CC BY
7
УДК 517.55
The Neumann Problem after Spencer
Azal Mera*
University of Babilon Babilon, Iraq Institute for Mathematics University of Potsdam Karl-Liebknecht-Str., 24/25, Potsdam (Golm), 14476
Germany
Nikolai TarkhanoV
Institute for Mathematics University of Potsdam Karl-Liebknecht-Str., 24/25, Potsdam (Golm), 14476
Germany
Received 19.03.2017, received in revised form 20.05.2017, accepted 06.07.2017 When trying to extend the Hodge theory for elliptic complexes on compact closed manifolds to the case of compact manifolds with boundary one is led to a boundary value problem for the Laplacian of the complex which is usually referred to as Neumann problem. We study the Neumann problem for a larger class of sequences of differential operators on a compact manifold with boundary. These are sequences of small curvature, i.e., bearing the property that the composition of any two neighbouring operators has order less than two.
Keywords: elliptic complexes, manifolds with boundary, Hodge theory, Neumann problem. DOI: 10.17516/1997-1397-2017-10-4-474-493.
Introduction
In the theory of elliptic linear partial differential equations the term coercive is used to describe a certain class of boundary value problems for elliptic systems Lu = f, in which, for functions u satisfying the boundary conditions, it is possible to estimate in relevant norm all the derivatives of u of order equal to the order m of L in terms of the norm of Lu and in terms of suitable norms for the given boundary data. That is, there is no loss in derivatives - in going from Lu to u we gain precisely m derivatives. Nowadays such boundary value problems are called simply elliptic, where the ellipticity refers to the invertibility of both interior and boundary symbols, the last condition being also known as the Shapiro-Lopatinskij condition.
In connection with the study of inhomogeneous overdetermined systems of partial differential equations, Spencer [16] proposed a method which leads in some cases to well determined elliptic boundary value problems which are however not coercive. In case the systems consists of the inhomogeneous Cauchy-Riemann equations for differential forms the resulting boundary value problem is called the d -Neumann problem. Extending a basic inequality of [8] this problem was solved in [6] for forms on strongly pseudo-convex domains on a complex manifold. The elliptic operator L in the d-Neumann problem is of second order, and in going from Lu to u, in a pseudo-convex domain, one gains only one derivative instead of two. This makes the problem more difficult than a coercive one, the main difficulty occuring in the proof of regularity at the
*mera@math.uni-potsdam.de; azalmera@gmail.com ttarkhanov@math.uni-potsdam.de © Siberian Federal University. All rights reserved
boundary. The regularity proof in [6] is rather complicated. A simpler proof was found in [8]. In [7] is also presented a simpler proof which yields a raher general theorem for elliptic equations, Theorem 5 of Sec. 2. The result for the d-Neumann problem is a very special case of this theorem.
In [7], the results are expressed in a fairly general form which may eventually prove useful in carrying out Spencer's attack on overdetermined equations. For functions u and v with values in Cki or in a smooth vector bundle F1 over a compact manifold with boundary X one considers a sesquilinear form Q(u,v) which is an integral over X of an expression involving derivatives of u and v. For functions u, v lying in a linear space V determined by certain boundary conditions one is looking for a solution u gV of Q(u, v) = (f, v) for all v G V, where f is a given function with values in Fl and (■, ■) denotes the L2 scalar product of sections in X. The form Q is primarily assumed to be almost Hermitean and that K Q(u,u) > 0 for u G V. The paper [7] is aimed at obtaining solutions that are regular in X up to the boundary. The solutions then lie in V and satisfy also "free" or "natural" boundary conditions.
It was Sweeney, a PhD student of Spencer, who developed the approach of [7] within the framework of overdetermined systems, see [11], [12-15]. A differential operator A0 is said to be overdetermined if there is a differential operator A1 = 0 with the property that A1 A0 = 0. Then, for the local solvability of the inhomogeneous equation A0u = f it is necessary that the right-hand side satisfies A1 f = 0. The above papers deal with sesquilinear forms Q(f,g) = (Alf,Aig) + (A1-1 * f, Ai-1*g) + (f,g) called the Dirichlet forms. This work is intended as an attempt at motivating an interesting class of perturbations of the Neumann problem after Spencer. It corresponds to "small" perturbations of complexes of differential operators which are are known as quasicomplexes, see [18].
Assume that X is a compact n -dimensional manifold with boundary. For each nonnegative integer i let Fl be a vector bundle over X, and let A1 be a first order differential operator which maps Csections of F1 to Csections of Fi+1. Suppose that the compositions A1 A1-1 are all of order not exceeding 1 so that the operators A1 form a sequence
0 C^(X, F0) —+ C^(X, F1) —» C^(X, Fn ) 0 (0.1)
whose curvature A1 A1-1 evaluated in appropriate Sobolev spaces is compact at each step.
The assumption that all of A1 have order 1 simplifies the notation essentially. This will usually not be the case in practice. However, this assumptions is fulfilled for classical complexes of differential operators which arise in differential geometry, see [17, Ch. 1].
As but one example of quasicomplexes of purely geometric origin we mention the sequence related to any connection on a smooth vector bundle over X, see for instance [19, Ch. III].
Example 0.1. Let F be a smooth vector bundle of rank k on X. For i = 0,1,..., n, we denote by Ql(X, F) the space of differential forms of degree i with Ccoefficients on X taking on their values in F. Pick a connection d on F. Consider the sequence
d0 -I d1
0 ^ C^(X,F) n1(X,F) ... ttn(X,F) ^ 0,
where d0 = d, d1 is a natural extension of d0 to one-forms under preservation of the Leibniz rule, etc. Since 3l+13l is a differential operator of order 0, the sequence is a quasicomplex. The principal symbols of the (formal) Laplacians Ala are given by
)(x,0= I Fx ® ^m^o,
where A"1 are the Hodge-Laplace operators. We thus conclude that &2(Aq)(x,£) is invertible for all (x,£) G T*X \ {0}. Hence, Ala is a second order elliptic differential operator on X.
Note that the quasicomplex of connections is a complex if and only if the associated bundle is trivial.
Another quasicomplex of great importance in complex analytic geometry is related to certain "small" perturbations of the Dolbeault complex.
Example 0.2. Assume that X is a complex (analytic) manifold of dimension n. As usual, we denote by O°'i(X) the space of all differential forms of bidegree (0, i) with Ccoefficients on X, where 0 ^ i ^ n. Locally such a form can be written as
f (z) = Y. fj (z)dzJ,
where z = (z1,..., zn) are local coordinates, dzJ = dzjl A... A dzji and fi are Cfunctions of z with complex values. Analogously to the exterior derivative d one defines the Cauchy-Riemann operator d which maps the differential forms of bidegree (0, i) to differential forms of bidegree (0, i +1) on X, see for instance [17,19]. Moreover, 32 = 0, i.e., the spaces O0,i(X) are gathered together to constitute a complex of first order differential operators on X called the Dolbeault complex. This complex is proved to be elliptic in (the interior of) X. Choose any differential form a of bidegree (0,1) with smooth coefficients on X and consider the sequence
0 —4 O 0'0(X) —4 O 0'1(X) —4 ...^+4 O 0n(X) —4 0 (0.2)
which is equipped with differential d + a given by (d + a)u = du + a A u for u G O°'q. Since
(z + a)2u = (z + a)(du + a A u)
= d2u + da A u — a A du + a A du + a A a A u = da A u,
the curvature of sequence (0.2) is equal to <9a. It follows that (0.2) is a quasicomplex. Moreover, it is a complex if the form a is <9-closed. The symbol sequence of (0.2) coincides with that of the Dolbeault complex, and so the quasicomplex is elliptic in X.
