Sharp Hodge Decompositions, Maxwell s Equations, and Vector Poisson Problems on Non-Smooth, Three-Dimensional Riemannian Manifolds

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1 Shar Hodge Decomositions, Maxwell s Equations, and Vector Poisson Problems on Non-Smooth, Three-Dimensional Riemannian Manifolds Marius Mitrea Dedicated to the memory of José F. Escobar Abstract We solve three basic otential theoretic roblems: Hodge decomositions for vector fields, Poisson roblems for the Hodge-Lalacian, and inhomogeneous Maxwell equations, in arbitrary Lischitz subdomains of a smooth, comact, three dimensional, Riemannian manifold. In each case we derive shar estimates on Sobolev-Besov scales and establish integral reresentation formulas for the solution. The roofs rely on tools from harmonic analysis and algebraic toology, such as Calderón-Zygmund theory and the de Rham theory. 1 Introduction In this aer we solve three basic otential theoretic roblems: (I) Hodge decomositions for vector fields, (II) Poisson roblems for the Hodge-Lalacian, and (III) inhomogeneous Maxwell equations, in Lischitz subdomains of a smooth, comact, boundaryless, three dimensional, Riemannian manifold M. They are all considered in the context of Sobolev-Besov saces, i.e. when the global smoothness of both the data and the solutions is measured on these scales. In hindsight, the roblems (I)-(III) above turn out to be closely related. A manifestation of this is that they share a common, (asymtotically) shar well-osedness region, stemming from necessary limitations on the indices s (smoothness) and (integrability), of the Sobolev-Besov saces for which these PDE s have unique solutions, continuously deendent on the given data. In turn, this region, call it R Ω, is entirely determined by the geometric characteristics of the underlying domain Ω M. More concretely, (s, 1/) R Ω 0 < 1 < 1, < s < 1 ( ), Ω < 1 s 3 < 1 3 ( 2 Ω + 1 Here, Ω is further defined in terms of the critical exonents intervening in the (regular) Dirichlet and Neumann roblems for the Lalace-Beltrami oerator in Ω (as well as its comlement), when otimal L estimates for the associated nontangential maximal function are sought. A recise definition is given in 4. For the urose of this introduction, we note that, generally seaking, 1 Ω < 2; cf. [11], [12], [31], [14], [65], for the flat, Euclidean setting and [46]-[49] for Lischitz subdomains of Riemannian manifolds. Partly suorted by NSF 2000 Mathematics Subject Classification. 31B10, 31C12, 35J25, 35Q60, 42B20, 46E35, 58A14, 58G20 Key words: Hodge decomositions, Maxwell s equations, Poisson roblems, Lischitz domains, Sobolev-Besov saces 1 ). (1.1)

2 One feature of Ω which influences the size of Ω is the local oscillations of the unit conormal ν to Ω, i.e. lim ε 0 su {dist (ν(x), ν(y)); dist (x, y) < ε} (cf. the comments in [5]). In fact, Ω tends to 1 as these local oscillations tend to zero. Moreover, Ω = 1 when Ω C 1 ([19]), and this continues to be the case even when the unit conormal only has vanishing mean oscillations. Finally, for a Lischitz olyhedron in the Euclidean setting, Ω can be estimated in terms of the dihedral angles involved; cf. [24]. The icture below reresents the two-dimensional region R Ω alluded to above, when considered in the (s, 1/)-lane (i.e., smoothness vs. recirocal integrability): 1/ 1 3 ( 2 Ω + 1) (2 2 Ω,1) (1,1) ( 1 Ω 1, 1 Ω ) (1, 1 2 ) (1 1 Ω, 1 1 Ω ) (-1,0) ( 2 O Ω 2,0) 2 3 (1 1 Ω ) s The (interior of the) dashed hexagon reresents the well-osedness region for the Hodge decomositions, Maxwell s equations, and vector Poisson roblems. We now roceed to describe our main results in greater detail (comlete definitions including those for the standard Sobolev-Besov saces L s, Bs,q, as well as various nonstandard versions such as H s,, THs, etc. are given in 2). I. Hodge Decomositions. Let Ω be an arbitrary Lischitz domain in M with outward unit normal ν. Call a 1-form monogenic, if it is both curl and divergence free. For 1 < <, s R, let L s stand for the usual Sobolev scale and, for 1 + 1/ < s < 1/, consider H s, (Ω) := {u L s(ω, Λ 1 T M); div u = 0, curl u = 0, ν u = 0}, (1.2) H s, (Ω) := {u L s(ω, Λ 1 T M); div u = 0, curl u = 0, ν u = 0}, (1.3) i.e., monogenic 1-forms in L s, with vanishing normal and tangential comonents on Ω. 2

3 Theorem 1.1 For each (s, 1/) R Ω, the following Hodge-tye decomositions hold: L s(ω, Λ 1 T M) = [L 1+s,0 (Ω)] curl[hs, (Ω; curl)] H s, (Ω), (1.4) L s(ω, Λ 1 T M) = [L 1+s (Ω)] curl[hs, (Ω; curl)] H s, (Ω), (1.5) where the direct sums (of closed subsaces) are toological. II. Poisson roblems for the Hodge-Lalacian on differential forms. Recall that the vector Hodge-Lalacian on M is defined as := curl curl + div. Theorem 1.2 Let Ω M be an arbitrary Lischitz domain. Then for each (s, 1/) in R Ω, the boundary value roblem (1st Poisson Problem) u = η L s(ω, Λ 1 T M), u H s, (Ω; curl), curl u H s, (Ω; curl), div u L s+1 (Ω), ν u = f B, s 1/ ( Ω), ν curl u = g TH s( Ω), (1.6) is Fredholm solvable of index zero. More secifically, (1.6) has a solution if and only if the data satisfy the linear constraints η, h = g, ν (ν h), h H s, (Ω), 1/ + 1/ = 1, (1.7) and the sace of null solutions is H s, (Ω). The latter is a finite dimensional sace, whose dimension is b 1 (Ω), the first Betti number of Ω. Similar results are valid for u = η L s(ω, Λ 1 T M), u H s, (Ω; curl), curl u H s, (Ω; curl), div u L s+1 (2nd Poisson Problem) (Ω), Tr (div u) = f B, s+1 1/ ( Ω), ν u = g TH s( Ω). (1.8) In this case, the necessary comatibility conditions read η, h = f, ν h, h H s, (Ω), 1/ + 1/ = 1, (1.9) and the sace of null solutions is recisely H s, (Ω), whose dimension is b 2 (Ω), the second Betti number of Ω. III. Maxwell s equations. Consider the time-harmonic version of the Maxwell system, governing the roagation of electromagnetic waves in a Lischitz domain Ω M. See, e.g., [7], [9], [10]. Theorem 1.3 For any Lischitz domain Ω M and any k C\{0}, the inhomogeneous Maxwell system 3

