Source document: https://docs-lawep.s3.us-east-2.amazonaws.com/1697749581652.pdf (Georgia)
Bulletin of TICMI Vol.

18, No.

2, 2014, 18–35 Some Applications of the Theory of Self-Conjugate Dierential Forms Pietro Caramutaa and Alberto Cialdeaa aUniversity of Basilicata, V.le dell’Ateneo Lucano, 10, 85100, Potenza, Italy (Received February 3, 2014; Revised June 5, 2014; Accepted October 31, 2014) The concept of self-conjugate diﬀerential forms provides a generalization in Rn of the notion of holomorphic functions of one complex variable.
In this framework several extensions of classical results hold.

In this paper we present a survey of this theory and diﬀerent applica- tions concerning Laplace series, the Brother Riesz Theorem, Boundary Value Problems and Cimmino system.

Key words: Self-conjugate diﬀerential forms, Laplace series, Brother Riesz theorem, Boundary value problem, Cimmino system.

AMS Subject Classication: 35E20, 35F45, 58A10.

1.
1.

Introduction ∑ n This paper deals with non homogeneous diﬀerential forms U = k=1 uk, uk being a diﬀerential form of degree k (brieﬂy a k-form).

Following [6], we say that U is self-conjugate if dU = δU, where d and δ are the diﬀerential and codiﬀerential operator respectively.

The equation dU = δU provides a generalization in Rn of the Cauchy-Riemann system.
The concept of self-conjugate diﬀerential forms includes classical “real” generalizations of holomorphic functions of one complex variable, like solutions of Moisil-Teodorescu, Fueter, Cimmino systems, harmonic vectors and harmonic forms.

The aim of the present work is to show some applications of the theory of self- conjugate diﬀerential forms.

The paper is organized as follows.

Section 2 presents a review of the basic notions concerning k-forms and k- measures.
Section 3 provides an overall treatment of the theory of self-conjugate diﬀerential forms.

In Section 4, conditions for the existence of conjugate harmonic forms in multiply connected domains of Rn are given.

Generalizations of the Brother Riesz Theorem are presented ﬁrst for Laplace series in Section 5, considering the concept of conjugate Laplace series introduced in [4], and then for conjugate diﬀer- ential forms in Section 6.

Section 7 is devoted to some BVPs for non homogeneous diﬀerential forms.
In Section 8 necessary and suﬃcient conditions for the resolu- bility of the Dirichlet problem for the Cimmino system are given.

Corresponding author.

Email: cialdea@email.it ISSN: 1512-0082 print © 2014 Ivane Javakhishvili Tbilisi State University Press Vol.

18, No.
18, No.

2, 2014 19 2. k-forms and k-measures Let V m ⊂ Rn be an oriented m-dimensional diﬀerential simple manifold of class p. A dierential form of degree k, or brieﬂy a k-form, on V m is represented in an admissible coordinate system (t1, .

.

.

, tm), as u = 1 k!

us1;:::;skdts1 .

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.

dtsk, where us1;:::;sk are the components of a k-covector, i.e.

the components of a skew- symmetric covariant tensor.
By Cq k(V m) we denote the space of the k-forms whose coeﬃcients are continuously up to the order q in a coordinate system of class Cq+1 (and then in every coordinate system of class Cq+1).

Moreover the symbol Lp k stands for the space of all k-forms whose coeﬃcients are Lp real valued functions.

Let Ω ⊂ Rn be a domain.

The representation of u ∈ C0 system is k (Ω) in the natural coordinate u = 1 k!

us1;:::;skdxs1 .

.

.

dxsk.

The adjoint of u is the (n − k)-form ∗u = 1 (n − k)!

1 k!
1 k!

δ1;:::;n s1;:::;sk;i1;:::;ink us1;:::;skdxi1 .

.

.

dxink. We remark that ∗ ∗ u = (−1)k(nk)u.

