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УДК 517.55
The de Rham Cohomology through Hilbert Space Methods
Ihsane Malass* Nikolai TarkhanoV
Institute for Mathematics University of Potsdam Karl-Liebknecht-Str. 24/25, Potsdam, 14476
Germany
Received 22.10.2018, received in revised form 06.12.2018, accepted 16.03.2019 We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler-Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer.
Keywords: De Rham complex, cohomology, Hodge theory, Neumann problem. DOI: 10.17516/1997-1397-2019-12-4-455-465.
Introduction
When looking for a natural representation of the de Rham cohomology on a compact manifold, one uses diverse homotopy formulas for differential forms. They are of the form u = Ru+P(du)+ +d(Pu), so that du = 0 implies d(Ru) = 0 and the cohomology classes of u and Ru coincide. While being a homomorphism of the de Rham complex, R fails to induce any mapping of the de Rham cohomology into closed differential forms unless it vanishes on exact differential forms. If n is a projection onto the subspace of exact forms, then P o (1 — n) already acts on cohomology classes since
u = R(1 — n)u + P (du) + d(Pu + Rd-lnu) (0.1)
for all differential forms u. This is precisely the problem treated by the second author in his bachelor thesis advised by Lev A. Aizenberg, see [12]. In the 1970s, Aizenberg encouraged his students Sh.Dautov, A.Kytmanov and others to study the Dolbeault cohomology of complex manifolds.
In general there is no canonical projection onto the subspace of exact forms. The classical approach to this topic invokes Hilbert space methods of elliptic theory. The variational formulation of the equation du = f consists in minimising the so-called energy functional
F (u) = 2 (|\du\\2 + \\d*uf) (f,u)
over the subspace of all square integrable forms u such that both u and du belong to the domain of the adjoint operator for d, see [6]. The minimal solution is thought of as canonical. In the case of compact closed manifolds it is given by the classical Hodge theory, see [13,15].
On compact manifolds with boundary the Euler-Lagrange equations for the energy functional constitute what is usually referred to as the Neumann problem. To the best of our knowledge, this
* ihsane malass@hotmail.com
1 tarkhanov@math.uni-potsdam.de © Siberian Federal University. All rights reserved
boundary value problem was first formulated explicitly in [11] within the more general context of complexes of differential operators.
For the de Rham complex, a coerciveness estimate in the Neumann problem after Spencer was actually proved in various settings in the middle 1950s. This was already sufficient to conclude on the Fredholm solvability and regularity of the problem. However, no one has given a direct proof of the ellipticity of the Neumann problem within the Boutet de Monvel algebra of boundary value problems, cf. Example 4.1.28 in [13].
The analysis of the Neumann problem after Spencer was undertaken by his PhD student W. Sweeney in a series of papers. In [10] he derived an algebraic condition for coerciveness in the problem. This condition is fulfilled for the de Rham complex, implying the ellipticity of the Neumann problem.
The study of W. Sweeney was well motivated all the more so since the Neumann problem for the de Dolbeault complex had been proved to be subelliptic, see [4,7]. The latter paper gave rise to [5] where subelliptic estimates in the Neumann problem were studied within the framework of general complexes of differential operators.
The purpose of this paper is to give a systematic presentation of Hodge theory for the de Rham complex which is based on the Neumann problem after Spencer. When using Hilbert space methods, we choose the L2 setting of classical variational calculus. The same technique is known to apply in the setting of Sobolev spaces, see [2]. Although this work leads to new minimal solutions to the inhomogeneous equation du = f, no surprising phenomena are found there while the presentation is voluminous.
1. Representation of the de Rham cohomology
Suppose that X is a compact Cmanifold with boundary of dimension n. Consider the de Rham complex
on X, where Ql(X) stands for the space of all differential forms of degree i with Ccoefficients on X and d for the exterior differentiation of forms. We have d2 = d o d = 0.
Given any f G Qi(X), the question of solvability of the inhomogeneous equation du = f is of crucial importance in analysis and geometry. Under what conditions on f does there exist a form u G Qi-1(X) satisfying du = f, and how does u depend on f? A necessary condition on f follows immediately from the integration by parts formula.
To formulate it we endow the manifold X with a Riemannian metric. The induced metric on the bundle of exterior forms of degree i is denoted by ( f,g)x, where x G X. Set
where dx is a volume form on X.