4 E, H H s, (Ω; curl), curl E ikh = K L s(ω, Λ 1 T M), (Inhomogeneous Maxwell) curl H + ike = J L s(ω, Λ 1 T M), ν E = f TH s( Ω), (1.10) is Fredholm solvable, of index zero. Moreover, there exists an unbounded, non-decreasing sequence of real, nonnegative numbers {k j } j such that for each k C \ {±k j } j, the boundary value roblem (1.10) is well-osed, as long as (s, 1/) R Ω. It is imortant to oint out that the validity range for Theorems when s = 0 (i.e. for L (Ω, Λ 1 T M) fields) becomes 3( 2 Ω + 1) 1 < < 3 2 (1 1 Ω ) 1, as is visible from the above icture. In articular, will do in any Lischitz domain (when s = 0). The counterexamles in [20] can then be used to rove that the range of indices s, described in Theorem 1.1 is in the nature of best ossible. To ut matters in the roer ersective, let us now briefly discuss the origins of our results. The crowning achievement of the classical theory of ellitic PDE s (sometimes referred to as the Shift Theorem; cf. [26], [61]) is that all regular ellitic roblems Lu = f + boundary conditions, in C domains are Fredholm solvable (i.e. existence and uniqueness hold modulo finite dimensional saces) on all Sobolev-Besov scales and any solution u is deg L units smoother than the datum f. The situation is radically different in less smooth domains. For instance, Dahlberg [13] has constructed a domain Ω in R n with a C 1 -boundary and f C ( Ω) such that u = f, u L 2 1(Ω), Tr u = 0 = 2 u x j x k / L (Ω), j, k = 1, 2,..., n, 1 < <. (1.11) In other words, for Dahlberg s domain Ω and each 1 < <, the Poisson roblem for the Lalacian with homogeneous Dirichlet boundary conditions fails to be well-osed for data in L (Ω) (in the sense that it is no longer reasonable to exect solutions with two derivatives in L (Ω)). This raises the fundamental issue of identifying those Sobolev-Besov saces within which the natural correlation between the smoothness of the data and that of the solutions is reserved when the domain in question is allowed to have a minimally smooth boundary (in the sense of [60]). In their ground breaking work [32], Jerison and Kenig were able to roduce such an otimal well-osedness region for the Poisson roblem with Dirichlet boundary conditions for the scalar, flat sace Lalacian in bounded, Euclidean Lischitz domains in the context of Sobolev-Besov saces. The basic estimate roved in [32] is that u L s+1/ (Ω) Tr u B, s ( Ω), (1.12) uniformly for u harmonic in Ω, for an (asymtotically) otimal range of indices s,. The methods in [32], though beautiful in their elegance and sharness, rely in an essential fashion on harmonic measure estimates (and, by extension, on ositivity and maximum rinciles) and, as such, do not readily adat to systems of PDE s, or to other natural boundary conditions, e.g. of Neumann tye. In fact, the latter issue makes the object of one of the oen roblems listed in Kenig s book [36]; cf. # , In the author s words (cf. Introduction, loc. cit.), these have been singled out as roblems which we find articularly challenging, and which we feel will lead to further imortant develoments in the subject. Relatively recently, in [20], [48], we have develoed an alternative strategy to roving (1.12) which relies on a systematic use of boundary integral methods. This aroach, in rincile, does 4

5 not distinguish between Dirichlet and Neumann tye conditions, or between a single equation and systems. At the heart of the matter is the fact that, with K standing for the harmonic (rincial value) double layer otential oerator on Ω and I denoting the identity oerator (see 4 for a discussion), (s + 1/ 1, 1/) R Ω = 1 2 I + K : B, s ( Ω) Bs, ( Ω) is an isomorhism. (1.13) The chief goal of this aer is to continue this line of work and initiate the study of systems of PDE s in Sobolev-Besov saces in Lischitz domains. For the urose of this introduction, let us oint out that one of the main estimates we establish in this aer is that whenever (s, 1/) R Ω and k is not an eigenvalue, then E L s (Ω,Λ 1 T M) + H L s(ω,λ 1 T M) ν E B, s 1/ ( Ω,Λ1 T M) + ν H B, ( Ω), (1.14) s 1/ uniformly for vector fields E, H solving (the homogeneous) Maxwell s equations curl E ikh = 0, curl H + ike = 0 (1.15) in the Lischitz domain Ω. Our aroach to the three roblems listed at the beginning of this section is constructive in nature, as is based on singular integral oerators, and we momentarily digress in order to exlain some of the difficulties which arise in this scenario. The essence of Hodge decomositions is the identity I = div G curl curl G + P, where G is the Green oerator associated with the Hodge-Lalacian (i.e., the solution oerator 1 for a vector Poisson roblems with aroriate boundary conditions), and P is the rojection onto monogenic forms. Consequently, the success of decomosing a vector field as in (1.4)-(1.5), where the individual summands have the same amount of regularity as the original field, deends to a large extent on the maing roerties of oerators like curl curl 1 and div 1. On a smooth domain, these are classical zero-order seudodifferential oerators, but in the resence of boundary irregularities they fail even to be of Calderón-Zygmund tye. In this latter scenario, one can only describe them in terms of boundary layer otentials, tyically of vector character, and their inverses (whenever meaningful). It has been understood for some time (cf. the discussion in [40]) that, in the context of differential forms on a subdomain Ω of an arbitrary Riemannian manifold M, there exists a natural hierarchy of boundary layer otentials {M l } l, 0 l dim M, (acting on boundary l-forms), so that M 0 K is simly the usual harmonic double layer acting on scalar functions (viewed as differential forms of degree zero). In the setting of L ( Ω), the invertibility of the entire scale { 1 2 I + M l } l is now understood, thanks to the work in [41], [40]. There we have shown that 2 ε < < 2 + ε will do, and this is asymtotically the best range if one insists on allowing arbitrary l s. On the technical side, the novel ste we take here is clarifying what the analogue of (1.13) is, both in terms of the range of indices s,, as well as the nature of the saces which should relace the scalar Besov scale in (1.13), at the next level u on this hierarchy, i.e. when l = 1. This is achieved by exloiting some remarkable intertwining identities involving, on the one hand, the oerators M l at the scalar level (l = 0) in concert with those at the vector level (l = 1) and, on the other hand, some natural first order (boundary) differential oerators. In this connection, a major ingredient required to carry out this rogram is understanding the Fredholm roerties of certain surface differential oerators. More secifically, recall that there 5