If u ∈ C1 k (Ω), the dierential of u is the (k + 1)-form du = 1 k!

∂ ∂xj us1;:::;skdxjdxs1 .

.

.

dxsk, while the codierential of u is the (k − 1)-form δu = (−1)n(k+1)+1 ∗ d ∗ u.

These operators are strictly related to the Laplacian; indeed if u ∈ C2 k −(dδ + δd)u = ∆u = 1 k!

∆us1;:::;skdxs1 .

.

.

dxsk, where ∆us1;:::;sk = n∑ ∂2 ∂x2 h us1;:::;sk.


h=1 Fore more details about diﬀerential forms see, e.g., [14, 15].

On a C1 manifold the concept of k-form has been generalized to the notion of current (see [12]).

A k-measure can be considered as a current of order 0.

Fichera [14] showed how a k-measure can be introduced in a direct and natural way on a diﬀerentiable manifold.

We recall his deﬁnition.

20 Bulletin of TICMI A k-measure is the object determined (in a ﬁxed coordinate system (t1, .

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.

.

, tm)) by the coeﬃcients µs1;:::;sk(B), where µs1;:::;sk is a measure deﬁned on the family {B} of all the Borel sets of V m, and µs1;:::;sk(B) depends skew-symmetrically on the indices s1, .

.

.

, sk.

Moreover if eµj1;:::;jk(B) are the components of the k-measure µ in the coordinate system (τ1, .

.

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, τm) we have eµj1;:::;jk(B) = ∫ ∂ts1 ∂τ j1 B .

.

.

∂tsk ∂τ jk det ∂τ ∂t µs1;:::;sk(B) for every Borel set B ∈ {B}.
Mk(V m) denotes the space of all k-measure on V m. If µ ∈ Mk(V m), the following Lebesgue-Radon-Nycodim decomposition holds µs1;:::;sk(B) = µ s1;:::;sk (B) + ∫ B fs1;:::;skdσ, k(V m) and µ is a singular measure in Mk(V m).

If µ = 0, we say where f ∈ L1 that µ is absolutely continuous.

If u ∈ Ck(V m) and µ ∈ Mh(V m) the exterior product β = u ∧ µ is the (k + h)- measure whose components are βi1;:::;ik+h(B) = 1 k!h!
δs1:::skj1:::jh i1:::ik+h ∫ B us1;:::skdµj1;:::;jh, for any Borel set B; µ ∧ u is by deﬁnition (−1)kh(u ∧ µ).

3.

Self-conjugate dierential forms We begin with some deﬁnitions.

Definition 3.1 ([5]) : We say that u ∈ C1 k (Ω) and v ∈ C1 k+2(Ω) are conjugate if du = δv, δu = 0, dv = 0.

This deﬁnition can be extended to a non homogeneous diﬀerential form U = n∑ uk, uk being a k-form.

We set k=0 dU = n1∑ k=0 n∑ n∑ duk, δU = δuk, ∆U = ∆uk.
k=1 k=0 Since d2 = 0 and δ2 = 0, we can write ∆ = (d − δ)2.

(1) Vol.

18, No.

2, 2014 21 0(Ω) ⊕ n(Ω) is the space composed by k-forms whose coeﬃcients are Lp real valued n(Ω); similarly Lp(Ω) = Lp 0 (Ω) ⊕ .

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⊕ Ck We denote by Ck(Ω) the space Ck .

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⊕ Lp functions in Ω.

Definition 3.2 ([5, 6]) : We say that U ∈ C1(Ω) is self-conjugate if dU = δU, in Ω, i.e.

δu1 = 0, duk1 = δuk+1 (k = 1, .

.

.

, n − 1), dun1 = 0.


From (1) it follows that if U is self-conjugate then U is harmonic, i.e.

all the coeﬃcients of uk are harmonic functions.

Let us consider now some examples of self-conjugate diﬀerential forms.