The purpose of this paper is to show how one obtains existence and regularity theorems for the Neumann problem after Spencer, see (4.1), if an estimate of the form
\\f 111/2 < c (№f ||2 + \\Ai-1*f ||2 + \\f ||2)
holds for all smooth f satisfying certain boundary conditions. In the case of zero curvature, i.e., AlAi-1 = 0, basic results are contained in [7]. If AiAi-1 ^ 0, however, the theorems of [7] do not immediately apply. Our contribution rests on a detailed study of the boundary conditions which settles the matter of "free boundary conditions".
A major part of the paper is concerned with solving equations of the type Q(u, v) = (f, v) for all v G D. The form Q(u,v) is an integral of a sum of squares. In [7] also more general forms are considered, admitting a mild non-Hermitean part. Since the problem is not assumed to be coercive, one must be rather careful in handling the error terms which usually arise from derivatives of the coefficients, when deriving estimates. On assuming that Q(u,u) > \\u\\2 for u in a subspace D (after adding (u, v) to Q), and that Q(u, u)1/2 is compact with respect to the L2 norm, i.e., that any sequence (uv) with Q(uv,uv) bounded has a convergent subsequence in L2, one shows that the equation can be solved, the space of solutions of the homogeneous equation is finite dimensional, and that the solution operator is compact. On assuming a gain of derivatives we present a regularity theorem for solutions.
Similarly to [7], our results are not complete in themselves, but are meant as a technical aid in obtaining more definitive results. For no indication is given when a priori estimates hold. Indeed it seems to be rather difficult to say in general when they can be established.
It is very easy to prove the existence of a Hilbert space solution of the equation Q(u, v) = (f, v) for all v G V. But we are interested in those solutions which are smooth in X. To do this we derive a priori estimates for the L2 norms of derivatives of u. Near the boundary we first estimate derivatives in directions tangential to the boundary by essentially setting v equal to tangential derivatives of u. To this end, we assume that the boundary conditions are, in some sense, invariant with respect to translation along the boundary. Then, assuming the boundary to be noncharacteristic, we estimate also the normal derivatives. Then we are faced with the standard problem of going from a priori estimates of derivatives to the proof of their existence.
There is, as yet, no general theorem which states that whenever one has a priori estimates for derivatives of a function then, in fact, these derivatives exist. In each individual case one has to prove this separately, and this is often the most tedious and technical aspect of existence theorems. One way which is often used is to apply a smoothing operator to the solution. In order to apply the a priori estimates to the resulting functions it is necessary to handle the term arising from the commutator of the differential operator and the smoothing operator. This is sometimes rather complicated. This method is used extensively in the book [3], where a number of special lemmas concerned with the commutators of differential and smoothing operators are given.
In [7] another method of smoothing is used. It is more closely related to differential operators, and has proved useful in a wide class of problems. It consists in adding e times an elliptic operator so that the resulting equation becomes elliptic and coercive under the given boundary conditions for e > 0, even if the original equation is not elliptic. Thus we rely on the fact that the differentiability theorems are well known for such problems and we wish to reduce the differentiability theorems to those for coercive elliptic problems. The new equation, being coercive elliptic, has a smooth solution uE in X and, if the elliptic term has been added in a suitable way, the method of obtaining a priori estimates applies as well to the new equation as to the original one, and yields estimates for the derivatives of uE which are independent of e. Letting e ^ 0 through a sequence ev, it follows that a subsequence of the uEv, together with derivatives, converges to a smooth solution of the original problem.
This method, therefore, does not show that a generalised solution u is smooth, but constructs a smooth solution. If there is uniqueness among generalised solutions, then one may also infer that u is smooth.
Part 1. The Neumann problem for quasicomplexes 1. Preliminaries
Corresponding to each point x G X and cotangent vector £ G TX X there is associated with (0.1) a sequence of linear mappings
0 F° * F1 °1{—$x* ... '1(A—$x* FN 0, (1.1)
where Fi is the fibre of the bundle Fi over x and a1(Al)(x, £) the principal homogeneous symbol of A1 at (x,£). Since A1 A1-1 = 0 it follows that a1 (Ai)a1 (Ai-1) = 0, i.e., the symbol sequence (1.1) constitutes a complex. A cotangent vector £ G T*X is said to be noncharacteristic for the quasicomplex (0.1) if the symbol complex is exact.
In what follows, functional methods are used to study quasicomplex (0.1), and it will be necessary to have L2 norms defined for sections of the vector bundles F\ Accordingly, we shall always consider X to have a Riemannian structure with volume element dv, and we shall assume that each Fl has a CHermitean inner product (■, ■) x defined along its fibres. For arbitrary
sections f,g G C™(X,Fi), we define
(f,g)= (f (x),g(x))xdv Jx
and \\f \\ = /(f, f). Then L2(X,Fi) can be defined as the completion of C^(X,Fi) in the norm \\ • \\.
In a similar way, we use the induced area element ds on the boundary S of X to introduce the space L2(S,Fi) with scalar product (•, •)S and norm \\ • \\S.
As usual, we write Ai-1* for the formal adjoint of Ai-1 as determined by the inner products in the spaces L2(X,Fi-1) and L2(X,Fi). Thus Ai-1* is the unique differential operator from sections of Fi to sections of Fi-1 of order 1, such that (Ai-1u,g) = (u,Ai-1*g) whenever u G C^(X, Fi-1) and g G C^(X, Fi) have support in the interior of X.
We will also use the Sobolev norms \\ • \\s defined for sections of Fi, where s is a real number. Remark that if X the closure of an open set in Rn, Fi = X x Cki and s is a nonnegative integer, then the norm \\ • \\s on C^(X, Fi) is equivalent to the norm
f 4 (£ \\daf \\2)1/2,
\a\^.s
where da = d^1 ... .
The construction of Sobolev spaces on the compact closed manifold S is more direct. We write \\ • \\s,s for the Sobolev norm on C^(S,Fi) and Hs(S,Fi) for the corresponding function space.
2. A boundary decomposition
The operators Ai = Ai*Ai + Ai-1Ai-1* are called the Laplacians of (0.1). The unit normal vector v(x) of the boundary dX is noncharacteristic for the quasicomplex at step i if and only if dX is noncharacteristic for the Laplacian Ai G Diff2(X; Fi) at x. Throughout the paper we make the standing assumption that the conormal bundle of the boundary is noncharacteristic for quasicomplex (0.1) at steps i — 1 and i.
We can assume without loss of generality that X is embedded into a larger smooth manifold X' without boundary. Choose a smooth function g in a neighbourhood U of dX in X' which is negative in U n (X\ dX), positive in U n (X' \ X) and whose differential does not vanish on dX. By shrinking U if necessary, we may actually assume that \dg(x)\ = 1 holds for all x G dX, for if not, we replace g by g/\dg\.
