Automorphic Forms and Dirac Operators on Conformally Flat Manifolds
aa r X i v : . [ m a t h . C V ] A p r Automorphic Forms and Dirac Operators onConformally Flat Manifolds
R.S. Kraußhar ∗ April 13, 2018
Abstract
In this paper we present a summarizing description of the connectionbetween Dirac operators on conformally flat manifolds and automorphicforms based on a series of joint work with John Ryan over the last fifteenyears. We also outline applications to boundary value problems.
Keywords : Dirac operators, automorphic forms, conformally flat manifolds
MSC Classification : 30G35; 11F55.To Professor John Ryan for his 60th birthday for the long collaboration andfriendship
A natural generalization to R n of the classical Cauchy-Riemann operator hasproved to be the Euclidean Dirac operator D . Here R n is considered as em-bedded in the real 2 n -dimensional Clifford algebra Cl n satisfying the relation x = −k x k for each x ∈ R n . The elements e , . . . , e n of the standard orthonor-mal basis of R n satisfy the relation e i e j + e j e i = − δ ij . The Dirac operator isdefined to be P nj =1 e j ∂∂x j . Clifford algebra valued functions f and g that satisfy Df = 0 respectively gD = 0 are often called left (right) monogenic functions.Its associated function theory together with its applications is known asClifford analysis and can be regarded as a higher dimensional generalizationof complex function theory in the sense of the Riemann approach. Indeed,associated to this operator there is a higher dimensional direct analogue ofCauchy’s integral formula and other nice analogues, cf. [8]. As in complexanalysis, also the Euclidean Dirac factorizes the higher dimensional EuclideanLaplacian viz D = − ∆. Indeed, the Euclidean Dirac operator has been used ∗ Lehrgebiet f¨ur Mathematik und ihre Didaktik, Erziehungswissenschaftliche Fakult¨at,Universt¨at Erfurt, Nordh¨auser Str. 63, D-99089 Erfurt, Germany. E-mail: [email protected] R n . See for instance [14, 27].On the other hand Dirac operators have proved to be extremely usefultools in understanding geometry over spin and pin manifolds. Basic aspects ofClifford analysis over spin manifolds have been developed in [2, 3]. Further in[18, 20, 19, 16] and elsewhere it is illustrated that the context of conformallyflat manifolds provide a useful setting for developing Clifford analysis.Conformally flat manifolds are those manifolds which possess an atlas whosetransition functions are M¨obius transformations. Under this viewpoint confor-mally flat manifolds can be regarded as higher dimensional generalizations ofRiemann surfaces.Following the classical work of N. H. Kuiper [22], one can construct examplesof conformally flat manifolds by factoring out a subdomain U ⊆ R n by a torsion-free Kleinian group Γ acting totally discontinuously on U .Examples of conformally flat manifolds include spheres, hyperbolas, realprojective space, cylinders, tori, the M¨obius strip, the Kleinian bottle and theHopf manifolds S × S n − . The oriented manifolds among them are also spinmanifolds. In [18, 20, 16] explicit Clifford analysis techniques, including Cauchyand Green type integral formulas, have been developed for these manifolds.Finally, in one of our follow-up papers [1] we also looked at a class of hy-perbolic manifolds namely those that arise from factoring out upper half-spacein R n by a torsion-free congruence subgroup, H , of the generalized modulargroup Γ p . Γ p is the arithmetic group that is generated by p translation matrices( p < n ) and the inversion matrix. In two real variables these are k -handledspheres. Notice that the group Γ p is not torsion-free, as it contains the negativeidentity matrix. Consequently, the topological quotient of upper half-space withΓ p has only the structure of an orbifold. To overcome this problem we deal withcongruence subgroups of level N ≥
2, which are going to be introduced lateron. In this paper we present an overview about some of our most importantjoint results. In the final section of this paper we also outline some applica-tions addressing boundary value problems modelling stationary flow problemson these classes of manifolds where we adapt the techniques from [14] to thismore general geometric context.