6 are two natural first-order oerators on the Lischitz manifold Ω: the tangential gradient and its adjoint, the surface divergence. As their comosition is zero, they give rise to a comlex 0 Scalar Functions tangential grad surface div Vector Fields Scalar Functions 0. (1.16) The technical accomlishment alluded to above is to find suitable smoothness saces guaranteeing that the comlex (1.16) is exact, and then to identify its cohomology grous. This is done via tools from algebraic toology and harmonic analysis. It should be ointed out that the intertwining identities, referred to two aragrahs above, are most useful when the dimension of the ambient sace is three, a restriction which aears inherent to our method. What the corresonding situation is in arbitrary sace dimensions and for other values of l remains an oen roblem at the moment. In broad outline, our lan is to rove first a simler version of Theorem 1.2, dealing with vector Poisson roblems with homogeneous boundary conditions (cf. 7), by emloying layer otentials (lus other tools develoed in 4-5). This, in turn, leads to a constructive aroach to Theorem 1.1. These Hodge decomositions are key ingredients used in 8-9, in the solution of the inhomogeneous Maxwell system in Theorem 1.2. With this in hand, the cycle is then comleted by returning to (and finishing the roof of) Theorem 1.2 in 10. The (asymtotic) sharness of our main results is a consequence of the counterexamles in [32], [20]. Let us conclude with a few historical notes. The subject of Hodge decomositions in the C context can be traced back to the work of Hodge, Kodaira and de Rham ([27], [37], [2], [16]) starting in the 1930 s. A modern account can be found in, e.g., [61], [58]. Subsequent develoments, emhasizing regularity asects and/or allowing less smooth structures can be found in [50], [51], [52], [62], [28], [59], [42], [40], [56]. The methods emloyed by these authors do not work in the resent context. The roblem II that we solve here goes back to a question osed to us by Eugene Fabes and Michael Taylor in the mid 90 s. Natural boundary roblems for the Hodge-Lalacian in Lischitz domains for L -boundary data have been systematically studied in [40], where otimal nontangential maximal function estimates have been roved. The study of the Maxwell system via integral equation methods has a long tradition, originating in the ioneering work of A. P. Calderón [4] and C. Müller [53], [54]. L estimates for the nontangential maximal function oerator (denoted in the sequel by N ) have been roduced in [45], [44]. There it is roved that for solutions of the homogeneous Maxwell s equations (1.15), there holds N (E) L 2 ( Ω,Λ 1 T M) + N (H) L 2 ( Ω,Λ 1 T M) E L 2 1/2 (Ω,Λ1 T M) + H L 2 1/2 (Ω,Λ1 T M) ν E L 2 ( Ω,Λ 1 T M) + ν H L 2 ( Ω), (1.17) whenever k is not an eigenvalue for the Lischitz domain Ω; this can be regarded as the limiting case = 2, s = 1/2 of (1.14). An extension to higher dimensions is in [30]; cf. also [40]. The novelty here is roducing shar estimates in Sobolev-Besov saces for an otimal range of indices. Acknowledgments. The author would like to thank Professors Ronald A. DeVore, Carlos E. Kenig and Michael E. Taylor for their interest in this work and for stimulating discussions. He is also grateful to the referee for carefully reading the manuscrit and for suggesting a number of changes. 6

7 2 Preliminary results In this section, which is further divided into four subsections, we review basic notation, collect definitions and rove some rearatory results. 2.1 The geometrical setting Let M be a smooth, comact, boundaryless, connected, oriented manifold of real dimension dim M = 3. As is customary, we denote by Λ l T M, l = 0, 1, 2, 3, the exterior ower bundles (i.e. differential forms of degree l), and by d and, resectively, the exterior derivative oerator and the exterior roduct of forms. When acting on scalar-valued functions, we denote d by. Assume next that M is endowed with a metric tensor g = j,k g jkdx j dx k, whose coefficients g jk are of class C 1,1. This induces a volume element dvol and, further, a ointwise inner roduct (u, v) u v = u, v in each Λ l T M, 0 l 3. Recall that the Hodge star oerator is the unique vector bundle morhism : Λ l T M Λ 3 l T M such that u ( u) = u 2 dvol. We also denote by δ the formal adjoint of d and set div u := δ u, curl u := du and u v := (u v), u, v Λ 1 T M. (2.1) In articular, the fact that d 2 = 0 entails curl = 0. Recall that the Lalace-Beltrami oerator on M is given in local coordinates, where the metric tensor reads g = g jk dx j dx k, by u := div( u) = (det (g jk )) 1/2 j j ( k ) g jk (det (g jk )) 1/2 k u, (2.2) where we take (g jk ) to be the matrix inverse of (g jk ). This oerator fits naturally into the framework of the Hodge Lalacian := dδ δd on l-forms, when l = 0. Here (and in the sequel), we make no secial notational distinction between the scalar Lalace-Beltrami oerator and the Hodge- Lalacian on 1-forms. As is well known, Also, curl curl = + div on 1-forms, and div = on scalar functions. (2.3) curl = curl and div = div. (2.4) Recall next that Ω M is called a Lischitz domain rovided Ω can be described in aroriate local coordinates by means of grahs of Lischitz functions. For each 1, we denote by L ( Ω) the Lebesgue sace of -ower integrable functions with resect to the surface measure dσ (with the usual convention when = ). If ν T M is the unit outward conormal to Ω, we let tan := ν (ν ) stand for the tangential gradient on Ω. Finally, throughout the aer,, will stand for either a ointwise inner roduct, or the natural airing between a sace and its dual. Also, if X Y are Banach saces, we will let X Y denote the annihilator of X. 2.2 Scalar Sobolev and Besov saces in Lischitz domains Denote by L 1 ( Ω) the Sobolev sace of functions in L ( Ω) with tangential gradients in L ( Ω), 1 < <. Saces with fractional smoothness can then be defined via comlex interolation, i.e. 7

8 ( L θ ( Ω) := [L ( Ω), L 1 ( Ω)] θ, 0 < θ < 1, 1 < <. We also set L s( Ω) := L s ( Ω)) for 0 s 1, 1 <, <, 1/ + 1/ = 1. Next, Besov saces with ositive smoothness on Ω can then be introduced via real interolation, i.e. B,q θ ( Ω) := (L ( Ω), L 1 ( Ω)) θ,q, with 0 < θ < 1, 1 <, q <. (2.5) An intrinsic definition for membershi to Bs,q (R 2 ), 1, q <, 0 < s < 1, is obtained by requiring that f B,q s (R 2 ) := f L (R 2 ) + ( f( + t) f( ) R 2 t 2+sq q L (R 2 ) dt ) 1/q < +. (2.6) For the same range of indices, when Ω is the region from R 3 above the grah of a Lischitz function φ : R 2 R, we define Bs,q ( Ω) as the sace of functions f for which the assignment x f(x, φ(x)) belongs to Bs,q (R 2 ). This definition then readily extends to the case of arbitrary Lischitz subdomains of M via a standard artition of unity argument. The case when = q = corresonds to the usual (non-homogeneous) Hölder saces. Also, for 1 < s < 0 and 1 <, q < or = q =, we set B,q s ( Ω) := (B,q s ( Ω)), 1/ + 1/ = 1, 1/q + 1/q = 1. (2.7) In the sequel, we shall also need to work with the Besov saces B 1,1 s ( Ω), s (0, 1). Insired by the corresonding atomic characterization from [21], we set { B s( Ω) 1,1 := L q ( Ω) + f = λ j ϑ j ; ϑ j -atom, (λ j ) j l 1}, (2.8) where the series converges in the sense of distributions, and q > 1 is arbitrary (different choices yield isomorhic saces). In this context, a B s 1,1 ( Ω)-atom, 0 < s < 1, is a function ϑ L ( Ω) with suort contained in a surface ball B r (x 0 ) Ω, x 0 Ω, 0 < r < diam Ω, and satisfying ϑ dσ = 0, ϑ L ( Ω) r s 2. (2.9) Ω Furthermore, for f B 1,1 s ( Ω), 0 < s < 1, f B 1,1 s ( Ω) := inf { g L q ( Ω) + λ j ; f = g + λ j ϑ j }, (2.10) where g L q ( Ω), q > 1 (different q s yield equivalent norms), ϑ j s and (λ j ) j are as in (2.8). Let us also oint out that B s 1,1 ( Ω) is local, in the sense that this sace is a module over Bα, ( Ω), for each α (s, 1), and that (B s 1,1 ( Ω)) = Bs, ( Ω), for 0 < s < 1. We now briefly discuss the case of Sobolev and Besov classes on an oen subset Ω of M. First, the Sobolev (or otential) scale L s(m), 1 < <, s 0, is obtained by lifting L s(r 3 ) := {(I ) s/2 f; f L (R 3 )} to M, and we denote by L s(ω) the restriction of elements in L s(m) to Ω. Similarly, for 1, q, s > 0, the Besov sace Bs,q (M) is defined by localizing and transorting via local charts its Euclidean counterart, i.e., Bs,q (R 3 ) (the latter is defined analogously to (2.5)-(2.6); see, e.g., [55], [1], [63], [33]). Then, Bs,q (Ω), 1, q, s > 0, consists of restrictions to Ω of functions from Bs,q (M). Both scales in Ω are equied with natural norms, defined by taking the infimum of the corresonding norms of all ossible extensions to M. Using Stein s extension oerator and then invoking 8