Lemma 2.1. For x G dX, the cotangent vector dg(x) G T*X is independent of the particular choice of g.
Proof. Let g1 and g2 be two functions with the properties described above. For each x G dX there is a neighbourhood Ux of this point in X', such that g2 = fg1 in Ux with some smooth function f in Ux. It is clear that f is positive in Ux \ dX. Furthermore, we get dg2 = fdg1 on Ux n dX whence f = 1 on Ux n dX, as desired. □
Write ai(x) for the principal homogeneous symbol of Ai evaluated at the point (x, dg(x)) of T*X. This is a smooth section of the bundle Hom(Fi,Fi+1) whose restriction to the surface dX does not depend on the particular choice of g, the latter being due to Lemma 2.1. The principal homogeneous symbol of Ai evaluated at (x,dg(x)) is ai(x)*ai(x) + ai-1(x)ai-1(x)*, which we denote by t(x) for short. Since the boundary is noncharacteristic for Ai, the map t(x) G Hom(Fx) is invertible for all x in some neighbourhood of dX in X', and similarly for the symbol li-1(x).
Theorem 2.2. The restriction of the bundle Fi to the surface dX splits into the direct sum
Fi \8X = Fi © ai-1 Fi-1, where Fi = ai*ai(li)-1Fi [dX is a smooth subbundle of Fi [dX■ Proof. For each x £ X close to the boundary, any f £ FX can be written in the form
f = t(f)+ °i-1 (x)n(f), (2.1)
where
t(f) = ai(x)*ai(x)(£i(x))-1f, n(f) = ai-1(x)*(t(x))-1f
prove to satisfy t o t = t, t o n = n, n o t = 0 and n o n = 0. This establishes the theorem. □
Note that if Fi = AiT*X is the bundle of exterior forms of degree i over X then Fi = i*Fi is the pullback of Fi under the embedding dX ^ X. It follows that Fi = AiT*(dX).
3. Green formula
To describe natural boundary value problems for solutions of Aiu = f in X, one uses a Green formula related to the Laplacian Ai. Such formulas are well understood in general, see for instance Lemma 3.2.10 in [17]. In this section we just compute explicitly the terms included into this formula, to get it in the form we need.
Theorem 3.1 (Green formula). For all smooth sections u and v of Fi over X it follows that
/ f(t(u),t£n(Av))X - (iln(u), t(A*v))X + (t(A*u), iln(v))x - (iln(Au),t(v))X) ds =
= (( Au,v)x - (u, Av)x^J dv, J X
where i = \J-1.
Proof. Let GA(*g, u) be the Green operator for a differential operator A = Ai, see § 2.4.2 of [17]. Here, * : Fi+1 ^ Fi+1' is the fibrewise Hodge star operator determined by (*g,f) = (f,g)x for all f £ Flx+1. An easy computation shows that the pullbacks of differential forms GA(*g,u) and Ga* (*u,g) under the inclusion dX ^ X amount to
i* GA(*g,u) = (t(u),iln(g))xds, i* Ga* (*u,g) = -(iln(g),t(u))xds
on dX for all smooth sections g and u of Fi+1 and Fi, respectively, cf. § 3.2.2 ibid. Applying Corollary 2.5.14 of [17] establishes the formula. □
Theorem 3.1 shows immediately that the quadrupel t(u), n(u), t(A*u) and n(Au) gives a representation of the Cauchy data of u on the surface dX relative to the Laplacian A. The tangential part of the Cauchy data, (t(u),t(A*u)), is usually referred to as the Dirichlet data, and the normal part of the Cauchy data, (n(u), n(Au)), is referred to as the Neumann data. This designation is due rather to the whimsical development of mathematics than to well-motivated choice, for, at the last step of the quasicomplex, the data (t(u),t(A*u)) reduce to t(A*u), which is the classical Neumann data, and (n(u), n(Au)) reduce to n(u), which is the classical Dirichlet data.
^ u = f in
n(u) = 0 on d X,
(Au) = 0 on d X,
4. The Neumann problem
In his paper [16], Spencer proposed a method of studying the cohomology of an elliptic complex similar to (0.1) at step i. The main step involves the boundary value problem
(4.1)
where f is a given section of Fi over X.
Example 4.1. In the special case of the de Rham complex and i = 0 problem (4.1) reduces to the classical Neumann problem. For n(du) amounts to the normal derivative of u at dX.
Even in the classical case, (4.1) is not solvable unless f satisfies additional conditions. Since (4.1) is a boundary value problem symmetric with respect to the Green formula, it is solvable only if f is orthogonal in the L2 sense to the space Hi(X) of all h G C^(X, Fi) satisfying the corresponding homogeneous problem, i.e., Aih = 0 in X and n(h) = 0, n(Ah) =0 on dX. The sections of Hi(X) are called harmonic.
Lemma 4.2. A section h G CX, Fi) is harmonic if and only if Ah = 0, A*h = 0 in X and n(h) =0 on dX.
Proof. The point here is that the boundary conditions of (4.1) allow us to integrate by parts without introducing integrals on the boundary. The sufficiency is obvious. To show the necessity, pick a section h G Hi(X). On integrating by parts we readily obtain
0 = (Aih, h) = \Aih\2 + \\Ai-1*h\\2,
and the lemma follows. □
The main step in the approach of [16] is to establish that Hi(X) is finite-dimensional and if f G C^(X, Fi) is orthogonal to Hi(X) then (4.1) can be solved for u G C^(X, Fi). Suppose that these solvability properties for problem (4.1) have been established. We introduce the subspace Ni(X) of C^(X,Fi) consisting of those sections u which satisfy the boundary conditions in
(4.1), i.e., n(u) = 0 and n(Au) =0 on dX. Given any f G C^(X,Fi), we denote by Hif the orthogonal projection of f into Hi(X). The difference f — Hif still belongs to C^(X, Fi) and is orthogonal to Hi(X), hence there is a section u G Ni(X) such that Aiu = f — Hif in X. Set Nif := u — Hiu, thus obtaining a linear operator from C^(X,Fi) to Ni(X). This operator is well defined, for from u1, u2 G Ni(X) and Aiu1 = Aiu2 it follows that u1 — Hiu1 = u2 — Hiu2. We see that any section f G C^(X, Fi) can be written as
f = Hif + Ai*Ai Nif + Ai-1Ai-1* Nif (4.2)
in X.
If the curvature of quasicomplex (0.1) vanishes at step i, i.e., AiAi-1 = 0, then the terms on the right-hand side of (4.2) are mutually orthogonal, as is easy to check. In this case formula
(4.2) furnishes an isomorphism between the cohomology of (0.1) at step i and the space Hi(X) of harmonic sections, see [17, 4.1] for more details.
Part 2. Subelliptic estimates 5. The main theorem
Quasicomplex (0.1) is said to be elliptic at step i if the symbol complex (1.1) is exact at step i for each x £ X and for each cotangent vector £ £ T*X different from zero. This is equivalent to the fact that the Laplacian A"1 is a second order elliptic operator on X.
Theorem 5.1. Suppose (0.1) is elliptic at steps i — 1 and i and there is a constant c such that
\\u\\l/2 < c (\Aiu\\2 + HA-1*u\\2 + M2) (5.1)
holds for all u £ C'(X, Fi) satisfying n(u) = 0. Then Hi(X) is finite-dimensional, and if f £ C'(X,Fi) is orthogonal to Hi(X) then there exists u £ C'(X,Fi) satisfying (4.1).