As mentioned in the introduction, we embed the R n into the real Clifford algebra Cl n generated by the relation x = −k x k . For details, see [3, 8, 14]. This rela-tion defines the multiplication rules e i = − i = 1 , ..., n and e i e j = − e j e i ∀ i = j . A vector space basis for Cl n is given by 1 , e , . . . , e n , e e , . . . , e n − e n , . . . ,e · · · e n . Each x ∈ R n \{ } has an inverse of the form x − = − x k x k . We also2onsider the reversion anti-automorphism defined by ˜ ab = ˜ b ˜ a , where ˜ e j = e j ∀ j = 1 , ..., n and the conjugation defined by ab = b a , where e j = − e j ∀ j = 1 , ..., n .Notice that e xe = − x e + x e + · · · + x n e n . The multiplication of e from the left and from the right realizes in a simple form a reflection inthe e -direction. More generally, one can say: If O ∈ O ( n ), then there arereflections R , . . . , R m such that O = R · · · R m . In turn for each R j thereexists a y j ∈ S n − such that R j x = y j xy j for all x ∈ R n . Summarizing, one canrepresent a general transformation of O ( n ) in the way Ox = y · · · y m xy m · · · y ,so Ox = ax ˜ a with a = y · · · y m .This motivates the definition of the pin group as P in ( n + 1) = { a ∈ Cl n | a = y · · · y m , y i ∈ S n } . Each transformation of O ( n ) can be written as Ox = ax ˜ a with an a ∈ P in ( n ). In view of ax ˜ a = ( − a ) x ( − ˜ a ), P in ( n ) is a double cover of O ( n ). Asubgroup of index 2 is the spin group defined by Spin ( n ) := { a = y · · · y m ∈ P in ( n ) | m ≡ } . Again,
Spin ( n ) is the double cover of SO ( n ). Here we summarize some basic results from [2, 9, 22].Let M be a connected orientable Riemannian manifold with Riemann metric g ij .Consider for x ∈ M all orthonormal-bases of the tangential space T M x ,which again are mapped to orthonormal-bases of T M x by the action of the SO ( n ). This gives locally rise to a fiber bundle.Gluing together all these fiber bundles gives rise to a principal bundle P over M with a copy of SO ( n ).This naturally motivates the question whether it is possible to lift each fiberto Spin ( n ) in a continuous way to obtain a new principal bundle S that covers P . However, the ambiguity caused by the sign may give a problem. If s : U → U × SO ( n ) is a section then there are two options of lifting s to a spinor bundle s ∗ : U → U × Spin ( n ), namely s ∗ and − s ∗ . So, it may happen that: • It is not always possible to choose the sign in order to construct in a uniqueway a bundle S over M , such that each fiber is a copy of Spin ( n ). • There also might be several possibilities. • The different spin structures are described by the cohomology group H ( M, Z ). 3 .3 The Atiyah-Singer-Dirac operator Let M be a Riemannian spin manifold. Let Γ be the Levi-Civita connec-tion. Then Γ g ij = 0. Stokes’s theorem tells us that Z ∂V h s ( x ) , n ( x ) s ( x ) i S dσ ( x )= Z V ( h s ( x ) D, s ( x ) i S + h s ( x ) , Ds ( x ) i S ) dV The arising differential operator here is the Atiyah-Singer-Dirac operator.In a local orthonormal basis e ( x ) , ...e n ( x ) the latter has the form D = n X j =1 e j ( x )Γ ∗ e j ( x ) . R n and R ⊕ R n Following [8] and others, the Dirac operator in R n has the simple form D = n P j =1 ∂∂x j e j . In the so-called space of paravectors R ⊕ R n it particularly has theform D = ∂∂x + n P j =1 ∂∂x j e j . The latter naturally generalizes the well-knownCauchy-Riemann operator ∂∂x + ∂∂x i in a straight forward way to higher di-mensions by adding the additional basis elements.As mentioned in the introduction, functions f : U → Cl n (where U ⊆ R n resp. U ⊆ R ⊕ R n ) that are in the kernel of D are often called monogenic. R n +1 Following for example [22], conformally flat manifolds are Riemannian mani-folds with vanishing Weyl tensor. In turn these are exactly those Riemannianmanifolds which have an atlas whose transition functions are conformal maps.In R conformal maps are exactly the (anti-)holomorphic functions. So, onedeals with classical Riemann surfaces. Up from space dimension n ≥ R ≥ however, the set of conformal maps coincides with the set of M¨obius transfor-mations, cf. [3] The latter set of functions then represent reflections at spheresand hyperplanes. This seems to be quite restrictive at the first glance. However,this is not the case as the abundance of classical important examples will showas outlined in the following.So, let us now turn to an explicit construction principle of conformally flatmanifolds in R n with n ≥