9 well known real interolation results (cf., e.g., [1]), it follows that for any Lischitz domain Ω M, L s(ω) and Bs,q (Ω) are comlex interolation scales, i.e. if 1 = 1 θ 0 [L 0 s 0 (Ω), L 1 s 1 (Ω)] θ = L s(ω) (2.11) + θ 1, s = (1 θ)s 0 + θs 1, 0 < θ < 1, 1 < 0, 1 <, s 0, s 1 0, and [B 0,q 0 s 0 (Ω), B 1,q 1 s 1 (Ω)] θ = B,q s (Ω) (2.12) if 1 = 1 θ 0 + θ 1, 1 q = 1 θ q 0 + θ q 1, s = (1 θ)s 0 + θs 1, 0 < θ < 1, 1 0, 1, q 0, q 1, s 0, s 1 > 0, s 0 s 1. As is well known, the Besov and Sobolev saces on the domain are also related via real interolation. For instance, we have the formula ( L (Ω), L k (Ω)) s,q = B,q sk (Ω) (2.13) when 0 < s < 1, 1 <, q < and k is a ositive integer. In this connection, let us also oint out that ( B,q 0 s 0 (Ω), B,q 1(Ω) ) s,q = B,q s (Ω) (2.14) s 1 if 1 q = 1 θ q 0 + θ q 1, s = (1 θ)s 0 + θs 1, 0 < θ < 1, 1, q 0, q 1, s 0, s 1 > 0, and s 0 s 1. For the remainder of this subsection we assume that Ω is a Lischitz subdomain of M. Following [32], for 1 < <, s R we define the sace L s,0 (Ω) to consist of distributions in L s(m) suorted in Ω (with the norm inherited from L s(m)). It is known that C o (Ω) is dense in L s,0 (Ω) for all values of s and. Recall (cf. [33]) that the trace oerator Tr : L s(ω) B, ( Ω) (2.15) s 1 is well-defined, bounded and onto if 1 < < and 1 < s < This also has a bounded right inverse whose oerator norm is controlled exclusively in terms of, s and the Lischitz character of Ω. Similar results are valid for Tr : B,q s (Ω) B,q s 1 ( Ω). In this latter case we may allow 1, q ; cf. [1]. If 1 < < and 1 < s < 1 + 1, the sace L s,0 (Ω) is the kernel of the trace oerator Tr acting on L s(ω). This follows from the Euclidean result (Proosition 3.3 in [32]). In fact, for the same range of indices, L s,0 (Ω) is the closure of C o (Ω) in the L s(ω) norm. For ositive s, L s(ω) is defined as the sace of distributions in Ω such that { } f L s (Ω) := su f, ϕ ; ϕ Co (Ω), ϕ L q s (M) 1 < +, (2.16) where tilde denotes the extension by zero outside Ω and q = 1. For all values of and s, C (Ω) is dense in L s(ω). Also, C o (Ω) is dense in L s(ω) if s 0. Next, for any s R, L q s,0 (Ω) = (L s(ω)) and L s(ω) = ( L q s,0 (Ω) ). (2.17) For later reference, let us oint out that for each 1 < <, 1 + 1/ < s < 1/, there exists a bounded, linear extension oerator L s(ω) u ũ L s(m), (2.18) with the roerty that su ũ Ω; cf. Theorem 3.5 in [64]. (Of course, when s 0, this is just the extension by zero outside Ω.) Its image is recisely L s,0 (Ω), allowing the identification 9

10 L s(ω) = L s,0 (Ω), (1, ), s ( 1 + 1/, 1/). (2.19) Thus, if, (1, ) are such that 1/ + 1/ = 1, then (L s(ω)) = L s(ω), s ( 1 + 1/, 1/). (2.20) Also, the restriction oerator to Ω, denoted by Ω in the sequel, has a linear, bounded realization Ω : L s(m) L s(ω), u Ω, ϕ := u, ϕ, ϕ C o (Ω), (2.21) for each 1 < <, s > 1 + 1/. Furthermore, the extension oerator to M by zero outside the natural domain denoted in the sequel by tilde extends to a bounded ma from L s(ω) into L s(m) for any 1 < <, 1 + 1/ < s < 1/; cf. [64]. In articular, u Ω, v = u, ṽ, u L s(m), v L s(ω), (2.22) if 1 <, <, 1/ + 1/ = 1, and 1 + 1/ < s < 1/. Another imortant asect (which follows from Corollary 13.5 in [21], or Proosition 3.5 in [32]; cf. also [64]) is that multilication by χ Ω, the characteristic function of Ω, mas L s(m) boundedly onto L s,0 (Ω) = L s(ω), for each 1 + 1/ < s < 1/, 1 < <. We also refer to [57], [63], [1], [17], [32], [64], [48], for a more detailed exosition of these and other related matters, such as embedding theorems (frequently used in the aer). Here we only want to alert the reader that L s(ω, Λ 1 T M) will stand for L s(ω) Λ 1 T M, i.e. the collection of 1-forms with coefficients in L s(ω). A similar convention is in lace for saces defined on Ω, and for the Besov scale. E.g., B,q s ( Ω, Λ 1 T M) := B,q s ( Ω) Λ 1 T M, etc. 2.3 Vector Sobolev saces. Tangential and normal traces In this aer we shall work with certain nonstandard Sobolev saces which are naturally adated to the tye of differential oerators we intend to study. Secifically, if Ω is an oen subset of M and if 1 < <, s R, we introduce H s, (Ω; div) := {u L s(ω, Λ 1 T M); div u L s(ω)}, (2.23) H s, (Ω; curl) := {u L s(ω, Λ 1 T M); curl u L s(ω, Λ 1 T M)}, (2.24) equied with the natural grah norms. Throughout the aer, all derivatives are taken in the sense of distributions. Let us now assume (as we shall do for the remainder of this subsection) that Ω M be an arbitrary Lischitz domain with outward unit conormal ν T M. If 1 < <, 1 + 1/ < s < 1/, and u H s, (Ω; div) then we can then define ν u B, ( Ω) by setting s 1 ν u, Tr ψ := [ψ div u + u, ψ ] dvol, (2.25) Ω for each ψ L 1 s (Ω), 1/ + 1/ = 1. It follows that the oerator is bounded, that is, ν : H s, (Ω; div) B, ( Ω) (2.26) s 1 10