If n = 2, U = u0 + u2, where u0 = u is a scalar function and u2 = vdxdy is a 2-form, we have dU = ∂u ∂x dx + ∂u ∂y dy, δU = ∂v ∂y dx − ∂v ∂x dy.

Then U is self-conjugate if and only if u + iv is holomorphic.
If n = 3, U = u0 + u2, where u0 = u and u2 = v1dx2dx3 + v2dx3dx1 + v3dx1dx2, U is self-conjugate if and only if (u, v1, v2, v3) is solution of the Moisil-Theodorescu system: { div(v1, v2, v3) = 0, gradu = curl(v1, v2, v3).
If n = 4, where U = u0 + u2 + u4, (2) u0 = f0, u2 = f1(dx1dx2 − dx3dx4) + f2(dx1dx3 − dx4dx2) + f3(dx1dx4 − dx2dx3), u4 = f0dx1dx2dx3dx4, it follows that U is self-conjugate if and only if (f0, f1, f2, f3) is solution of the 22 Bulletin of TICMI Fueter system    f0x1 f0x2 f0x3 f0x4 − f1x2 + f1x1 + f1x4 − f1x3 − f2x3 − f2x4 + f2x1 + f2x2 − f3x4 + f3x3 − f3x2 + f3x1 = 0, = 0, = 0, = 0.

It is known that Fueter system characterizes quaternionic hyperholomorphic functions.
Consequently, the form (2) is self-conjugate if and only if (f0, f1, f2, f3) are the components of a quaternionic hyperholomorphic function.

In a similar way it is possible to ﬁnd a relation between self-conjugate diﬀerential forms and solutions of the Cimmino system.
Namely, if n = 4, let be U = u0 + u2 + u4, where u0 = f0, u2 = f1(dx1dx2 + dx3dx4) − f2(dx1dx3 + dx4dx2) + f3(dx1dx4 + dx2dx3), u4 = −f0dx1dx2dx3dx4; U is self-conjugate if and only if (f0, f1, f2, f3) is a solution of the Cimmino system (see [9])    f0x1 f0x2 f0x3 f0x4 − f1x2 + f1x1 − f1x4 + f1x3 + f2x3 − f2x4 − f2x1 + f2x2 − f3x4 − f3x3 + f3x2 + f3x1 = 0, = 0, = 0, = 0.

U = u1, where u1 = whdxh, For any n ≥ 2 if (3) U is self-conjugate if and only if (w1, .

.

.
.

.

, wn) is a harmonic vector, i.e.

it is a solution of the following system More generally, if { div(w1, .

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, wn) = 0, curl(w1, .

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, wn) = 0.

U = uk, U is self-conjugate if and only if uk is a harmonic form, i.e.

Vol.

18, No.

2, 2014 23 { duk = 0, δuk = 0.

The use of diﬀerential forms points out the connection between Analysis and Ge- ometry, as we can remark by the ensuing result.
It is well known that in a simply connected domain of C any harmonic function is the real part of a holomorphic function.

The next theorem provides an extension of this result to Rn.

Here Hk(Ω) denotes the k-dimensional singular homology group of the domain Ω ⊂ Rn.

Theorem 3.3 ([6]) : Let us x 1 ≤ k ≤ n. Suppose    Hn1(Ω) = 0, Hk1(Ω) = 0 and Hnk1(Ω) = 0, Hn2(Ω) = 0, if k = 0 or k = n, if 2 ≤ k ≤ n − 2, if k = 1 or k = n − 1.
If uk is a harmonic form dened in Ω such that    δu0 = 0, δu1 = 0, δduk = 0, dun1 = 0, dδun1 = 0, dδun = 0, δdu1 = 0, dδuk = 0, if k = 0, if k = 1, if 2 ≤ k ≤ n − 2, if k = n − 1, if k = n, then there exists in Ω a self-conjugate form U such that uk is the k-th component of U.

