s on boundaryless manifolds.) In comparison, Levendorskii [Le85,90] can treat some abstractly defined realizations on domains with nonsmooth boundaries, obtaining θ < 1 3 in general and up to θ < 2 3 under the assumption that p0 (x,ξ) is microlocally smoothly diagonalizable. One consequence of (0.23) is the estimate for elliptic ps.d.o.s Q of negative order d having the transmission property at Γ, supplied with an arbitrary s.g.o. term G of order d and class 0: (0.24) N (t;q + + G ) = C (q 0,Ω)t n/d + O(t (n θ)/d ) for t ;

12 12 INTRODUCTION here θ is as above. (N (t;q + + G ) denotes the number of characteristic values 1/t.) Q + is also treated in works of Widom [Wi84,85] and later Andreev [An86,90]. Next, the symbol estimates for complex powers derived in Section 4.4 are used to show that the generalized s.g.o. term G (z) (cf. (0.21)) satisfies (0.25) s k (G (z) ) c z k d Re z /(n 1) for all k, when Rez < (2d) 1 and n > 2 (a slightly weaker estimate is shown for n = 2), by an application of the method in [G84]. In particular, G (z) is of trace class if Rez < (n 1)/d. Furthermore, the estimate (0.25) is used to obtain spectral estimates for the complex powers B z and for the positive and negative part of B in the case where P is not assumed strongly elliptic (as it was in [G78,78,83]). In Section 4.6, we show how the preceding results can, by use of some special reductions, be applied to implicit eigenvalue problems (of Pleijel type [Pl61]) (0.26) λa 1 u = A 0 u on H r+d (E), γ j u = 0 for j < (r + d)/2, where A 1 is selfadjoint elliptic and of higher order (r + d) than A 0 (symmetric and of order r 0); here A 1 and A 0 are of the form A 1 = P 1,+ +G 1, resp. A 0 = P 0,+ + G 0. It is found that the eigenvalues λ behave like the eigenvalues of an operator of order d, with a principal asymptotic estimate in general cases, and with a remainder estimate as above when A 0 is also elliptic. The results extend to multi-order systems. (They have some overlap with results of Kozlov [Kol79,83], Levendorskii [Le84 90], Ivrii [Iv84].) Finally, in Section 4.7 we show how our parameter-dependent theory can be applied to the study of singular perturbation problems. One considers the perturbed problem (0.27) ε d A 1 u ε + A 0 u ε = f in Ω, T 0 u ε = ϕ 0 at Γ, T 1 u ε = ϕ 1 at Γ, in relation to the unperturbed problem (0.28) A 0 u = f in Ω, T 0 u = ϕ 0 at Γ, where A 1 is of order d 0 + d and A 0 is of order d 0 (d 0 0 and d > 0), like in the preceding application, and T 1 and T 0 are trace operators, with T 1 normal and formed of strictly higher order operators than those in T 0. To treat this problem (in the differential operator case as well as for pseudodifferential generalizations), we describe a method consisting of a reduction

13 INTRODUCTION 13 by use of parameter-independent operators to a situation where the results of Chapter 3 can be applied, with µ = ε 1. This gives a simple and straightforward representation of the solution u ε and its relation to the unperturbed solution u, with natural estimates of the convergence as ε 0. The problem has been studied earlier by Vishik and Lyusternik [V-L57], Huet [Hu66 85], Greenlee [Grl69], Demidov [De75], Eskin [E81], Frank and Wendt [F-W82 85], [We83] and many others. In comparison with the treatment of (0.27) (0.28) in the works of Frank and Wendt using pseudodifferential considerations, our method has the advantage, for the cases it can treat, of eliminating the difficulties occurring in problems of negative regularity. Moreover, it extends readily to general (nonrational) ps.d.b.p.s in vector bundles. 6. Further developments. The results in the first edition of this book on the operator family exp( tb) have been followed up by a more extensive treatment of parabolic problems (0.15) in [G-So90] and [G95]. Purmonen [Pu89,90] has treated cases with higher powers of t (corresponding to polynomially parameterdependent symbols). The results of [G-So90] were applied to Navier-Stokes problems in [G-So91]; shorter presentations were given in [G-So87,89]. The present book gives results in L 2 Sobolev spaces, but extensions to L p -based generalizations are of interest too, especially for nonlinear problems. [R-S82] contains some remarks on L p theory that do not fully achieve the goal for Poisson operators. We therefore worked out an L p calculus in [G90] but were informed, during the publication procedure, of the work of Franke (a summary [Fra85] and a not yet published thesis [Fra86]) treating the ps.d. boundary operator calculus in Besov and Triebel-Lizorkin spaces Bp,q s and Fp,q s with p,q ]0, ] (p < for the F-scale). [G90] treats smaller scales (p > 1, q = 2 in the F-case) but goes further in some respects, e.g., including operators P + + G of negative class, which allows a sharp discussion of mapping properties (there are also some improvements concerning range complements in the elliptic case, and order-reducing operators). Since [Fra86] builds on unavailable material on the treatment of Poisson operators, the questions were taken up again by Johnsen, who worked out a full calculus in [Jn95] in the Bp,q s and F p,q s scales, generalizing the results of [G90] and [Fra86]. For parabolic problems, an L p theory with 1 < p < has been worked out in [G-K93] and [G95]. The parameter-dependent calculus in [G-K93] is set up for Bessel-potential spaces Hp s,µ and Besov spaces Bp s,µ, and the parabolic theory in [G95] is for anisotropic Bessel-potential and Besov spaces. Applications to the Navier-Stokes problem have been given in [G91 ], [G94] and [G95 ]. To our knowledge, a parabolic theory with p 1 for ps.d.b.p.s has not yet been established.

14 14 INTRODUCTION In the question of resolvent trace expansions (generalizations of (0.19)), a full expansion with logarithmic terms was obtained in [G-Se95] for the special case of Dirac-type operators with ps.d. boundary conditions. This was based on a mixture of several types of asymptotic expansions of the symbols but used in an essential way that the interior operator was differential. We have worked out some elements of a calculus allowing the interior operator to be a ps.d.o., but not yet in a publishable form. Other treatments of ps.d.b.p.s have been developed by, e.g., Schulze [Sl91,94], Schrohe and Schulze [S-S94] and Melrose [Me92,95], considering manifolds with singularities, and Schrohe [Sr91,94], considering operators in global, weighted Sobolev spaces.