Lemma 1.1. In order that equation du = f may be solvable, it is necessary that (f, h) = 0 for all h G Qi(X) satisfying d* f = 0 in X and n(h) =0 on dX.
Here, d* stands for the formal adjoint of d and n(f) for the normal part of f on the boundary, see [13, 3.2.2].
Proof. Indeed, if u G Qi-l(X) and h G Q1 (X), then we get by Stokes' formula
0 ^ Q 0(X ) A n 1(X ) 4 ... 4 Q n(X ) ^ 0
(1.2)
(1.3)
where t(u) is the tangential part of u and ds the area form on dX. From (1.3) the lemma follows. □
On choosing h = d*w with w G O1+1 (X) of compact support in the interior of X we see that df = 0, i.e., f is a closed form. Write W(X) for the subspace of Oi(X) consisting of all h such that dh = 0 and d*h = 0 in X and n(h) = 0 on dX. The differential forms of Hl(X) are said to be harmonic. This concept generalise that of harmonic forms on a compact closed manifold, see for instance [13,15] and elsewhere.
Lemma 1.2. The natural mapping H1 (X) ^ HiR(X) is injective. Proof. Assume that f G Hl(X) and f = du for some form u G Oi-1(X). Then we get
\\f ||2 = (u,d*f )=0,
i.e., f = 0. □
In connection with the problem of global solvability of overdetermined inhomogeneous systems of differential equations Spencer [11] suggested an approach that in certain cases allowed one to prove the surjectivity of the natural mapping W(X) ^ HlAR(X). It may be thought of as an attempt to extend Hodge's theory for elliptic complexes on compact closed manifolds to the case of compact manifolds with boundary. The second order differential operator A = d*d + dd* taking forms of degree i into forms of the same degree is called the Laplacian. It is elliptic and formally selfadjoint. For i = 0, this is precisely the Laplace-Beltrami operator in differential geometry.
Lemma 1.3. The space Hi(X) just amounts to the subspace of Oi(X) which consists of all forms u satisfying Au = 0 in X, and n(u) = 0 and n(du) = 0 on the boundary.
Proof. We only need to show that if u G Oi(X) satisfies Au = 0 in X, and n(u) = 0 and n(du) =0 on dX, then du = 0 and d*u = 0 in X. To this end, use integration by parts as in (1.3), showing
0 = (Au,u) = \\du\\2 + \\d*u\\2, and the lemma follows. □
By the Neumann problem after Spencer is meant the following boundary value problem in X. Given a form f G Or(X), under what conditions does there exist a form u G Oi(X) satisfying
(1.4)
and how does u depend on f ?
Lemma 1.4. For the existence of a solution u G Ol(X) to problem (1.4) it is necessary that the form f should satisfy (f, h) = 0 for all h G Hi(X).
Proof. As is known, for problem (1.4) to be solvable, it is necessary that f should be orthogonal to the space of solutions of the homogeneous boundary value problem which is (formal) adjoint to (1.4) with respect to a Green formula. An easy verification based on formula (1.3) shows that (1.4) is formally selfadjoint. It remains to invoke Lemma 1.3. □
Au = f in
n(u) = 0 on dX,
i(du) = 0 on dX,
Suppose that the necessary condition of Lemma 1.4 is also sufficient, i.e., problem (1.4) has a solution u G Qi(X) for each form f G Qi(X) orthogonal to Hi(X). Then, given any f G Qi(X),
the inhomogeneous equation du = f has a solution u G Qi-1(X) if and only if
df = 0 and (1.5)
\ f ± Hi(X). ( )
To see this, we only need to show that for any f G Qi(X) satisfying (1.5) there is a u G Qi-1 (X) such that du = f. By hypothesis, there exists a form w G Q1 such that Aw = f in X, and n(w) = 0 and n(dw) =0 on the boundary. If we have in addition df = 0, then the equality Aw = f implies dd*dw = 0 because d2 = 0. Since n(dw) = 0, we derive from here that ||d*dw||2 = (dd*dw, dw) = 0 whence d*dw = 0. Using now the equality n(dw) = 0 again we obtain ||dw||2 = (w,d*dw) = 0, and so dw = 0. Hence it follows that f = dd*w, i.e., u = d*w is a solution of du = f, as desired. Note that the solution u = d*w constructed in this way is minimal in the sense that it is orthogonal to the subspace of Q1-1 (X) consisting of all closed forms.