Let us consider now the double k-form introduced by Hodge sk(x, y) = ∑ j1<:::<jk s(x − y)dxj1 .

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dxjkdyj1 .

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dyjk, where s(x − y) =    1 2π − log |x − y|, if n = 2, 1 (n − 2)ωn |x − y|2n, if n > 2 (ωn being the hypersurface measure of the unit sphere of Rn) is the fundamental solution of the Laplace equation.

It satisﬁes the following identities for x ̸= y (see [10]): dysk(x, y) = δxsk+1(x, y), k = 0, .

.

.
.

.

, n − 1, from which one can prove that    dysk(x, y) = 0, δx ∗ y dxδysk(x, y) = 0, δxdysk(x, y) = 0, dx ∗ y δysk(x, y) = 0, Ω n∑ If U = 24 Bulletin of TICMI { Moreover, dysk(x, y) = −δx ∗ y dx ∗ y dxdysk(x, y) = −δxδysk+2(x, y).

δysk+2(x, y), ∗ x dxsk(x, y) = (−1)nk+1 ∗ y dysn1k(x, y), sk(x, y) = (−1)(nk)k ∗ ∗ y x snk(x, y).
Let now Ω be a regular domain; this means that Ω is a bounded domain, its boundary Σ is an orientable (n − 1)-dimensional C1 diﬀerentiable manifold and for any u ∈ C0 n(Ω) the Stokes formula holds n1(Ω) ∩ C1 n1(Ω) such that du ∈ C0 ∫ ∫ This implies ∫ ∫ du = u.

+Σ Ω ∫ du ∧ ∗v = u ∧ ∗v + δv ∧ ∗u, ∀u ∈ C1 k (Ω), v ∈ C1 k+1(Ω).
+Σ Ω uk ∈ C0(Ω) ∩ C1(Ω) is self-conjugate, we may write    k=0 ∫ ∫ ∫ Ω dv ∧ ∗u1 = v ∧ ∗u1, +Σ [dv ∧ ∗uk+2 − δv ∧ ∗uk] = ∫ Ω ∫ ∫ ∀v ∈ C1 0 (Ω), [uk ∧ ∗v + v ∧ ∗uk+2] , +Σ ∀v ∈ C1 k+1(Ω) k = 0, .

.

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, n − 2, − δv ∧ ∗un1 = un1 ∧ ∗v, Ω +Σ ∀v ∈ C1 n(Ω).
Theorem 3.4 ([3, 6]) : that dU − δU = F ∈ C0(Ω), then the following Cauchy integral formula holds If Ω is a regular domain and U ∈ C0(Ω) ∩ C1(Ω) is such ∫ − [dysk(x, y) ∧ ∗Fk+1(y) − δysk(x, y) ∧ ∗Fk1(y)] Ω ∫ + +Σ [ uk(y) ∧ ∗ y dysk(x, y) − δysk(x, y) ∧ ∗uk(y) ] +dysk(x, y) ∧ ∗uk+2(y) − uk2(y) ∧ ∗ y δysk(x, y) { = uk(x), 0, x ∈ Ω, x /∈ Ω, k = 0, .

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, n, (4) n∑ k=0 where U = −1, n + 1. uk, F = n∑ k=0 Vol.

18, No.
18, No.

2, 2014 25 Fk, uk ≡ 0, k = −2, −1, n + 1, n + 2; Fk ≡ 0, k = We remark that in the case n = 2 formula (4) gives − 1 2π ∫ Ω d log |z − ζ| ∧ ∗F1(ζ) − 1 2π ∫ [ Σ u(ζ) ∂ ∂n log |z − ζ| ] log |z − ζ| ds = −v(ζ) ∂ ∂s { u(z), 0, z ∈ Ω, z /∈ Ω, ∫ Ω 1 2π − ∗ d log |z − ζ| ∧ ∗F1(ζ) − 1 2π ∫ [ v(ζ) ∂ ∂n log |z − ζ| Σ ∂ ∂s +u(ζ) ] log |z − ζ| { ds = v(z), 0, z ∈ Ω, z /∈ Ω.
Putting f (z) = u(z) + iv(z) we have the well known formula ∫ 1 2πi +Σ f (ζ) ζ − z dζ − 1 π ∫ f(ζ) ζ − z Ω { dζ = f (z), 0, z ∈ Ω, z /∈ Ω.