11 ν u B, s 1 ( Ω) C( u L s(ω,λ 1 T M) + div u L ), (2.27) s(ω) granted that 1 < < and 1+1/ < s < 1/. Similarly, if u H s, (Ω; curl) for some 1 < < and 1 + 1/ < s < 1/ then we can define ν u B, ( Ω, Λ 1 T M) by s 1 ν u, Tr ϕ := [ curl u, ϕ u, curl ϕ ] dvol, (2.28) Ω for any ϕ L 1 s (Ω, Λ1 T M), 1/ + 1/ = 1. Furthermore, it follows from (2.24), (2.28) that the oerator is bounded, i.e. ν : H s, (Ω; curl) B, ( Ω, Λ 1 T M) (2.29) s 1 ν u B, s 1 ( Ω,Λ 1 T M) C( u L s (Ω,Λ 1 T M) + curl u L s (Ω,Λ 1 T M)) (2.30) as long as 1 < < and 1 + 1/ < s < 1/. Parenthetically, let us also oint out that if, e.g., u C 1 (Ω, Λ 1 T M) then ν u, ν u coincide with the usual (ointwise) cross roduct and dot roduct with ν, resectively. Our immediate goal is to describe the images of the oerators (2.26) and (2.29). Proosition 2.1 Let Ω M be a connected Lischitz domain and fix 1 < <, 1 + 1/ < s < 1/. Then for any f B, s 1/ ( Ω) with Ω f dσ = 0 there exists u Hs, (Ω; div) with div u = 0 and ν u = f. Furthermore, matters can be arranged so that u L s (Ω,Λ 1 T M) C f B, s 1/ ( Ω) for some constant C = C( Ω,, s) > 0. As a corollary, the oerator in (2.26) is always onto. In order to rove the above result we need a lemma of indeendent interest. Lemma 2.2 Let Ω M be Lischitz and suose that 1 < <. Then, for each s > 1 + 1/, the gradient is well-defined, bounded, and has closed range. : L s(ω) L s 1 (Ω, Λ1 T M) (2.31) Proof. The well-definiteness and boundedness when s 1 are known (cf. Proosition 2.18,. 173 in [32]). Furthermore, when 0 s 1, we can interolate between the standard cases s = 0 and s = 1. Thus, as far as the first art of the claim in the lemma is concerned, we are left with analyzing the situation when 1 + 1/ < s < 0. In this case, for each u L s(ω) = (L s(ω)), with 1/ + 1/ = 1, ψ Co (Ω, Λ 1 T M), and with the airing, considered in the distributional sense, we write u, ψ = u, div ψ u (L s (Ω)) div ψ L s (Ω) C u (L s (Ω)) ψ L 1 s (Ω,Λ1 T M). (2.32) 11

12 Since Co (Ω, Λ 1 T M) L 1 s,0 (Ω, Λ1 T M) densely and for the range of indices we are considering this latter sace is the closure of the former in the L 1 s-norm, we conclude from (2.32) that u belongs to the sace (L 1 s,0 (Ω, Λ1 T M)) = L s 1 (Ω, Λ1 T M), as desired. Turning our attention to the last claim in the lemma, it suffices to treat the case when 1+1/ < s 1 (cf., e.g.,. 173 in [32] for s 1). Now, since dim Ker ( ; L s(ω)) <, the closedness of the range of (2.31) is equivalent to the validity of the estimate u L s (Ω) C u L s 1 (Ω,Λ1 T M) + Com u, (2.33) uniformly for u L s(ω), where Com denotes a comact oerator on L s(ω). In turn, this estimate localizes so there is no loss of generality assuming that Ω is a bounded, Euclidean Lischitz domain, which is starlike with resect to some ball B. In this context, as it follows from the discussion in the Aendix, if θ Co (B), θ = 1, then there exists a linear, bounded oerator J : L q α,0 (Ω) L q α+1,0 (Ω), for 1 < q <, 2 + 1/q < α < 1/q, such that J ϕ C o (Ω, Λ 1 R 3 ) for any ϕ Co (Ω) and div J ϕ = ϕ θ( ϕ) for any ϕ Co (Ω). Note that 1/+1/ = 1 and 1+1/ < s < 1+1/ ensures that J : L s,0 (Ω) L 1 s,0 (Ω). Then, for any ϕ C o (Ω), we have u, ϕ u, div J ϕ + u, θ ϕ, 1 u, J ϕ + Com u ϕ L s,0 (Ω) u L s 1 (Ω,Λ1 T M) J ϕ L 1 s,0 (Ω,Λ1 T M) + Com u ϕ L s,0 (Ω) C( u L s 1 (Ω,Λ1 T M) + Com u ) ϕ L s,0 (Ω). (2.34) ( ) Since Co (Ω) is dense in L s,0 (Ω), we see that u L s 1,0 (Ω) = L s (Ω) and (2.33) holds. The desired conclusion follows. For further use, let us oint out that the above reasoning also shows that for any distribution u in the Lischitz domain Ω, the following imlication holds: 1 < <, s > and u L s 1 (Ω, Λ1 T M) = u L s(ω). (2.35) Now we are ready for the ( Proof of Proosition 2.1. Since f B, s 1/ ( Ω) = B, 1 s 1/ ( Ω)) with 1/ + 1/ = 1, and ) Tr : L 1 s (Ω), B 1 s 1/ ( Ω), it follows that Φ := f Tr belongs to the sace (L 1 s (Ω) = L s 1,0 (Ω) and satisfies Φ, 1 = 0. In other words, Φ is a bounded linear functional on L 1 s (Ω)/R. In turn, this can be thought of as a closed subsace in L s(ω, Λ 1 T M) via the identification u u. That this works is guaranteed by Lemma 2.2. Thus, by Hahn-Banach theorem, there exists u (L s(ω)) = L s (Ω) such that u, ϕ = Φ, ϕ = f, Tr ϕ, ϕ L 1 s (Ω). (2.36) Choosing ϕ Co (Ω) in the above identity entails div u = 0, so u H s, (Ω; div) is divergence-free. Then another look at (2.36) reveals that ν u = f, as desired. 12

13 Later on, it is going to be imortant to extend elements from H s, (Ω; div), H s, (Ω; curl) to distributions in H s, (M; div) and H s, (M; curl), resectively. For the time being, we rove the following. Lemma 2.3 Let Ω, D be two arbitrary Lischitz subdomains of M and assume that 1 < <, 1 + 1/ < s < 1/. Then u H s, (Ω; div), ν u = 0 on D Ω = (div ũ) D = ( div u) D, (2.37) u H s, (Ω; curl), ν u = 0 on D Ω = (curl ũ) D = ( curl u) D. (2.38) Proof. This follows from (2.25), (2.28). While the oerator in (2.26) is onto, this is clearly not the case for (2.29). Since the range of this latter oerator is going to be of fundamental imortance for us in the sequel, for 1 < < and 1 + 1/ < s < 1/, we introduce TH s( Ω) := Now, if we define { } f B, ( Ω, Λ 1 T M); f = ν u for some u H s, (Ω; curl). (2.39) s 1 H s, (Ω; div) := {u H s, (Ω; div); ν u = 0}, (2.40) H s, (Ω; curl) := {u H s, (Ω; curl); ν u = 0}, (2.41) (i.e. the null saces of (2.26) and (2.29), resectively) then, at the level of quotient saces, / ν : H s, (Ω; curl) H s, (Ω; curl) TH s( Ω) (2.42) becomes an algebraic isomorhism. Thus, it is natural to endow the sace (2.39) with the norm for which (2.42) is also toological, i.e. } f TH s ( Ω) { u := inf L s (Ω,Λ 1 T M) + curl u L s(ω,λ 1 T M) ; f = ν u, u Hs, (Ω; curl). (2.43) Some of the basic roerties of the saces just introduced are summarized in the following roosition. Proosition 2.4 Let Ω be an arbitrary Lischitz subdomain of M and assume that 1 < <, 1 + 1/ < s < 1/. Also, let 1 < < be the conjugate exonent of. Then the following hold. (i) TH s( Ω) is a reflexive Banach sace, continuously embedded into B, ( Ω, Λ 1 T M). s 1 (ii) ν C ( Ω, Λ 1 T M) TH Ω s( Ω) continuously and densely. 13