Theorem 1.5. Assume that problem (1.4) has a solution u G Ql(X) for each form f G Qi(X) orthogonal to W (X). If moreover the space W (X) is finite dimensional, then
top
Hdr(X) - W(X).
Proof. First we notice that the space W(X) is finite dimensional if and only if it is closed in L2(X, A1), where A1 is the vector bundle of exterior forms of degree i over X. Indeed, as far as the sufficiency is concerned, we may apply the open mapping theorem to conclude that two topologies in W(X) induced from Ql(X) and L2(X, A1), respectively, coincide. Hence it follows by the Arzela-Ascoli theorem that the identity mapping of W(X) is compact. Fix now any f G Qi(X). Denote by Hf the orthogonal projection of f into the space W (X). Then the difference f — Hf is orthogonal to the space W(X), so there is a form w G Qr(X) such that Aw = f — Hf in X, and n(w) = 0 and n(dw) =0 on the boundary. Therefore, we get
f = H f + d* dw + dd*w
on X. If df = 0, then dd*dw = 0. This implies just as above that dw = 0. It follows that f = Hf + d(d*w), so the natural mapping W(X) HlAR(X) is surjective. We now use Lemma 1.2 to see that the natural mapping is an isomorphism of vector spaces without topologies. By the above, the space Hi(X) is separated, and so the natural mapping is actually a homeomorphism by the open mapping theorem. This finishes the proof. □
2. Variational approach
The solution u = Gf to the Neumann problem after Spencer constructed in Section 1. is minimal in the sence that it is orthogonal to solutions of the corresponding homogeneous problem. It has minimal energy HduH2 + ||d*u||2 = (f,u), as is easy to see.
To recover this solution within the framework of variational calculus, consider the problem of local minima of the functional
F (u) = 2 iUduf + Ud*uf) (f,u) (2.6)
over the space of all u G H 1(X, A1) satisfying n(u) = 0 on dX. Pick any form v G H1 (X, A1) such that n(v) =0. If F takes on a local minimum at an admissible form u, then the Gateaux derivative F' of F in direction v vanishes at u. It follows that
d_F(u + tv) |t=0 = K ((du, dv) + (d*u, d*v) — (f, v)) = 0.
On substituting iv for v we readily conclude that also the imaginary part of the expression on the right-hand side vanishes. Hence
(du,dv) + (d*u,d*v) = (f,v) (2.7)
for all v G H 1(X, A1) satisfying n(v) = 0. This is the weak form of Euler-Lagrange equations for functional (2.6).
Assuming that u is moreover of Sobolev class H2(X, A1), we can use the integration by parts formula, to get
(du,dv) + (d*u,d*v) = (Au,v) + / (n(du),t(v))xds = (f,u).
JdX
On choosing v with compact support in the interior of X we conclude that Au = f in X. Therefore,
/ (n(du),t(v))xds = 0 JdX
for all differential forms v G H 1(X, A1) such that n(v) = 0, and so n(du) =0 on the boundary. We thus get
Theorem 2.1. Let f G L2(X, A1). If u is a solution of class H2(X, A1) to the variational problem F(u) ^ min, then u satisfies the Neumann problem after Spencer of (1.4).
This theorem shows that the Neumann problem after Spencer just amounts to the Euler-Lagrange equations for functional (2.6).
At extreme steps the Neumann problem after Spencer is quite classical. At step 0 it coincides with the usual Neumann problem for the inhomogeneous Laplace-Beltrami operator. The Neumann problem at step n amounts to the classical Dirichlet problem. Simple calculations show that the Neumann problem at an arbitrary step i reduces locally to the Dirichlet and Neumann problems for coefficients of a differential form u. Thus the Neumann problem is a regular elliptic boundary value problem and it may be investigated by standard techniques. By ellipticity is meant the ellipticity in the Boutet de Monvel algebra of boundary value problems on X, see [1]. Along with the ellipticity of the Laplace operator in the interior of X this requires the invertibility of a boundary symbol. The latter property admits an equivalent algebraical description which is usually referred to as Shapiro-Lopatinskij condition. Analytically it implies a coercive estimate for the functional F which is a far reaching generalisation of A. Korn's (1908) inequality. Such an estimate for differential forms on compact manifolds with boundary was first proved in [3].
Lemma 2.2. There are positive constants c and k with the property that
\\du\\2 + \\d*u\\2 > c\\u\\2Hl{xtAi) - k \\u\\2 (2.8)
for all u G H 1(X, A1) satisfying n(u) =0 on the boundary of X.