As is mentioned in the introductory remarks, this theorem is contained in [7] if the curvature of (0.1) vanishes at step i.
The first step in proving the theorem is to extend the Laplacian Ai to a closed operator Li on the Hilbert space L2(X,Fi). To this end we apply a classical method of (Kurt) Friedrichs, cf. [2]. In functional analysis, by the Friedrichs extension is meant a canonical self-adjoint extension of a nonnegative densely defined symmetric operator. This extension is particularly useful in situations where an operator may fail to be essentially self-adjoint or whose essential self-adjointness is difficult to show. The definition of the Friedrichs extension is based on the theory of closed positive forms on Hilbert spaces. If T is a nonnegative operator in a Hilbert space H, then Q(u,v) = (u,Tv) + (u,v) is a sesquilinear form on DomT and Q(u,u) > \\u\\2. Thus Q defines an inner product on Dom T. Let H1 be the completion of Dom T with respect to Q. This is an abstractly defined space. For instance its elements can be represented as equivalence classes of Cauchy sequences of elements of Dom T. It is not obvious that all elements in H1 can be identified with elements of H. However, the canonical inclusion Dom T ^ H extends to an injective continuous map H1 ^ H. We regard H1 as a subspace of H. Define an operator T1 in H whose domain consists of all u £ H1 such that v ^ Q(u,v) is a bounded conjugate-linear functional on H1 . Here, bounded is relative to the topology of H1 inherited from H. Pick u £ Dom T1. By the Riesz representation theorem applied to the linear functional v ^ Q(u,v) extended to all of H, there is a unique f £ H such that Q(u,v) = (f,v) for all v £ H1. Set T1u := f. Then T1 is a nonnegative self-adjoint operator in H, such that T1 — I extends T. The operator T1 — I is called the Friedrichs extension of T.
The operator Ai in L2(X, Fi) with domain Ni(X) is nonnegative, densely defined and symmetric. The sesquilinear form Q(u,v) = (u, Aiv) + (u,v) on Ni(X) reduces readily to
D(u, v) := (Aiu, Aiv) + (Ai-1 *u, Ai-1*v) + (u, v),
which is known as the Dirichlet scalar product on C'(X, Fi). When completing Ni(X) in the norm D(u) := \JD(u,u), one can scarcely retain the boundary condition n(Au) =0 at dX. Hence, one disregards this condition from the very beginning and considers the Dirichlet inner product on the subspace of C'(X,Fi) which consists of all u satisfying n(u) =0 on dX. We write Di for its completion to a Hilbert space. It is not difficult to see that Di can be thought of as a subspace of L2(X, Fi). We now define Li + I to be the operator whose domain consists of all u £ Di such that v ^ D(v,u) extends to a bounded linear functional on L2(X, Fi) and whose rule of correspondence is given by D(u, v) = ((Li + I)u, v), for all sections v £ Di. Then Li + I is a self-adjoint operator on L2(X, Fi), and (Li + I)u = (Ai + I)u if u £ Ni(X). Also, Li +1 is surjective, and (Li + I)-1 is bounded as an operator from L2(X, Fi) to Di. It follows by (5.1) and Rellich's theorem that (Li +1)-1 is a compact operator from L2(X,Fi) to itself,
and hence Li = (Li + I) — I must have closed range and finite-dimensional null space. Since Li is self-adjoint, its null space is the orthogonal complement of the range of Li. Hence, any f G L2(X, Fi) can be written in the form f = h + Liu, where h belongs to the null space of Li and u is in the domain of Li. The proof of Theorem 5.1 will now be complete when we establish two facts. The first of the two is that if u lies in the domain of Li and if Liu is C^, then u is C The second fact is that every smooth section u in the domain of Ll must satisfy the boundary conditions n(u) = 0 and n(Au) =0 on dX. If f is C, then the first statement will imply that the sections h and u in f = h + Liu are CIts proof will occupy the next three sections. The second statement will then imply that h is in Hi(X), that u is in Ni(X), and that Liu = Aiu. We turn to the proof of the second statement right now.
Lemma 5.2. Every Csection u in Vi satisfies the boundary condition n(u) =0 on dX.
Proof. Since u G Vi, there exists a sequence {uj} in C^(X,Fl) such that n(uj) =0 on dX and D(u — uj) ^ 0 as j ^ x>. Since n(uj) =0 on dX, integration by parts yields the equality (A*uj,p) = (uj,Ay>) for every ^ G CX(X,Fl-1). Since D(u — uj) ^ 0, we may pass to the limit in the equality to obtain (A*u, p) = (u, Ap) for every p. In view of the integration-by-parts formula (see the proof of Theorem 3.1), this means that
/ (iln(u),t(p))xds = 0 JdX
for all p G C^(X, Fi-1). Hence the lemma holds. □
Lemma 5.3. Suppose the boundary is noncharacteristic for quasicomplex (0.1) at step i — 1. Then every u G C^(X, Fl) which belongs to the domain of L1 satisfies n(Au) = 0 on dX.
Proof. If u G C^(X, Fl) belongs to the domain of L1, then for every Csection v in V1 we get
0 = D(u,v) — ((L + I )u,v)
= ((Aiu, Aiv) — (Ai*Aiu, v)) + ((Ai-1 *u, Ai-l*v) — (Ai-1Ai-1*u, v))
= / (iln(Au),t(v))x ds — (t(A*u), iln(v))x ds, JdX JdX
the last equality being due to the integration-by-parts-formula. Since n(v) = 0 on the surface dX, the second term on the right-hand side vanishes, which gives readily
(itn(Au),t(v))x ds = 0
JdX
for all v G C^(X,Fi) satisfying n(v) = 0 on dX. On applying Theorem 2.2 we conclude that n(Au) =0 on dX, as desired. □
6. A priori estimates
To complete the proof of Theorem 5.1 we must prove that u is CTO, whenever Liu is. In this section we derive certain a priori estimates which help establish this result. In what follows, c will denote a generic constant.