3. To proceed in this direction take a torsion free4iscrete subgroup G of the orthogonal group O ( n + 1) that acts totally dis-continuously on a simply connected domain D . Next define a group action G × D → D . Then the topological quotient space D / G is a conformally flatmanifold. As also shown in the original work of N.H. Kuiper in 1949 [22], theuniversal cover of a conformally flat manifold possesses a local diffeomorphismto S n . Conformally flat manifolds of the form D / G are exactly those for whichthis local diffeomorphism is a covering map D → D ⊂ S n .Let us present a few elementary examples: • Take G = T p := Z + Z e + · · · + Z e p − , D = R n +1 and consider the action( m , . . . , m n ) ◦ ( x , . . . , x n ) ( x + m , . . . , x p − + m p − , x p , . . . , x n ) . Then the topological quotients D / G represent oriented p cylinders. Theseare spin manifolds with 2 p many different spinor bundles. In the particularcase p = n + 1 one deals with a compact oriented n + 1-torus, cf. [18, 20]. • Take again as group T n +1 and the same domain, but consider a differentaction of the form( m , . . . , m n − , m n ) ◦ ( x , . . . , x n − , x n ) ( x + m , . . . , x n − + m n − , ( − m n x n + m n ) . Now D /G is the non-orientable Kleinian bottle in R n +1 . Due to the lack oforientability it is not spin. However, it is a pin manifold with 2 n +1 manypinor bundles. For some Clifford analysis on these manifolds we refer thereader to our recent works [16, 17].Further interesting examples can be constructed by forming quotients withnon-abelian subgroups of M¨obius transformations in R n , in particular with arith-metic subgroups of the so-called Ahlfors-Vahlen group discussed for instance in[10] and many other papers. To make the paper self-contained we recall itsdefinition: Definition 1. (Ahlfors-Vahlen group)A Clifford algebra valued matrix M = (cid:18) a bc d (cid:19) ∈ M at (2 , Cl n ) belongs to thespecial Ahlfors Vahlen group SAV ( n ) , if:- a, b, c, d are products of paravectors from R ⊕ R n - a ˜ c, c ˜ d, d ˜ b, b ˜ c ∈ R ⊕ R n and a ˜ d − b ˜ c = 1 . The use of Clifford algebras allow us to describe M¨obius transformations in R ⊕ R n in the simple way M < x > = ( ax + b )( cx + d ) − , similarly to complexanalysis. For our needs we consider the special hypercomplex modular group[15] defined byΓ p := * (cid:18) (cid:19) , (cid:18) e (cid:19) , . . . , (cid:18) e p (cid:19)| {z } =: T p , (cid:18) −
11 0 (cid:19)| {z } =: J + p < n , then by applying the same arguments as in [12], Γ p acts totallydiscontinuously on upper half-space discussed in [10] H + ( R ⊕ R n ) := { x + x e + · · · + x n e n ∈ R ⊕ R n | x n > } . In two dimensions it coincides with the classical group SL (2 , Z ). To get a largeramount of examples we look at the following arithmetic congruence subgroups:Γ p [ N ] := ( (cid:18) a bc d (cid:19) ∈ Γ p | a − , b, c, d − ∈ N O p ) , where O p := P A ⊆ P ( { ,...,p } ) Z e A are the standard orders in Cl n .If we now take D = H + ( R n +1 ), G = Γ p [ N ] with N ≥ M, x ) M < x > := ( ax + b ) · ( cx + d ) − , where · is the Clifford-multiplication, then for N ≥ D / G realizes a class of conformally flat orientable manifold with spin structuresgeneralizing k -tori to higher dimensions, cf. [1]. In the sequel let us make the general assumption that M := D / G is an orientableconformally flat manifold. Let furthermore f : D → R n +1 be a function with f ( G < x > )) = j ( G, x ) f ( x ) ∀ G ∈ G where j ( G, x ) is an automorphic factorsatisfying a certain co-cycle relation. In the simplest case j ( G, x ) ≡ G . Then thecanonical projection p : D → M induces a well-defined function f ′ := p ( f ) onthe quotient manifold M . Now let D := ∂∂x + n P i =1 ∂∂x i e i be the Euclidean Dirac-or Cauchy-Riemann operator and let ∆ be the usual Euclidean Laplacian. Thecanonical projection p : D → M in turn induces a Dirac operator D ′ = p ( D )resp. Laplace operator ∆ ′ = p ( D ) on M . Consequence : A G -invariant function on D from Ker D (resp. from Ker∆) induces functions on M in Ker D ′ resp. in Ker ∆ ′ . More generally, oneconsiders for f ′ monogenic resp. harmonic spinor sections with values in certainspinor bundles. Following classical literature on automorphic forms, for instance [12], let G bea discrete group that acts totally discontinuously on a domain D by G × D →D , ( g, d ) → d ∗ , Roughly speaking, automorphic forms on G are functions on6 that are quasi-invariant under the action of G . The fundamental theory ofholomorphic automorphic forms in one complex variable was founded around1890, mainly by H. Poincar´e, F. Klein and R. Fricke. The associated quotientmanifolds are holomorphic Riemann surfaces.The simplest examples are holomorphic periodic functions. They serve asexample of automorphic functions on discrete translation groups. Further veryclassical examples are the so-called Eisenstein series G n ( τ ) := X ( c,d ) ∈ Z \{ (0 , } ( cτ + d ) − n n ≡