14 (iii) The maing ν : TH s( Ω) (TH s( Ω)) (2.44) defined by ν f, g := curl u, w u, curl w (2.45) for u H s, (Ω; curl) with f = ν u, and w H s, (Ω; curl) with g = ν w is well-defined and bounded, in fact an isomorhism. (iv) The inverse of the oerator (2.44) is ν : (TH s( Ω)) TH s( Ω), defined by ν Φ := f if Φ (TH ( Ω)) is of the form Φ = ν f, f TH s( Ω). In articular, we have (ν ) 1 = ν and (ν ) = ν both on the scale (TH s( Ω)) as well as on the scale TH s( Ω). Finally, with the above conventions, for any u H s, (Ω; curl), v H s, (Ω; curl). curl u, v = u, curl v ν u, ν (ν v) (2.46) Proof. That (2.39) is Banach follows from (2.42) and definitions, whereas the fact that (2.39) is reflexive will be a consequence of (iii). Going further, (ii) is a direct consequence of the following aroximation result: u H s, (Ω; curl) = u ε C ( Ω, Λ 1 T M) with u ε u in L s(ω, Λ 1 T M) and curl u ε curl u in L s(ω, Λ 1 T M). (2.47) This, so we claim, is valid as long as 1 < < and 1 + 1/ < s < 1/. In order to see this, observe that, by using a artition of unity argument, there is no loss of generality to assume that Ω R 3 is a bounded, Euclidean Lischitz domain and su u Ω is small enough so that it is ossible to construct an oen cone Γ centered at the origin for which Γ + ( Ω su u) R 3 \ Ω. (2.48) Next, let ϕ be a smooth, comactly suorted function in R 3, having integral one and such that su ϕ Γ. As usual, set ϕ ε := ε 3 ϕ( ε 1 ) for ε > 0, and let U L s(r 3, Λ 1 R 3 ) be so that U Ω = u, su U Ω. Finally, introduce u ε := (ϕ ε U) Ω C ( Ω, Λ 1 R 3 ). Relying on the boundedness of the oerator (2.21), it follows that u ε u in L s(ω, Λ 1 R 3 ). Now, if W L s(r 3, Λ 1 R 3 ) is such that W Ω = curl u, su W Ω, then curl U = W + Σ, where Σ is a distribution suorted on Ω. Thus, by (2.48), su (Σ ϕ ε ) Γ + ( Ω su u) R 3 \ Ω. (2.49) Consequently, curl u ε = (ϕ ε curl U) Ω = (ϕ ε W ) Ω curl u in L s(ω, Λ 1 R 3 ), once again thanks to the boundedness of (2.21) for the range of indices we are considering here. This, in concert with (2.19), justifies (2.47) and finishes the roof of (ii). As for (iii), it is clear that the ma (2.44) is well-defined, linear, bounded and one-to-one. There remains to shows that it is also onto. With this goal in mind, let θ (TH s( Ω)) be arbitrary and consider ˆθ : H s, (Ω; curl) R defined by ˆθ(u) := θ(ν u). Via the identification 14

15 H s, (Ω; curl) u (u, curl u) L s(ω, Λ 1 T M) L s(ω, Λ 1 T M), (2.50) we can regard H s, (Ω; curl) as a (closed) subsace of L s(ω, Λ 1 T M) L s(ω, Λ 1 T M). In this scenario, the Hahn-Banach theorem and (2.50) allow us to conclude that there exist v 1, v 2 L s(ω, Λ 1 T M) such that ˆθ(u) = v 1, curl u v 2, u, u H s, (Ω; curl). (2.51) Note that choosing u Co (Ω, Λ 1 T M) yields curl v 1 = v 2. In articular v := v 1 H s, (Ω; curl). Utilizing this back in (2.51) gives that θ(ν u) = ν (ν v 1 ), ν u, for each u which, in turn, entails ν (ν v 1 ) = θ. Hence, the ma (2.44) is onto and this finishes the roof of (iii). Finally, (iv) is imlicit in what we have roved so far. An imortant issue whose discussion is ostoned at the moment is whether the sace TH s ( Ω) ultimately deends only on Ω (and not on Ω itself, as (2.39) may initially suggest). This asect will be addressed in The surface divergence and related oerators In this subsection, we discuss the surface divergence oerator. First, at the L -level with 1 < <, we set and introduce by requiring L tan ( Ω) := {f L ( Ω, Λ 1 T M); ν, f = 0 a.e. on Ω}, (2.52) Ω Div : L tan ( Ω) L 1 ( Ω), (2.53) g (Div f) dσ = f, tan g dσ, (2.54) Ω for each f L tan ( Ω), and g L 1 ( Ω) = (L 1 ( Ω)), 1/ + 1/ = 1. A sace which is going to be imortant for us in the sequel is 1 < <, which we equi with the natural norm A closed subsace of (2.55) is L,Div tan ( Ω) := {f L tan ( Ω); Div f L ( Ω)}, (2.55) f L,Div tan ( Ω) := f L ( Ω,Λ 1 T M) + Div f L ( Ω). (2.56) L,o tan ( Ω) := {f L,Div( Ω); Div f = 0}. (2.57) tan A convenient way to extend the definition of the surface divergence in context of (2.39) is to let act according to Div : TH s( Ω) B, ( Ω), 1 < <, 1 + 1/ < s < 1/, (2.58) s 1 15

16 Div f := ν (curl u), (2.59) if f TH s( Ω) is of the form f = ν u, for some u H s, (Ω; curl). From definitions, it is immediate that the oerator Div above is well-defined, linear and bounded. Moreover, Div defined in (2.53)-(2.54) is comatible with Div defined in (2.58)-(2.59). Another oerator of interest for us is defined by ν tan : B, 1 1 +s( Ω) TH s( Ω), 1 < <, 1 + 1/ < s < 1/, (2.60) (ν tan )f := ν ( u), (2.61) if f B, 1 1 +s( Ω), and if u L s+1 (Ω) is so that f = Tr u on Ω. Again, one can see that ν tan in (2.61) is well-defined, linear and bounded. Also, from (2.54), we have the factorization Div = (ν tan ) (ν ). The kernel of the oerator (2.58)-(2.59) is the sace { } TH,o s ( Ω) := f TH s( Ω); Div f = 0. (2.62) Clearly, since curl = 0, we get from (2.59) and (2.61) that and, further, Div (ν tan ) = 0 (2.63) ν tan ( B, 1 1 +s( Ω) ) TH,o s ( Ω). (2.64) However, a deeer issue is that of comuting the index of the inclusion in (2.64); we lan to address this, along with other related roblems, in the next section. 3 Fredholm first order differential oerators The main aim of this section is to identify certain convenient functional analytic settings in which some of the (first order, differential) oerators introduced in 2 become Fredholm. We start by reviewing some concets and results from algebraic toology. 3.1 Singular homology and sheaf theory For a toological sace X, we set Hsing l (X ; R) for the l-th singular homology grou of X over the reals, l = 0, 1,... (cf., e.g., [38]). Then b l (X ), the l-th Betti number of X, is defined as the dimension of Hsing l (X ; R). As is well known, b l(x ), l = 0, 1,..., are toological invariants of X. In fact, b 0 (X ) is simly the number of connected comonents of X. The most imortant case for us is when X is a Lischitz subdomain Ω of the (three dimensional) manifold M, or its boundary. Then b l (Ω), b l ( Ω) are all finite, nonnegative integers and, according to Poincaré s duality theorem, b 0 ( Ω) = b 2 ( Ω). Next, we include a brief synosis of some basic terminology together with some fundamental results from sheaf theory. Recall that a sheaf F on a toological sace X is a double collection {F(U), ρ U V } V U X, indexed by oen subsets of X, such that: 16