We shall need the norms \\f ||(rjS) when f is a Cfunction with compact support in the closed half-space R"0 consisting of all x G 1" with xn ^ 0. For the definition of these norms in terms of Fourier transform we refer to Section 2.5 of [3]. We only remark that if r and s are nonnegative integers, then \\ • \\(rjS) is equivalent to the norm
f E f \daf (x)?dv)
an<r
So \\f \\(r,s) controls the L2 norms of those partial derivatives of f which are of total order < r + s and are of order ^ r in the normal derivative d/dxn. We list the main properties of the norms
\\ • \\(r,s) in
Lemma 6.1. As defined above, the scale \\ • \\(r,s) bears the following properties:
1) \\f \\(r,0) = \\f \\r, the Sobolev r -norm on R>0;
2) \\f\(r,s) < \\f \\(r',s') if r < r' and r + s < r + s';
3) \\Pf \\(r,s) ^ c\\f \\(r+m,s) holds with some constant c independent of f, if P is a differential operator of order m;
4) \\f \\(r,s) ^ c (\\Pf \\ (r—m,s) + \\f \\(r',s')) holds with a constant c independent of f, if P is an elliptic differential operator of order m and r + s = r' + s';
5) \\f \\s,s ^ \\f \\(i,s-i), where \\ • \\s,s is the Sobolev s -norm on {xn = 0};
6) 2 ^ (f,9) < \\ f\((0,s)\g\((0,-s) for anV s-
Proof. Assertion 4) is Lemma 2.1.1 in [5]. The rest of the lemma is contained in Sec. 2.5 of [3]. □
Let U be a coordinate neighbourhood in X such that the bundles F1-1, Fand F1+1 are trivial over U. Assume that the coordinate x = (x1,... ,xn) on U maps U into the closed halfspace K"0. Then any Cfunction with support in U can be considered as a function on M"0, and hence the norms \\f\\(r,s) are defined for f e C^mp(U). Now fix a frame in Fl\u, that is, choose sections e1,...,eki in Cx(U,Fl) with the property that for each x e U the elements e1(x),..., eki(x) form a basis for the fibre over x. Then each u e C^mp(U, Fl) has component functions defined by
and we may define
1 k* U = U C\ + ... + U Ck*
Huiler = (Ê U\\¡r,s))1/2. 3 = 1
It is easy to check that the assertions in Lemma 6.1 continue to hold for these norms.
1 d
Let D' = (D\,..., Dn-1), where Dj = ^__d— ■ Consider the pseudodifferential operator
A3 = x(D') (1 + \D'\2)s/2
on I"-1, where x G Cœ(Rn-1) is 0 on a neighbourhood of the origin and 1 outside a slightly larger set. On letting As act along the first n _ 1 coordinate directions we define Asf when f is a Cfunction on X with compact support in U. And with a fixed choice of frame in F1 over U we can define Asu for u G C^mp(U, F1 ) by letting As act on the component functions of u as determined by the frame. If y G C^mp(U) and
Ts = yAsy. (6.1)
then Ts is an operator which acts on C(X) and also, with a choice of local frame, an arbitrary smooth sections of Fi-1, Fl, or Fl+1.
If an appropriate frame is used to define Ts on sections of F1, then Ts becomes a formally self-adjoint operator. In fact, let e'1,...,eki G C^(X,Fl) be such that for each x G U the elements e[(x),..., e'ki(x) form an orthonormal basis for the fibre, and let the volume element be given by dv = v(x)dx in the coordinate x on U. Then define ej = ej/\/v, for j = 1,... , so that if u = uj ej and v = vj ej have support in U, then
(u,v) = / y ui(x)vi(x)dx.
Ju 3=1
If we define Tsu = (Tsuj)ej for u = ujej G C'(U,Fi), then (Tsu,v) = (u,Tsv) for all C' sections u and v. When letting Ts operate on sections of a bundle, we shall assume that the frame being used makes Ts self-adjoint.
Lemma 6.2. Suppose p, 0, w are C' functions with compact support in U and p = 1 on the support of w, 0 = 1 on the support of p. Let Ts be the operator defined by (6.1). Then,
1) for each r, t there is a constant c such that \\Tsf \\(rjt) ^ c\\0f \\(rjt+s);
2) if moreover P is a differential operator of order m, then for each r, t there exists a constant c such that
\\[P,Ts]f \\(rit) < c\\0f \\(r+mit+s_i), \\[[P,Ts],Ts]f\\(rit) < c\\0f\\{r+m,t+2s-2);
3) for each t there is a constant c such that
\\wf \\(o,t+s) < c (\\Tsf \\(0,t) + \\f \\t+s-i) •
As usual, the bracket [P, Q] of two operators denotes their commutator PQ — QP.
Proof. Assertions 1) and 2) are well-known properties of classical pseudodifferential operators. 3) holds because Ts is tangentially elliptic on the support of w, see Theorem 4.7 in [4]. □
Lemma 6.3. Assume that quasicomplex (0.1) is elliptic at Fi and let u G C'(X,Fi) satisfy n(u) =0 on dX. Then there exist v,u',u'' G C'(U,Fi) with support in supp p such that
1) TsTsu = v + Tsu' + u'';
2) n(v) =0 on dX;
3) for each t there is a constant c such that
\\u'\\(1,t) < c\\0u\\(1,t+s-1), \\u''\\(1,t) < c Uu\\(1,t+2s-2)•
Proof. We follow the proof of Lemma 4 in [12]. Theorem 2.2 shows immediately that the homo-topy formula an(u) + n(au) = u holds for all u G C'(dX, Fi), where n2 = 0. Hence, the results of [12] apply with A = n, B = n and R = a(x). Consider
w = a(x) n(TsTsu)
= a(x) Ts [n, Ts]u + a(x) [n, Ts]Tsu
= T sw' + w'',
where w' = 2a(x) [n,Ts]u and
w'' = [a(x)[n,T s],T s]u + [a(x),T s][n,T s]u = a(x) [[n, T s],T s]u + 2 [a(x), Ts ] [n, T s]u•
Using Lemmata 6.1, 6.2 and inequality \\<7(x)u\\s,s < c\\u\\s,s with c a constant independent of u, we infer
\\w '\\s ,t+1/2 < c\\[n,T s\u\\s,t+1/2
< c\\0u\\s,t+s-1/2
< c\\0u\\(1,t+s-1)
and
\\w''\\s,t+1/2 < c (\\m Ts], Ts]u\\s,t+1/2 + \\ [n, Ts]u\\s,t+s-1/2)
< c (\\0u\\s,t+2s-3/2 + \\0u\\s,t+2s-3/2 )
< c\\0u\\(1,t+2s-2) •
By Theorem 2.5.7 in [3] we can choose u',u'' G C' (U, Fi) such that
w ,
on the boundary of X and
u'll(M) < c ||w'||5,t+1/2, 1,t) < c ||w''lls,t+1/2-
//
In view of the estimates for w and w which have already been obtained we get
u'N(i,t) < cy^u|(i,t+s-i),
u''|(1,t) < c ^A^^s^),
as required. Since
T su' + u'' = T sw' + w''
= a(x) n(TsTsu)
on dX, we can define v = TsTsu — Tsu' — u'', and the proof is complete. □
In [7] the boundary condition n(u) =0 on dX is assumed to be invariant with respect to action in the directions parallel to the boundary. This means, in particular, that if n(u) = 0 on dX then also n(Tsu) = 0, in which case Lemma 6.3 is trivial. How can the condition a(x)*u = 0 imply a(x)*Ts =0 on the boundary? This can be achieved only in the case if n(u) = 0 just amounts to saying that several components of the section u of Fi vanish on dX. Since quasicomplex (0.1) is elliptic at the step i, this can certainly be achieved by choosing special local frames for the bundle Fi. The decomposition of Theorem 2.2 actually gives such a vector bundle Fi which is a direct summand of Fi. Technically this means that all norms under consideration are independent up to equivalent norms of the particular choices of local frames, which is an ungrateful exercise in functional analysis of sections of smooth vector bundles over dX.