Proof. See Theorem 4.2 in [6] which is a simple consequence of Gaffney's inequality [3]. □
Since the Neumann problem after Spencer is elliptic in Boutet de Monvel's agebra on X, it possesses a parametrix in the algebra. When focused on the left upper corners of (2 x 2) -matrices in the algebra, this means that there is a pseudodifferential operator G of order —2 on X, which maps L2(X, A1) into the subspace of H2(X, A1) consisting of all forms u with n(u) = 0 and n(du) = 0 on dX, and satisfies GA = I and AG' = I up to operators of order —1 on X. But we will not develop this point here.
3. Weak form of the Neumann problem
By the above, (2.7) is a weak formulation of the Neummann problem. If the functional F takes on a local minimum at a form u G H 1(X, A®), then equality (2.7) is fulfilled, and so f is orthogonal to all h G H1 (X, A1) satisfying dh = 0 and d*h = 0 in X, and n(h) =0 on dX. To handle problem (2.7) we introduce as usual the so-called Dirichlet norm on Q® (X) by
D(u) = (\\duf + yd* u\\2 + ||u||
2\1/2
Denote by V® the completion of the space of all u G Q®, such that n(u) =0 on dX, under the Dirichlet norm. Since D(u) ^ ||u||, we can identify V® with a subspace of L2(X, A1). Lemma 2.2 shows readily that V® coincides with the domain of the energy functional F.
Let f G L2(X, A*). We look for a differential form u G V1 which satisfies the equality (du,dv) + (d*u,d*v) = (f,v) for all v G V®. As mentioned above, if u G V® is a solution to (2.7), then Au = f weakly in the interior of X. If w G H 1(X, A®+1) and (d*w,v) = (w,dv) for all v G V®, then n(w) =0 on the boundary, and conversely. However, this is no longer true for differential forms of the form w = du, where u G V®, for n(du) need not be defined on the boundary. (The normal part n(du) can be still defined for those u G V® which satisfy d*du = 0 weakly in the interior of X.)
Lemma 3.1. If u G V® n H2(X, A®) is a solution to equation (2.7), then n(du) =0 on dX. Proof. Indeed, using (2.7) yields
(d*du,v) = (Au,v) — (dd*u,v) =
= (f, v) — (d* u, d* v) = = (du, dv)
for all v G V®, and the lemma follows. □
Thus, if f G Q®(X) and for a solution u gV® of (2.7) we could prove the regularity up to the boundary, i.e., u G Q®(X), then u would be a solution of problem (1.4).
Lemma 3.2. The space of solutions of the homogeneous equation, corresponding to (2.7), is finite-dimensional.
Proof. Denote the space in question by H®, i.e., H® consists of all forms u gV® satisfying (2.7) with f = 0. This is a closed subspace of V® because
I(du,dv) + (d*u, d*v) | < D(u) D(v)
for all u,v G V®. Hence, when endowed with the Dirichlet norm, H® is a Banach space. If u G H®, then choosing v = u in (2.7) yields ||du||2 + ||d*u||2 = 0, i.e., du = 0 and d*u = 0 in X. It follows that the Dirichlet norm on H® coincides with the L2 -norm. By Lemma 2.2,
k
M2m{X< - ||u||2
for all u G H®. Thus, the topology on H® induced from H 1(X, A®) coincides with that induced from L2(X, A®). Since the embedding H 1(X, A®) L2(X, A®) is compact, which is due to Rellich's theorem, we conclude that the identity mapping of H® is compact. □
In fact problem (2.7) is hypoelliptic in the sense that. for each right-hand side f G Q®(X), all solutions u G V® of the problem are infinitely differentiable up to the boundary in X. More
precisely, if f G Hs(X, A1) with a nonnegative integer s, then every solution u G Di to problem (2.7) is of Sobolev class Hs+1(X). Since the scale of Sobolev spaces shrinks to C^(X), the above assertion follows. As usual, the proof of regularity in Sobolev spaces is very technical. We dwell on regularity in Section 5. Having disposed of this step, we can now proceed with constructing a canonical solution to (2.7).
Corollary 3.3. The space of solutions of the homogeneous equation corresponding to weak problem (2.7) coincides with Hi(X).