17 1. For each U oen subset of X, F(U) is a vector sace (over the reals) whose elements are called sections of F over U; 2. For each air V U of oen subsets of X, we have that ρ U V : F(U) F(V ) is a vector sace homomorhism, called the restriction ma, subject to the following two axioms. Firstly, ρ U U is the identity homomorhism of F(U), for any oen set U. Secondly, for any trilet W V U of oen sets in X, ρ U W = ρ U V ρ V W. (3.1) In order to lighten notation, for each ω F(U) and any V U oen, we may write ω V lace of ρ U V (ω). By virtue of (3.1), this is without loss of information. in 3. For each U, oen subset of X, any oen covering {U i } i I of U, and any family {ω i } i I, ω i F(U i ), satisfying the comatibility condition ω i Ui U j = ω j Ui U j, for any i, j I (3.2) there exists a unique section ω F(U) such that ω Ui = ω i for any i I. Given two sheaves F, G over X, a sheaf homomorhism ϑ : F G is a collection of vector sace homomorhisms {ϑ(u) : F(U) G(U)} U X, indexed by oen subsets of X, which commute (in a natural way) with the restriction maings. We define su (ϑ) as the smallest closed set outside of which ϑ is the null sheaf homeomorhism. A sheaf F over X is said to be fine if for each oen, locally finite cover {U i } i I of X there exists a family of sheaf homomorhisms ϑ i : F F, i I, such that su (ϑ i ) U i, i I, ϑ i = identity F. (3.3) i Next, assume that F 0, F 1,... are sheaves over the toological sace X and that, for l = 0, 1,..., the maings ϑ l : F l F l+1 are sheaf homomorhisms. Then 0 F 0 ϑ 0 F 1 ϑ 1 F 2 ϑ 2 (3.4) is called an exact comlex rovided the following two conditions are true: 1. (the comlex condition) ϑ l+1 ϑ l = 0 for l = 0, 1,...; 2. (the exactness condition) for each fixed index l = 1, 2,..., each oint x 0 X, each oen neighborhood U of x 0 and any section ω F l (U) such that ϑ l (U)(ω) = 0, there exist V U, oen neighborhood of x o and a section ω F l 1 (V ) for which ϑ l 1 (V )(ω ) = ω V. To each sheaf F over a toological sace X one can associate the so called cohomology grous H l (X ; F), l = 0, 1,... For a recise definition as well as for more extensive discussion we refer the reader to, e.g., [66], [25]. In this resect, two results are going to be of basic imortance for us in the sequel. To state the first one, for each oen set O X consider R O := {f : O R; locally constant function}, (3.5) and introduce the sheaf of locally constant functions on X 17

18 } LCF X := {R O O oen in. (3.6) X Then, for any reasonable toological sace X, the l-th cohomology grou of LCF X is isomorhic to the Hsing l (X ; R), the classical l-th singular homology grou of X over the reals. The other result referred to above is the dee theorem of de Rham which we resent below in an abstract form, well suited for our uroses. Theorem 3.1 (The Abstract de Rham Theorem) Let X be a Hausdorff, ara-comact toological sace, and let F be a sheaf over X. Also, let L 0, L 1,.. be fine sheaves over X and, for l = 0, 1,..., let ϑ l : L l L l+1 be sheaf homomorhisms such that the following is an exact comlex: (hereafter, ι denotes inclusion). Then 0 F ι L 0 ϑ 0 L 1 ϑ 1 L 2 ϑ 2 (3.7) H l (X ; F) = Ker (ϑ l : L l (X ) L l+1 (X )) Im (ϑ l 1 : L l 1 (X ) L l, l = 1, 2,... (3.8) (X )) See [66], Theorem 5.25,. 185 for a roof; cf. also [22]. 3.2 The calculation of the index: boundary oerators Here we rove that, when considered between aroriate saces, Div and ν tan are Fredholm oerators with indices deending exclusively on the toology of Ω (and its boundary). We debut with the following theorem. Theorem 3.2 Assume that Ω is an arbitrary Lischitz domain in M and suose that 1 < <, 1 + 1/ < s < 1/. Then Div : TH s( Ω) ( Ω) TH,o s B, ( Ω), (3.9) s 1 ν tan : B, ( Ω) TH,o 1+s 1 s ( Ω), (3.10) are Fredholm oerators with indices b 0 ( Ω) and b 0 ( Ω) b 1 ( Ω), resectively. Proof. First, we localize the definition of Div. More secifically, if 1 < <, 1 + 1/ < s < 1/, for U an arbitrary, fixed oen subset of Ω, we define TH s(u) as the subsace of functions f locally belonging to Bs, (U, Λ 1 T M) and which enjoy the following roerty: for each x U there exists u H s, (Ω; curl) such that f and ν u coincide in some oen neighborhood of x in U. Clearly, each TH s(u) is an additive Abelian grou and also a module over Li( Ω), the algebra of Lischitz functions on Ω. It follows that the family TH s := (TH s(u)) U, indexed by oen subsets in Ω, is a sheaf on the toological sace Ω. Given its local character and the existence of a (smooth) artition of unity, this sheaf is fine. We shall also work with the fine sheaf Bs, := (Bs, (U)) U, again indexed by oen subsets U of Ω, and where 1 < <, 0 < s < 1. This time, the grou Bs, (U) is the collection of distributions f (Li( Ω)) such that fϕ Bs, ( Ω) for each ϕ Li(M) with Ω su ϕ U. Thus, if 1 + 1/ < s < 1/, 1 < <, we can then define 18