Lemma 6.4. For all u G C'(X, Fi),
D(u,TsTsu) = D(Tsu, Tsu) + O (HHkM^) •
Proof. Since Ts is formally self-adjoint, the lemma reduces to Lemma 3.1 in [7]. The proof is essentially algebraic, using only self-adjointness and those properties of Ts which are mentioned in Lemma 6.2. □
Lemma 6.5. Assume that quasicomplex (0.1) is elliptic at steps i — 1 and i. Let the estimate \\u\\l/2 ^ cD(u, u) hold for all u G C'(X, Fi) satisfying the boundary condition n(u) =0 on dX. Then for each s ^ 1/2 there is a constant c with the property that
\\Tsu\\\/2 < cD(Tsu,Tsu) < c(\\(4 + I)u\\2-1/2 + \\u\\2) (6.2)
holds for all u G C'(X, Fi) in the domain of Li.
Proof. Since u is in the domain of Li, we have D(u,v) = ((A + I)u,v) for all v G C'(X,Fi) satisfying the boundary condition n(v) =0 on dX. Hence, this equality holds in particular for the section v = TsTsu — Tsu' — u'' described in Lemma 6.3. Thus,
D(u, TsTsu) = D(u, Tsu') + D(u, u'') + ((A + I)u, TsTsu) + ((A + I)u, Tsu' + u''). (6.3)
u
u =
We shall treat the terms on the right of (6.3) one by one. To treat the first term we first claim that
D(u, Tsu') = D(Tsu, u') + O(\\M(1,s-1)).
In fact, to prove this we must majorise two terms like
(Au, ATsu') - (ATsu, Au') = (Au, [A, Ts]u') + ([Ts,A]u, Au'),
and by the preceding lemmata this expression is bounded by
\\A^u)\\(0,s-1)\\[A,Ts]u'\\(0,-s+1) + \\[Ts,A]u\\ \\Au'\\ < c\\V>u\\(1,s-1)\\u'\\(1,0)
(6.4)
< c W^Ahis-l).
Therefore, (6.4) holds, and since
\D(Tsu,u ' )\ < VD(T su,T su)^f D(u ',u ' )
< 4 D(Tsu,Ts u)+ c Wu ' \\l
< ! D(Tsu,Ts u) + c УФу^^),
we get
\D(u,Tsu' )\ < 4 D(Tsu,Tsu)+ c УФА2^
^ ,s-1) ■
As for the second term in (6.3) we claim that \D(u,u'')\ < cH^u^ s-1). In fact, for a typical term, we have
\(Au,Au '')\ < c \\A^u)\\(o,s-l) Уи '^(l.-s+l) < c W'ФuW21,s-1),
and hence the above estimate holds.
The third term in (6.3) is majorised as
\((Л + I )u,T sT su)
= \(^(Л + I)u, Tsu)\
< \\^(Л + I )u\\(0--l/2)WTsuW(0A/2)
< c У(Л + I)u\\s-l/2\\Tsu\\l/2
< c (e2\\Tsu\\2/2 + e-2 У (Л + I)u\\2-l/2)
< ce2D(Tsu,Tsu) + ce-2\\^ + I)u\\2s_
1/2,
l
where e > 0 is taken so small that ce2 < -.
4
The remaining term in (6.3) can now be estimated by
|((4 + I)u, Tsu' + u'')\ < U(A + I)u\\(0s-1/2)\\Tsu' + u''\\(0,-s+1/2)
< c(\\(л + I)u\\2-l/2 + \\Фи\\2^-1)),
and thus we have proved that
l
D(u, TsTsu) < - D(Tsu, Tsu) + c (\\(Л + I)и\2-1/2 + \\ФЧ\2М-1)).
On using Lemma 6.4 and substracting the term 1 D(Tau,Tau) from both sides we get
1 D(Tau, Tsu) < с(||(Л + Iyut-^ + UuWls-D).
To complete the proof it suffices to show that W^u|(1 a-1) is majorised by the right-hand side of (7.1). But since quasicomplex (0.1) is elliptic at step i, the operator Лг +1 is elliptic, and so, by part 4) of Lemma 6.1,
WMlhs-i) < с (11(Л + I^uWUs-i) + ||u||2)
< с (||(Л + I^uHl_V2 + ||u|2)
< с (||(Л + I)uH2s_3/2 + Ц^ФЫ2-^ + ||u|2)
< с (||(Л + I)uH2s_3/2 + ||u|2 ),
as desired. □
Recall that by ш we mean a Сж function with compact support in U, such that ф = 1 on the support of ш.
Lemma 6.6. Suppose the quasicomplex (0.1) is elliptic at step i. Then for each s ^ 1/2 there is a constant с such that
ЦшА^ < с(H^uHi/a + Ц(Л + I)uHa-3/2 + ||u||s-V2) holds for all u G C^(X,Гг).
Proof. Since quasicomplex (0.1) is elliptic at step i, the operator Лг +1 is elliptic, and part 4) of Lemma 6.1 yields
\\uu\\s + 1/2 < С (\\(A + I )uu\\s-3/2 +
< С (\\(A + I )u\\s-3/2 + \\[A,w]u\\s-3/2 + \M\(0,
< С (\\(A + I)u\s-3/2 + \\u\\s-l/2 + \^u\(0,s+1/2))-
The desired estimate now follows from part 3) of Lemma 6.2. □
Theorem 6.7. Assume that quasicomplex (0.1) is elliptic at steps i — 1 and i. Let the estimate \\u\\l/2 ^ cD(u, u) hold for all u G C^(X, Fl) satisfying the boundary condition n(u) =0 on dX. Then for each s ^ 1/2 there is a constant с such that the estimate
\\v\\s+i/2 < С\\(A + I)v\\s-i/2 (6.5)
holds for all u G C^(X, Fl) in the domain of Ьг.
Proof. Choose a finite covering {Uv} of X by coordinate neighbourhoods of the form used above. For each v, let , , and TS be as described in Lemma 6.2. We can assume that } forms a partition of unity on X. Then, by Lemmata 6.5 and 6.6, we get
\\u\\s+1/2 < c(£\KSuu\1/2 + \\(A + I )u\s-3/2 + \\u\\s-1/2)
V
< c (\\(A + I)u\\s-1/2 + \\u\\s) for all smooth u in the domain of Li. Using the interpolation inequality
\\u\\s < 4u\\s+1/2 + C(e) \\u\\
with e > 0 sufficiently small, we obtain
\\u\\s + 1/2 < c (\\(A + I)u\\s-1/2 + C(e) \\u\) + | \\u\\ s+1/2
whence
M + 1/2 < c (\\(A + I)u\\s-1/2 + \\u\\). (6.6)
Since
\\u\\2 < D(u,u) = ((A + I)u,u) < \\(A + I)u\\ \\u\\ for all u in the domain of L1, we obtain
\\u\\ < \\(A +1)u\\ < \\(A +1)u\\s-1/2.
Estimate (7.1) now follows from (6.6) and the last inequality, as desired. □
7. Elliptic régularisation
Following [7], we use the techniques of elliptic regularisation in this section to prove that u is Cwhenever Liu is CTO. This will complete the proof of Theorem 5.1.