Proof. By hypoellipticity, if u G Di is a solution to (2.7), then v G Sli(X). On applying Lemma 3.1 we see that n(du) =0 on dX. To complete the proof it suffices to apply Lemma 1.3. □
By Lemma 3.2, the space Hi(X) is finite-dimensional. Hence, it is a closed subspace of L2(X, Ai). The orthogonal projection H of L2(X, Ai) onto Hi(X) is given by
ft
Hu = J2(u, hj) hj
j=1
for u G L2(X, Ai), where {hj}j=1,..,pi is an orthonormal basis in Hi(X). This is a smoothing operator.
Theorem 3.4. Given any form f G L2(X, A1) orthogonal to Hi(X), the variational problem F(u) ^ min has a uniques solution u GVi orthogonal to Hi (X).
As the Euler-Lagrange equations show, the condition f ± Hi(X) is also necessary for the variational problem to possess a solution u GDi.
Proof. We first observe that if u GDi is a solution to the variational problem and h G Hi(X) an arbitrary form then u + h is also a solution, for
F (u + h) = 1 (\\d(u + h)\\2 + \\d*(u + h)\\2) — ft (f,u + h) = = F (u),
as desired. On the other hand, if F takes on a local minimum at two forms u1,u2 G Di, then the Euler-Lagrange equations imply (du, dv) + (d*u, d*v) = 0 for all v G Di, where u := u2 — u1. Taking v = u we deduce that du = 0 and d*u = 0, i.e., u G Hi(X). So, u2 = u1 + h, where h G Hi(X). It follows that the problem F(u) ^ min has at most one solution orthogonal to Hi(X). We now proceed as follows.
Write R1 for the orthogonal complement of Hi(X) in Di. We give R1 the Hilbert space structure induced from Di, i.e., that determined by the Dirichlet scalar product D(u,v) = = (du, dv) + (d*u, d*v) + (u, v). Since Hi is closed in the norm D(u), we get Di = Hi(X) © Ri.
The restriction of the energy functional F to Ri is continuous. Our next objective is to show that F is bounded from below on Ri. To this end, we invoke the estimate
\\u\\2 < C (\\du\\2 + \\d*u\\2)
for all u G R, with C a constant independent of u. This estimate is a consequence of Gaffney's inequality (2.8) and the closed graph theorem, cf. Lemma 4.2.15 in [13]. Therefore,
f(u) z 7C INI2 -Wf IIIMI
2 {jcM-^IIfO2- C
(3.9)
2
for all u
By (3.9), F is bounded from below on Ri by -(C/2) \\f ||2. Set
m = inf F(u)
and choose any minimising sequence {uv} in R®, i.e., F(uv) ^ m as v ^ to. The sequence {F(uv)} is bounded. Using (3.9) once again we see that {uv} is bounded in L2(X, A®), for
||u|| < C f || + yqcMyT^Fiu)
whenever u G R®. Substituting this estimate into Gaffney's inequality (2.8) yields readily F(uv) > c Huv||Hi(x Ai) — Q for all v = 1,2,..., where Q > 0 is a constant independent of v. Thus, the sequence {uv} is actually bounded in H 1(X, A®), and the rest of the proof runs within the direct methods of variational calculus, see [8]. □
Since (2.7) constitute the Euler-Lagrange equations of the variational problem F(y) ^ min, Theorem 3.4 concerns the solvability of the weak Neumann problem after Spencer, too.
Corollary 3.5. There is a bounded linear operator G in L2(X, A®) with the property that
1) G maps L2(X, A®) continuously into V®, preserves the property of being Cup to the boundary, and satisfies HG = 0 and GH = 0.
2) Each form f G L2(X, A®) splits weakly as the sum of three pairwise orthogonal terms
f = Hf + d*dGf + dd*Gf. (3.10)
In classical papers the operator G was referred to as the Green operator. In [4] and subsequent papers G is called the Neumann operator (denoted by N). When acting on f G L2(X, A®), the operator G need not fulfill n(dGf) =0 on the boundary. This is precisely what is meant by 'weakly' in 2). The decomposition of (3.10) just amounts to saying that Gf satisfies (2.7) with f — Hf in place of f on the right-hand side. If f G H 1(X, A®), then Gf is of Sobolev class H2(X), and so n(dGf) =0 by Lemma 3.1. In this case all of three summands in (3.10) are pairwise orthogonal.
Proof. Pick any f G L2(X, A®). The difference f — Hf is orthogonal to H®(X). Therefore, there is a form u G V® satisfying equation (2.7) with f — Hf in place of f. Set Gf = u — Hu. This operator is well defined because, if u1 and u2 are two solutions of (2.7) with f — Hf in place of f, then the difference u = u2 — u1 is in H®(X) whence
(u2 — Hu2) — (u1 — Hu{) = u — Hu = 0.