19 Div : TH s(u) B, s 1/ (U) (3.11) by setting Div f := ν curl u near x U, if f TH s(u) is locally given by ν u near x, for some u H s, (Ω; curl). In the same context, we can also introduce ν tan : B, 1+s 1/ (U) TH s(u) (3.12) by asking that (ν tan )g := ν ( w) near x U, if g B, 1+s 1/ (U) is locally given by Tr w near x, for some w L s+1 (Ω). Going further, observe that we have a natural sequence of sheaf morhisms ι 0 LCF Ω B, ν tan 1+s 1/ TH s Div B, s 1/ 0. (3.13) Here LCF Ω stands for the sheaf of germs of locally constant functions on Ω, and ι is the natural inclusion oerator. Since Div (ν tan ) = 0, the above is a comlex. In fact, so we claim, (3.13) rovides a fine resolution of the sheaf LCF Ω. The essential ingredient in the roof of this claim is the acyclicity of the comlex (3.13). Granted this, de Rham s theorem alies to our context and gives that TH,o s ( Ω) ( ) = H 1 ν tan B, 1+s 1/ ( Ω) sing ( Ω; R), B, s 1/ ( Ω) Div(TH s( Ω)) = H 2 sing( Ω; R), (3.14) for each 1 < < and 1 + 1/ < s < 1/. With this at hand, we may conclude that the oerators (3.9)-(3.10) have finite dimensional kernels and cokernels. To show that they also have closed ranges, we rely on a general functional analytic result (erhas folklore), to the effect that if T : Dom (T ) X Y is a closed oerator between two Banach saces, with the roerty that Im T, the image of T, has finite codimension in Y, then Im T is a closed subsace of Y. Thus, we are left with roving the acyclicity of the sheaf (3.13). It is not hard to see that this is equivalent to the following two claims: and x o Ω and for any f B, s 1/ ( Ω) = u H s, (Ω; curl) with ν curl u = f near x o, (3.15) x o Ω and u H s, (Ω; curl) with ν curl u = 0 near x o, = ϕ L 1+s (Ω) so that ν u = ν ( ϕ) near x o. (3.16) Consider first (3.15). Since the statement is local, there is no loss of generality assuming that Ω f dσ = 0. In this context, Proosition 2.1 works and gives some w Hs, (Ω; div) such that div w = 0 in Ω and f = ν w on Ω. Let now D Ω be a sufficiently small Lischitz subdomain so that D Ω is an oen neighborhood of x o in Ω. Furthermore, we can assume that the Poincaré tye results roved in the Aendix are valid on D. This lus an alication of the Hodge star isomorhism, and in concert with (2.35), then give that there exists u L s+1 (D, Λ1 T M) H s, (D; curl) such that curl u = w D. In articular, ν curl u = ν w = f near x o, which justifies (3.15). As for (3.16), assume that u H s, (Ω; curl) satisfies ν curl u = 0 near x o Ω. Fix now a suitably small Lischitz domain D in M such that x o D and, in fact, ν curl u = 0 on D Ω. 19

20 Lemma 2.3 and (2.18) ensure that ( curl u) D H s, (D; div) and div ( curl u) D = 0. Going further, the Poincaré tye results with reservation of smoothness and suort roved in the Aendix (together with an alication of the Hodge star oerator) give, much as before, that there exists ω L s+1 (D, Λ1 T M) with su ω D Ω such that curl u = curl ω in D. Thus, Tr ω = 0 on D Ω so that, further, ν ω = 0 on D Ω. (3.17) Observing that curl (u ω) = 0 in D Ω and, once again aealing to the results roved in the Aendix, yields that there exists ϕ L s+1 (D Ω) with u ω = ϕ in D Ω. Hence, by invoking (3.17), we see that ν u = ν ( ϕ) near x o, which roves the claim (3.16). Theorem 3.3 Assume that Ω is an arbitrary Lischitz domain in M. Then, for each 1 < <, the oerators Div : L,Div tan ( Ω) L,o tan ( Ω) L ( Ω), ν tan : L 1 ( Ω) L,o tan ( Ω) (3.18) are Fredholm with indices b 0 ( Ω) and b 0 ( Ω) b 1 ( Ω), resectively. Also, for 1 < <, the oerator is Fredholm with index b 0 ( Ω). Div : L tan ( Ω) L,o tan ( Ω) L 1 ( Ω) (3.19) Proof. Once again, the treatment of the oerators (3.18) relies on a convenient alication of de Rham theory (cf. Theorem 3.1). The arguments closely arallel those in the roof of Theorem 3.2 and are somewhat simler since, this time, the necessary Poincaré lemma is at the level of L saces; cf. Aendix. We omit the details of this ste. Consider next the oerator in (3.19). Clearly, this is one-to-one. To comute its range, observe that it suffices to consider the situation when the original domain is relaced by the sace L tan ( Ω)/ν tan(l 1 ( Ω)). Assuming that this is the case, we claim that Div in (3.19) factorizes as Div : L tan ( Ω) ν tan (L 1 ( Ω)) Φ (L q,o tan ( Ω)) ( (ν tan) L q 1 ( Ω) R Ω ), (3.20) where 1/ + 1/q = 1, and Φ is the oerator defined by Φ([f]), g := ν f, g dσ, (3.21) Ω for any [f] L tan ( Ω) ν tan (L 1 ( Ω)), g Lq,o tan ( Ω). (3.22) Note that Φ is well-defined since, by (2.54), the right side of (3.21) always vanishes when one airs elements of the form f = ν tan h, for h L 1 ( Ω), with g Lq,o tan ( Ω). To justify the factorization (3.20), for f L tan ( Ω) and g Lq 1 ( Ω) we write 20

21 Div([f]), [g] = f, tan g dσ = Φ([f]), ν tan g) Ω = (ν tan ) Φ([f]), [g], (3.23) i.e., Div = (ν tan ) Φ, and (3.20) is roved. Next, we claim that Φ is surjective. To see this, fix an arbitrary functional ξ (L q,o tan ( Ω)). By Hahn-Banach s extension theorem and Riesz s reresentation theorem there exists f L tan ( Ω) which satisfies ξ(h) = Ω f, h dσ for any h Lq,o tan ( Ω). Thus, Φ sends [ ν f] into ξ and, hence, Φ is onto. We now examine the second arrow in (3.20). Recall the definition (3.6). Thanks to what we know already from the first art of the current theorem, the oerator ν tan : Lq 1 ( Ω) R Ω L q,o tan ( Ω) (3.24) is one-to-one and Fredholm. Thus, its adjoint is onto. Combining all these facts, we see that the image of Div in (3.20) is (L q 1 ( Ω)/R Ω), i.e. the sace {f L 1 ( Ω); f, χ = 0, χ R Ω}. The desired conclusion follows. 3.3 An intrinsic descrition of the tangential Sobolev saces The aim of this subsection is to rovide an alternative descrition of the sace (2.39) which is intrinsic, in the sense that it only deends on, s and Ω (and not on Ω itself). The starting oint is the following Fredholmness result. Proosition 3.4 Let Ω M be an arbitrary Lischitz domain, and assume that 1 < <, 1 + 1/ < s < 1/. Then the oerator ν tan : B, s 1/+1 ( Ω) B, s 1/ ( Ω, Λ1 T M) (3.25) has closed range and finite dimensional kernel; in articular, it is semi-fredholm. Proof. We shall show that (3.25) is bounded from below, modulo comact oerators. Given that we are dealing with a first-order differential oerator (so that the commutator between this and multilication by a smooth cut-off function is a comact maing in the current context), makes this articular roblem local in nature so there is no loss of generality in assuming that Ω has trivial toology. In this latter context, the roof of (3.15) shows that if g (B, s 1/+1 ( Ω)) = B, s 1/ ( Ω), 1/ + 1/ = 1, is such that g, 1 = 0, then there exists u L 1 s (Ω, Λ1 T M) so that ν curl u = g and u L C g 1 s (Ω,Λ1 T M) B, s 1/ ( Ω). Thus, if f B, s 1/+1 ( Ω) is arbitrary and φ L s+1 (Ω) is so that Tr φ = f, we may write Consequently, for each g as above, f, g = f, ν curl u = φ, curl u = (ν tan )f, Tr u. (3.26) f, g C (ν tan )f B, s 1/ ( Ω,Λ1 T M) g (B, ( Ω)), (3.27) s 1/+1 which, since g is arbitrary, further entails 21