0( mod 2) n ≥ H + ( C ) := { z = x + iy ∈ C | y > } . For all z ∈ H + ( C ) they satisfy : f ( z ) = ( f | n M )( z ) ∀ M = (cid:18) a bc d (cid:19) ∈ SL (2 , Z )where ( f | n M )( z ) := ( cz + d ) − n f (cid:16) az + bcz + d (cid:17) .The Eisenstein series G n are the simplest non-vanishing holomorphic auto-morphic forms on SL (2 , Z ). Further important examples are Poincar´e series:For n > , n ∈ N let P n ( z, w ) = X M ∈ SL (2 , Z ) ( cz + d ) − n ( w + M < z > ) − n These functions have the same transformation behavior as the previously de-scribed Eisenstein series, namely: P n ( w, z ) = P n ( z, w ) = ( cz + d ) − n P n ( M < z >, w ) . In contrast to the Eisenstein series, these Poincar´e series have the special prop-erty that they vanish at the singular points of the quotient manifold resp. orb-ifold. n real variables Already in the 1930s C.L. Siegel developed higher dimensional analogues ofthe classical automorphic forms in the framework of holomorphic functions inseveral complex variables. The context is Siegel upper half-space where oneconsiders the action of discrete subgroups of the symplectic group.More closely related to our intention is the line of generalization initiated byH. Maaß and extended by J. Elstrodt, F. Grunewald, J. Mennicke [10] and A.Krieg [21] and many followers in the period of 1985-1990 and onwards.These authors looked at higher dimensional generalizations of Maaß forms which7re non-holomorphic automorphic forms on discrete subgroups of the Ahlfors-Vahlen group (including for example the Picard group and the Hurwitz quater-nions) on upper half-space H + ( R n ). As important analytic properties they ex-hibit to be complex-valued eigensolutions to the scalar-valued Laplace-Beltramioperator ∆ LB = x n (cid:16) n X i =0 ∂ ∂x i (cid:17) − ( n − x n ∂∂x n In the case n = 1 one has ∆ LB = x ∆. Therefore, in the one-dimensional caseholomorphic automorphic forms simply represent a special case of Maaß forms.A crucial disadvantage of Maaß forms behind the background of our par-ticular intentions consists in the fact that they do not lie in the kernel of theEuclidean Dirac or Laplace operator. Additionally, they are only scalar-valued. Some milestones in the literature
To create a theory of Clifford algebra valued automorphic forms that are inkernels of Dirac operators remained a challenge for a long time. A serious ob-stacle has been to overcome the problem that neither the multiplication northe composition of monogenic functions are monogenic again. However, the setof monogenic functions is quasi-invariant under the action of M¨obius transfor-mations up to an automorphic factor that fortunately obeys a certain co-cylerelation. The latter actually provides the fundament to build up a theory ofautomorphic forms in the Clifford analysis setting.The first contribution in the Clifford analysis setting is the paper [25] whereJ. Ryan constructed n -dimensional monogenic analogues of the Weierstraß ℘ -function and the Weierstraß ζ -function built as summations of the monogenicCauchy kernel over an n -dimensional lattice. Here, the invariance group is anabelian translation group with n linearly independent generators.In the period of 1998-2004 the author developed the fundamentals for amore general theory of Clifford holomorphic automorphic forms on more generalarithmetic subgroups of the Ahlfors-Vahlen group, cf. [15]. The geometriccontext is again upper half-space. However, the functions are in general Cliffordalgebra valued and are null-solutions to iterated Dirac equations. In fact, theframework of null-solutions to the classical first order Dirac opertor is too smallfor a large theory of automorphic forms, because the Dirac operator admits onlythe construction of automorphic forms to one weight factor only. To considerautomorphic forms with several automorphic weight factors a more approriateframework is the context of iterated Dirac equations of the form D m f = 0 tohigher order integer powers m . In fact, higher order Dirac equations can also berelated to k -hypermonogenic functions [24] which allows us to relate the theoryof Maaß forms to Clifford holomorphic automorphic forms. This is successfullyexplained in [5, 7]. In this context it was possible to generalize Eisenstein-8nd Poincar´e series construction which gave rise to the construction of spinorsections