Choose a bundle F and a differential operator d : C^(X,F*) ^ C^(X,F) of order 1 such that ||du|| > ||uy1 for all u G C^(X,F*). Define
AÎ = Ai © ed : C^(X, Fi) ^ C^(X, Fi+1 ) © C^(X, F)
for e ^ 0. Except for the fact that the composition A\Ai-1 need not be of order 1 when e > 0, the operators Ai-1 and A\ share most of the properties of Ai-1 and Ai which were used in the last two sections. In particular, we can use the sesquilinear form
De(u,v) = (Aieu, Aiev) + (Ai-1*u, Ai-1*v) + (u, v) = D(u,v) + e2 (du, dv)
to define a self-adjoint operator LiE on L2(X,Fi) such that DE(u,v) = ((LI +I)u,v) for all u in the domain of L* and all Csections v satisfying n(v) =0 on dX.
We still give De(u,v) the domain that consists of all u,v G CX,Fi) whose normal parts vanish on dX. The only problem is on the additional boundary condition for A\u for smooth sections u G C^(X, Fi) lying in the domain of L\. An easy verification using the Green formula shows that this free boundary condition reduces to
li(x)n(Au) + e2(a1(d)(x, dg(x)))*du = 0
on dX.
Lemma 7.1. Assume that quasicomplex (0.1) is elliptic at steps i _ 1 and i. Let the estimate ||u| 1/2 ^ cD(u, u) hold for all u G C^(X, Fi) satisfying the boundary condition n(u) =0 on dX. Then for each s ^ 1/2 there is a constant c with the property that
M + 1/2 < c H(Le + I)uUs-1/2 (7.1)
holds whenever u G C(X, Fi) is in the domain of L* and 0 ^ e ^ 1.
Proof. All the arguments used to prove (7.1) continue to be valid when A1 is replaced by A\, and it is easy to see that the constant c in each of the various estimates can be chosen independently of e. □
The reason for introducing A\ is that when e > 0 then the coercive estimate
e2\\u\\\ < De(u,u) (7.2)
holds for all u G CX(X, Fi), and it is fairly easy to obtain a regularity theorem for LiE. In fact, we have
Theorem 7.2. Suppose that quasicomplex (0.1) is elliptic at steps i — 1 and i. Let the estimate \\u\\1/2 ^ cD(u,u) hold for all u G C^(X,Fi) satisfying the boundary condition n(u) =0 on dX and let 0 < e < 1. Then for every f G C^(X, Fi) there is a unique section u G C^(X, Fi) in the domain of LI such that (Lie + I)u = f.
Proof. The operator LiE was constructed in such a way that LiE +1 automatically maps its domain onto L2(X, Fi) in a one-to-one fashion. Hence, to prove the theorem, it will suffice to show that if u is in the domain of L* and if (Lie + I)u is C, then u is also C. We shall use the method of difference quotients which occurs, e.g., in [9] and [1].
If f is a function on the closed upper half-space in 1", if 1 < j < n and h > 0, then we write
1 f (x1,...,xj + h,...,xn) — f (x1,...,xj — h,...,xn)
Oh n f (x) = , -
hjJK ' 2h
and, for any multi-index a = (a1,..., a.n) with an = 0, we set = S• • • Sa'"Z11. After choosing a coordinate system x : U ^ 1n on X, which maps U into the closed upper half-space, and after choosing a function f G C^mp(U) we can use a local orthonormal frame to define
T^u = fsa (fu),
when u is a section of one of the vector bundles Fi or of F. For details we refer to the discussion just above Lemma 6.2.
If, in Lemma 6.2, the operator Ts is replaced by the operator with |a| = s, then statements 1), 2), and 3) continue to hold even if the constants c are required to be independent of h for 0 < h < 1. Consequently, Lemmata 6.3 and 6.4 also hold for the operators T£, where again the constants can be chosen independent of h. Using (7.2) and the arguments in the proof of Lemma 6.5, one can show that for each e > 0 and every integer s > 1 there is a constant c such that
\\Tau\\1 < c (U(Li + I)u\ho,s-1) + \\H\(1,s-1)) , (7.3)
provided |a| = s, 0 < h < 1, u belongs to the domain of L*, ^u G H(1's-1)(X, Fi) and (LI+I)u G C^(X,Fi). Now, if a and u satisfy these conditions, then (7.3) shows that (T^u)o<h<1 is a bounded subset of H 1(X,Fi). Hence, there is a sequence hv converging to zero such that T^u converges weakly to some element f of H 1(X, F*). Since Ta u converges in the distribution sense to fDa(fu) as h ^ 0, we infer that f = fDa(fu) an hence fDa(fu) G H 1(X,Fi). Thus, if f = 1 on the support of w G C££ (U), we conclude that wu G H(1,s)(X, F*).
Now let u be in the domain of L®, such that (L*e +1)u G C^(X, F*), and let p be a fixed point of dX. Then the argument just given shows that if u is in H (1,s-1) on a neighbourhood U of p, then u is in H (1,s) on a slightly smaller neighbourhood. Thus, for each integer s there exists a function w G C^mp(U) such that wu G H(1's)(U, F*) and hence, by Theorem 2.5.7 in [3], the restriction of wu to the boundary belongs to Hs(dX,F*). It follows that u G Hs(dX,F*) for each s, and so u \qx must be Cby Sobolev's lemma. Since both (L* +1)u and u \dx are Cthe regularity theorem for the Dirichlet problem implies that u is Calso (see for instance Theorem 9.9 in [1]). The proof of the theorem is thus complete. □
Corollary 7.3. Suppose that quasicomplex (0.1) is elliptic at steps i — 1 and i. Let the estimate \\u\\1/2 ^ cD(u, u) hold for all u G C™(X, Fi) satisfying the boundary condition n(u) = 0 on dX and let u belong to the domain of L1. Then,
1) u is C™ if (Li + I)u is C™;
2) u G Hs+1(X,Fi) if (Li + I)u G Hs(X, Fi);
3) u G Hs+1(x,F*) if Liu G Hs(X, Fi);
4) u is C™ if Liu is C™.
Proof. To prove 1) assume that (Li +1)u is C™ and for each 0 < e < 1 let uE be the unique C™ section satisfying (Lie + I)u£ = (Li +1)u. If s > 1/2, then (7.1) shows that (uE)0<E^1 is bounded in the norm \\ ... \\s+1/2, and by Rellich's theorem there is a sequence ev converging to zero, such that uEv converges in the norm \\ • \\s to an element u0 of Hs(X, Fi). On passing to the limit in De(ue, v) = ((L1 + I)u, v) we obtain
D(u0,v) = ((Li + I )u,v)
for all v G C™(X, Fi) satisfying n(v) =0 on the boundary. Thus, u0 is in the domain of Li and (Li +1)u0 = (Li +1)u. Since Li +1 is one-to-one, we conclude that u0 = u and so u G Hs(X, Fi). Since s > 1/2 can be arbitrarily large, it follows that u is C™.
To prove 2), let s > 0 and assume (Li + I)u is in Hs(X,Fi). Choose a sequence fv of C™ sections which converges to (Li +1)u in the norm \\ • \\s, and let uv be the unique C™ section satisfying (Li +1)uv = fv. Then, by (7.1), the sequence uv converges in the norm \\ • \\s+1 to some element u0 of Hs+1(X, Fi). Since Li + I has closed graph, we get (Li + I)u0 = (Li + I)u and hence u = u0. Thus, u belongs to Hs+1(X, Fi), as required.