Clearly, Gf G V® satisfies equation (2.7) with f — Hf in place of f. Moreover, we get both HG = 0 and GH = 0 by the very construction.
Consider now two Hilbert spaces U = L2(X, A®) Q H®(X), endowed with the norm || • ||, and V = V®, endowed with the norm D(^), and the mapping M : U ^ V defined by M(f) = u, where u gV® is the solution of (2.7) orthogonal to H®(X). The graph of M is easily verified to be closed in U x V. By the closed graph theorem, M is bounded, i.e., there is a constant c > 0 such that D(Mf) < cf || for all f G R®. On applying this estimate to the form f — Hf we conclude readily that
D(Gf) < cf — Hf|| < cf ||,
i.e., G is a bounded operator from L2(X, A®) to V®.
By construction, Gf is the unique solution in V® n (H®(Xto the equation (du,dv)+ + (d*u, d*v) = (f — Hf, v) for all v G V®. If f G Q®(X), then the difference f — Hf is in Q®(X). Since problem (2.7) is hypoelliptic, it follows that Gf is also in Q®(X). We now use Lemma 3.1 to see that Gf satisfies not only n(Gf) = 0 but also n(dGf) =0 on dX. □
We put off studying further properties of the Green operator G to the next section. Meanwhile, we notice that decomposition (3.10) allows one to get a continuous homotopy between the identity and H endomorphisms of the de Rham complex. Such a homotopy is given by the operator P = d*G because df = 0 implies dGf = 0 for all f G Oi(X).
4. The Green operator
While the variational approach leads to the weak formulation of the Neumann problem after Spencer, the direct study of problem (1.4) by Hilbert space methods suggests to consider the closure of the Laplace operator under stronger boundary conditions n(u) = 0 and n(du) = 0. To the best of our knowledge the direct study first appeared in [3].
Set L = L2 (X, Ai). Denote by T the maximal operator from L into Li+1 generated by the exterior derivative acting from Oi(X) into Oi+1(X). This is an unbounded closed densely defined operator, and so its adjoint T* : Li+1 ^ Li is well defined. Write DTi and DTi* for the domains of these operators acting on Li and Ci+1, respectively. We introduce an operator L on L with a domain DLi which has the property that Lu = Au for all u G DLi n Oi(X). Namely, let Dl be the space of all u G DTi n DTi-i* such that Tu G DTi* and T*u G DTi-i. Then L is defined by Lu = T* Tu + TT* u. The Neumann problem after Spencer reads as follows. Given a form f G Li, is there u G DLi such that Lu = f, and how does u depend on f? Gaffney's inequality applies to show that DLi actually consists of all u G H2(X, Ai) satisfying n(u) = 0 and n(du) =0 on the boundary. Moreover, the range LDLi is closed for all i = 0,1,... ,n. This is typical for elliptic boundary value problems.
Let f G Li. Then we get f = Hf + Lu, where u G DLi. The Green operator G : L ^ DLi is defined by Gf = u — Hu. Obviously, G is well defined. The properties of the Green operator generalise those of the Green operator from Hodge theory, see for instance [15].
Theorem 4.1. As defined above, the Green operator G possesses the following properties:
1) G is bounded, selfadjoint, HG = 0 and GH = 0, and each f G Li admits the orthogonal decomposition f = Hf + T*TGf + TT*Gf.
2) If f G DTi and Tf = 0, then TGf = 0. Moreover, G and T commute on the domain of Ti.
3) If f G DTi-i* and T* f = 0, then T*Gf = 0. Moreover, G and T* commute on the domain of Ti-1*.
Proof. 1) Both equalities HG = 0 and GH = 0 and the orthogonal decomposition follow immediately from the definition of the Green operator. Further, by the closed graph theorem there exists a constant c > 0 with the property that \\Lu\\ > c \ \u\\ for all u GDLi which are orthogonal to Hi(X). Applying this estimate to Gf, we obtain
\\Gf \\ < - \\LGf \\ = ± \\f — Hf \\ < - \\f \\
c c c
for all f G Li. Hence G is bounded. Finally, the selfadjointness of G follows immediately from that of L because
(Gf, g) = (Gf, Hg + LGg) = (Gf, LGg) = (LGf, Gg) = (f, Gg) whenever g G Li.