with values in certain spinor bundles on the related quotient manifolds.In fact, as shown in [7], it is possible to decompose the Clifford module ofClifford holomorphic automorphic forms as an orthogonal direct sum of Cliffordholomorphic Eisenstein- and Poincar´e series. As shown in our recent paper [13],both modules turn out to be finitely generated.The applications to solve boundary value problems on related spin manifoldsstarted with our first joint paper [18] where we applied multiperiodic Eisensteinseries on translation groups to construct Cauchy and Green kernels on confo-mally flat cylinders and tori associated to the trivial spinor bundle.In [20] we extended our study to address the other spinor bundles on thesemanifolds. Furthermore, we also looked at dilation groups instead of translationgroups, too, and managed to give closed representation formulas for the Cauchykernel of the Hopf manifold S × S n − . Already in this paper we outlinedthe construction of spinor sections on some hyperbolic manifolds of genus ≥ k -hypermonogenic functions [11] and holomorphic Clifford functions ad-dressing null-solutions to D ∆ m f = 0. This is the function class considered byG. Laville and I. Ramadanoff introduced in [23].Summarizing, the theory of Clifford holomorphic automorphic form providesus with a toolbox to solve boundary value problems related to the EuclideanDirac or Laplace operator on conformally flat spin manifolds or more generallyon manifolds generalizing classical modular curves.It also turned out to be possible to make analogous constructions for somenon-orientable conformally flat manifolds with pin structures. Belonging to thatcontext, in [20] we addressed real projective spaces and in [16, 17] we carriedover our constructions to the framework of higher dimensional M¨obius stripsand the Klein bottle. • The simplest non-trivial examples of Clifford-holomorphic automorphicforms on the translation groups T p are given by the series ǫ ( p ) m ( x ) := X ω ∈ Ω p q m ( x + ω ) , Ω p := Z ω + Z ω + . . . Z ω p | m | ≥ p + 2. Here, q m ( x ) := ∂ | m | ∂x m q ( x )where q ( x ) := x k x k m +1 and where m := ( m , · · · , m n ) is a multi-index and | m | := m + m + · · · + m n and x m := x m · · · x m n n is used as in usualmult-index notation.These series ǫ ( p ) m ( x ) can be interpreted as Clifford holomorphic generaliza-tions of the trigonometric functions (with singularities) and of the Weier-straß ℘ -function. The projection p ( ǫ ( p ) m ( x )) then induces a well-definedspinor section on the cylinder resp. torus R n +1 / Ω p with values in thetrivial spinor bundle. Other spinor bundles can be constructed by intro-ducing proper minus signs and the following decomposition of the periodlattice in the way Ω p := Ω l ⊕ Ω p − l where 0 ≤ l ≤ p . The proper analoguesof ǫ ( p ) m ( x ) then are defined by ǫ ( p,l ) m ( x ) := X ω ∈ Ω l ⊕ Ω p − l ( − m + ··· + m l q m ( x + ω ) , Ω p := Z ω + Z ω + . . . Z ω p . In total one can construct 2 p +1 different spinor bundles. The Cauchykernel is obtained by the series that one obtains in the case m = .In that case it may happen that the associated series above does notconverge. To obtain convergence in those cases one needs to apply specialconvergence preserving terms. For the technical details we refer to ourpapers [18, 20, 19]. • Let us now turn to examples of hyperbolic manifolds that are generatedby quotient forming with non-abelian groups. The simplest non-trivialexample in that context are the monogenic Eisenstein series for Γ n − [ N ](and also for Γ p [ N ]) with p < n − E ( z ) = lim σ → + X M : T n − [ N ] \ Γ n − [ N ] x n k cx + d k ! σ q ( cx + d ) . Notice that we here applied the Hecke trick (cf. [12]) to get well-definedness.In fact, these Eisenstein series provide us with the simplest examples ofnon-vanishing spinor sections defined on the hyperbolic quotient manifolds H + ( R n +1 ) / Γ p [ N ]. However, these series do not vanish at the cusps of thegroup. They serve as examples but they don’t reproduce the Cauchy inte-gral. This property can be achieved by the following monogenic Poincar´eseries, introduced in our papers [5, 1]. The latter have the form P p ( x, w ) = lim σ → + X M ∈ Γ p [ N ] x n k cx + d k ! σ q ( cx + d ) q ( w + M h x i ) . The series P p ( x, w ) are indeed monogenic cusp forms, in particularlim x n →∞ P p ( x n e n ) = 0 . Some practical motivations for the following studies are to understand weatherforecast and flow problems on spheres, cylindrical ducts or other geometriccontexts fitting into the line of investigation of [28, 4].