If s = 0, then 3) follows immediately from 2). Let m be a positive integer and assume that 3) holds for all s with 0 ^ s ^ m — 1. Let m — 1 < s ^ m, and assume that Liu is in Hs(X, Fi). Then, since Liu G Hs-1(X,Fi), we conclude that u G Hs(X,Fi) by the inductive hypothesis, and so (Li +1)u belongs to Hs(X, Fi). Thus, by 2), we see that u is in Hs+1(X, Fi), as desired. The assertion 4) follows obviously from 3) by Sobolev's lemma, and the proof is complete. □
8. A regularity theorem
In this section we assume that the curvature of quasicomplex (0.1) vanishes at step i, i.e., AiAi-1 0. In this case, the inhomogeneous equation Ai 1u = f might be locally solvable only for those f which satisfy Aif = 0. This is a starting point of [16].
Let T denote the operator from L?(X,Fi-1) to L2(X,Fi) obtained by closing the graph of A : C™(X, Fi-1) ^ C™(X, Fi). Thus, u is in the domain of T and Tu = f if and only only if there is a sequence (uv) in C™(X, Fi-1) such that uv ^ u and Auv ^ f in the L2 -norm. Our aim in this section is to prove
Theorem 8.1. Assume that the quasicomplex (0.1) is elliptic at steps i — 1, i and i + 1, and assume that the estimate \\f \\2/2 ^ cD(f,f) holds for all f G C™(X,Fi) satisfying n(f) = 0 on dX. Let u be in the domain of T, let u be orthogonal to the kernel of T, and let Tu G Hs(X, Fi) for some s > 0. Then u belongs to Hs+1/2(X,Fi-1).
Such a theorem has proved useful in studying counterexamples for a priori estimates like \\f\\2/2 < cD(f,f), see, e.g., [10].
Lemma 8.2. Under the assumptions of Theorem 8.1, for each s there is a constant c such that \Mu\\s + \M*u\\s < c (\\wAu\\(0,s) + \M*u\\(0is) + \\Au\\s-1 + \\u\\s) (8.1)
is valid for all u G C™(X,Fi).
Proof. Using the ellipticity of the quasicomplex at Fi 1 and Fi+1, one checks readily that
(g, h) ^ (A*g,Ag + A*h,Ah)
is an elliptic operator from sections of Fi-1 © Fi+1 to sections of Fi-2 © Fi © Fi+2. Hence, by Lemma 6.1, part 4),
||wA*u||s + ||wAu||s < c(||wA*u||(o,s) + |Mw||(0jS) + + ||A>A*u)||s-i + HA(wA*u) + A*(uAu)!s-i + ||A(wAu)||s-i),
and since the commutators [A*,w], [A, w], etc., have order zero, and the operators A*A*, AA have order one, we get
||A*(wA*u)||s-i < c Ms, HA(wA*u) + A*(wAu)Hs-i < c(HAuHs-i + ||u||s), ||A(wAu)|s-i < c || u| s •
Estimate (8.2) now follows. □
Lemma 8.3. Under the assumptions of Theorem 8.1, for each s ^ 1/2 there is a constant c such that
HA*u||s < cH(A + I)uHs-i/2 (8.2)
holds for each u £ C'(X, Fi) in the domain of Li. Proof. By Lemma 6.2 and Lemma 8.2 we have
HwA*ut < c (|M*u||2o,s) + |Mu||2o,s) + HAuH2a-i + HuH2)
< c(HTsA*uH2 + HTsAuf + HAuH- + ||u||2)
< c (D(Tsu,Tsu) + || Au12-1 + HuHl)
for all u £ C'(X, Fi). If u belongs to the domain of L1, then by Lemma 6.5
IMHI2 < c (\\(A + IMU/2 + ||uM2)
< c \\(A + I)uMl-1/2.
Now cover X with a finite number of neighbourhoods Uv of the kind used in Lemma 6.2 and choose the corresponding functions to form a partition of unity on X. Then
||A*u||s A*uHs < c H(A + I)uHs-i/2
V
for all u £ C'(X, Fi) in the domain of Li, as desired. □
Lemma 8.4. Under the assumptions of Theorem 8.1, let u £ L2(X, Fi) belong to the domain of Ll, and assume that Llu £ Hs-1/2 for some s ^ 1/2. Then u is in the domain of T* and T*u belongs to Hs(X,Fi-1).
Proof. In view of part 2) of Corollary 7.3 we get u £ Hs+1/2(X,Fi) and hence (Ll + I)u £ Hs-1/2(X, Fi). Choose a sequence (fV) in C'(X, Fi) which converges to (Ll + I)u in the norm H • ||s-1/2 and let uV £ C'(X,Fi) be the unique solution to
(Li + I)uv = fv.
Then by (7.1) the sequence (uv) converges in the norm \\ • \\s+1/2, and since Li +1 gas closed graph, the limit must be u. Now Lemma 5.2 and the Green formula show that each uv is in the domain of T*, and T*uv = A*uv. The estimate (8.2) now implies that (T*uv) converges in the norm \\ • \\s. Since T* has closed graph, we conclude that u = lim uv is in the domain of T* and T*u = limT*uv is in Hs(X, Fi-1). The proof is complete. □
As is remarked in Section 5., any f G L2(X, Fi) can be written as f = h + Liu, where h lies in the null space of Li and u is in the domain of Li. If we require that u be orthogonal to the null space of Li, then f determines u uniquely and the correspondence f ^ u defines an operator Ni : L2(X, Fi) ^ L2(X, Fi) which, as one easily sees, is self-adjoint and bounded.
Proof of Theorem 8.1. Let u be in the domain of Ti-1, let u be orthogonal to the kernel of Ti-1, and assume that Tu is in Hs(X,Fi) for some s ^ 0. Then, since Tu = h + Li(NTu), where h G Hi(X) is C™ on X, Lemma 8.4 shows that NTu is in the domain of T* and T*NTu belongs to Hs+1/2(X, Fi-1). To complete the proof we show that
u = T* NTu.
In fact, if v G C™(X,Fi-1) is an arbitrary section with support in the interior of X, then
Av = h + ANAv, where h G Hi(X). Hence,
Av — AA*NAv = h + A* AN Av.
Since AiAi-1 = 0, the terms on the right-hand side of this equality are orthogonal to the terms on the left-hand side. It follows that A(I — A*NA)v = 0 and so (I — A*NA)v is in the null space of T. Since u is orthogonal to the null space of T, we obtain
0 = (u, (I — A*NA)v) = ((I — T* NT)u,v),
and u = T* NTu now follows. □
The first author gratefully acknowledges the financial support of the Ministry of High Education of Iraq.
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Задача Неймана по Спенсеру
Азал Мера Николай Тарханов
Потсдамский университет Карл-Либкнехт-Штр., 24/25, Потсдам, 14476
Германия
Попытка распространить теорию Ходжа для эллиптических комплексов на компактных замкнутых многообразиях на случай компактных многообразий с краем приводит к краевой задаче для лапласиана комплекса, которая обычно называется задачей Неймана. Мы изучаем задачу Неймана для более широкого класса последовательностей дифференциальных операторов на компактном многообразии с краем. Это последовательности малой кривизны, т.е. обладающие свойством, что композиция любых двух соседних операторов имеет порядок меньший, чем два.
Ключевые слова: эллиптические комплексы, многообразия с границей, теория Ходжа, задача Неймана.