2) Suppose f G DTi satisfies Tf = 0. From TDTi c DTi+i and the orthogonal decomposition we get T*TGf G DTi, and so Tf = 0 implies TT*TGf = 0. Hence it easily follows that TGf = 0. Moreover, for any form f G DTi, we obtain Tf = TT*TGf on the one hand, and Tf = tt*GTf on the other hand. Therefore, we get L(TGf — GTf) = 0, and since TGf — GTf is orthogonal to Hi+1(X) then TGf — GTf = 0, as desired.
3) Part 3) is proved by analogy with part 2). □
The spectral invariance of Boutet de Monvel's algebra shows that G is an operator of order —2 in the algebra, see [9]. Since the operators G and T commute, the orthogonal decomposition of 1) becomes f = Hf + (T*G)Tf + T(T*G)f for all f G VTi. Thus, on sufficiently smooth forms P = T*G is a very special parametrix of the de Rham complex.
5. Spectral problems
Let U be a complex Hilbert space of functions on X and A a mapping of U with domain VA. Given any f G U, we look for a solution u G VA to the equation Au = f under an additional condition Bu = 0. This latter can be e.g. a boundary condition. From the viewpoint of variational calculus the condition Bu = 0 should determine a convex subset of VA. If allowing convex domains we can include the additional condition into the definition of Va. Then one can pass on to a variational formulation of the problem and invoke the Euler-Lagrange equations for its study. Consider the eigenvalue problem Au = f + Xu under the same additional conditions u G Va, where X is a complex parameter. This is a perturbation of the original equation, and if it is small or large is determined by the order of A relative to some scale of function spaces with compact embedding. For instance, the order of a pseudodifferential operator on X is evaluated relative to the scale of Sobolev spaces Hs on X, where s > so is an integer. The order is defined to be the least real number m with the property that the mapping Va n Hs ^ U n Hs-m generated by A is continuous for all s satisfying s > s0 + m. Then the perturbation A — XI of A is small, provided m > 0, and large, provided m < 0. The perturbation is said to regular or singular, respectively. Any singular perturbation results in changing the order. Conversely, A = (A — XI) + XI is a perturbation of A — XI. The inverse RA(X) = (A — XI)-1, if exists, is called the resolvent of the operator A. It is thought of as a function of X with values in operators acting from U to VA. The most intricate case is that of operators of order m = 0, where spectral theory requires additional scales of function spaces. For an operator A of positive order, A — XI is a perturbation of A by a compact operator. Such perturbations preserve the class of Fredholm operators. Assume that A — XI is Fredholm for some particular value X = X0. Then there is a parametrix P for A under the additional condition u G VA. This means that P maps U continuously into the domain of A and the equalities P(A — X0I) = I and (A — XI)P = I are fulfilled up to compact operators on VA and U, respectively. From the parametrix construction for an elliptic pseudodifferential operator one sees that P has order —m. If P is nonlinear, PA = P((A — X0I) + X0I) need not be equal to P(A — X0I) modulo compact operators. Hence, P fails to be a left parameterix of A which is responsible for uniqueness and regularity of solutions. On the other hand, the second equality implies readily that AP = I + X0P reduces to I up to a compact operator, i.e., P is still a right parametrix for A which is responsible for existence. This suggests to distinguish those nonlinear operators A which possess linear parametrices P. Another way is to mean by parametrices alone linear mappings. Then, if A — XI has a parametrix P for a particular value X = X0, then P is a parametrix of A — XI for all X G C. This is also true for existence and regularity theorems.
The first author gratefully acknowledges the financial support of the Islamic Center Association for Guidance and Higher Education.
References
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Когомологии де Рама посредством методов гильбертовых пространств
Исан Малас Николай Тарханов
Институт математики Потсдамский университет Карл-Либкнехт-Str. 24/25, Потсдам, 14476
Германия
Обсуждаются канонические представления когомологий де Рама на компактном многообразии с краем. Они получены путем минимизации интеграла энергии в гильбертовом пространстве дифференциальных форм, которые наряду с внешней производной принадлежат области присоединенного оператора. Соответствующие уравнения Эйлера-Лагранжа сводятся к эллиптической краевой задаче на многообразии, которую обычно называют проблемой Неймана после Спенсера.
Ключевые слова: комплекс де Рама, когомология, теория Ходжа, проблема Неймана.