Monogenic generalization of the Weierstraß ℘ -function ε ( p ) n give rise to mono-genic sections on p -cylinders R n +1 / T p . So, monogenic automorphic forms onΓ p ( I )[ N ] define spinor sections on the k -tori H + ( R n +1 ) / Γ p ( I )[ N ]. The Poincar´eseries give us the Cauchy kernel on these manifolds. Summarizing, for p < n − C ( x, y ) = X M ∈ Γ p ( I )[ N ] cx + d | cx + d | n ( y − M < x > ) k y − M < x > k n +1 . Each monogenic section f ′ on M then satisfies f ′ ( y ′ ) = Z ∂S ′ C ′ ( x ′ , y ′ ) dσ ′ ( x ′ ) f ′ ( x ′ )which is the reproduction of the Cauchy integral, cf. [1]. Let M := D /G be a conformally flat spin manifold for which we know theCauchy kernel C ( x, y ) to the Dirac operator (concerning a fixed chosen spinorbundle F ).Next let E ⊂ M be a domain with suffiently smooth boundary Γ := ∂E .Now we want to consider the following Stokes problem on M : − ∆ u + 1 η p = F in E (1)div u = 0 in Eu = 0 on Γ , where u ∈ W ( E, F ) is the velocity of the flow and where p ∈ W ( E, R ) is thehydrostatic pressure. The explicit knowledge of the Cauchy kernel on M allowsus to set up explicit analytic representation formulas for the solutions.11o meet this end we define in close analogy of [14] the Teodorescu transformand the Cauchy transform on M by( T E f )( x ) := − Z E C ( x, y ) f ( y ) dV ( y )( F Γ f )( x ) := Z Γ C ( x, y ) dσ ( y ) f ( y ) . where now C ( x, y ) stands for the Cauchy kernel associated to the chosen spinorbundle F on the manifold. The associated Bergman projection P : L ( E ) → L ( E ) ∩ Ker ( D ) then can be represented in the usual form P = F Γ ( tr Γ T E F Γ ) − tr Γ T E . An application of the Clifford analysis methods provide us with explicit ana-lytic formulas for the velocity and the pressure of the fluid running over thismanifold.An application of T E to (1) yields:( T E D )( Du ) + 1 η T E Dp = T E F. Next, an application of the Borel-Pompeiu formula (Cauchy-Green formula)leads to: Du − F Γ Du + 1 η p − η F Γ p = T E F If we apply the Pompeiu projection Q := I − P , then one obtains QDu − QF Γ Du + 1 η Qp − η QF Γ p = QT E F ( ∗ )Since F Γ Du, F Γ p ∈ Ker D , one further gets QF Γ Du = 0 and QF Γ p = 0 . Thus,
QDu + 1 η Qp = QT E F. (2)Further, another application of T E and the property QDu = Du leads to T E Du + 1 η T E Qp = T E QT E F ⇔ u − F Γ u |{z} =0 + 1 η T E Qp = T E QT E F. In view of u | Γ = 0 we obtain the following representation formula for the veloc-ity: u = T E ( I − P ) T E F − η T E ( I − P ) p. u = 0 allows us to determinate the pressure( ℜ ( I − P )) p = η ℜ (( I − P ) T E ( I − P ) F )where ℜ ( · ) stands for the scalar part of an element from the Clifford algebra. Remark . An extension of this approach to the parabolic setting in whichinstationary flow problems are considered are treated in our follow up paper [4].
The work of the author is supported by the project
Neue funktionentheoretischeMethoden f¨ur instation¨are PDE , funded by Programme for Cooperation in Sci-ence between Portugal and Germany, DAAD-PPP Deutschland-Portugal, Ref:57340281.
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