This shows that (4) generalizes this classical result to Rn.

4.

Conjugate harmonic forms in multiply connected domains Let us consider a domain Ω of the form Ω = Ω0 \ m∪ j=1 Ωj, where Ωj (j = 0, .

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, m) are bounded connected domains of Rn, whose boundaries Σj are connected Lyapunov surfaces, such that Ωj ⊂ Ω0 and Ωj ∩ Ωk = ∅, j, k = 1, .

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.

.

, m, j ̸= k. For brevity, we shall call such a domain an (m + 1)-connected domain.

If Ω ⊂ R2 is a simply connected domain, it is well known that if u is a harmonic function, there exists a conjugate function v (i.e.

u + iv is holomorphic).

One can prove that if Ω ⊂ R2 is a (m+1)-connected domain and u ∈ C1(Ω)∩C2(Ω) is a harmonic function, there exists a conjugate harmonic function v, if and only if, ∫ ∂u ∂n Σj ds = 0, j = 0, 1, .

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.

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, m. 26 Bulletin of TICMI The next theorem shows that a similar result holds in Rn ([8, Th 6.1]).

Theorem 4.1 : Let u ∈ C1(Ω) be a harmonic function, where Ω ⊂ Rn is a (m+1)- connected domain.

There exists a 2-form v conjugate to u in Ω (see Denition 3.1), if and only if, ∫ ∂u ∂n Σj dσ = 0, j = 0, 1, .

.

.
.

.

, m. Moreover the 2-form v is given by ∫ v(x) = − ∗ ψ(y)dysn2(x, y) + ω(x), Σ where ψ is the density of the double layer potential representing u and ω is an arbitrary closed and co-closed 2-form.

5.

Conjugate Laplace series Let us consider a trigonometric series a0 2 1∑ + (ak cos kθ + bk sin kθ).

k=1 It can be seen (at least formally) as the trace on the unit circle of the function harmonic in the unit disk u(ρ, θ) = a0 2 + 1∑ k=1 ρk(ak cos kθ + bk sin kθ).
The following series v(ρ, θ) = − 1∑ k=1 ρk(bk cos kθ − ak sin kθ) (5) is a harmonic function which is conjugate to u, i.e.

(u, v) is a solution of the Cauchy-Riemann system ux = vy, uy = −vx.

The conjugate trigonometric series is the series which we obtain taking ρ = 1 in (5) − 1∑ k=1 (bk cos kθ − ak sin kθ).

In [18] F. and M. Riesz proved the important result Theorem 5.1 : If a trigonometric series and its conjugate series a0 2 1∑ + (ak cos kθ + bk sin kθ), k=1 b0 2 − 1∑ k=1 (bk cos kθ − ak sin kθ) Vol.
18, No.

2, 2014 27 are both Fourier-Stieltjes, then they are ordinary Fourier series.

In other words, if we have two real measures α, β deﬁned on the Borel sets of [0, 2π] such that ∫ 2 ∫ 2 cos kθdα = sin kθdβ, 0 0 ∫ 2 0 sin kθdα = − ∫ 2 0 cos kθdβ (k = 1, 2, .

.

.

), then these measures have to be absolutely continuous.

The notion of conjugate Fourier series and the Brother Riesz Theorem have been generalized in Rn in diﬀerent ways (see for instance [2, 11, 16, 17]).
We describe here an extension of Theorem 5.1 hinging on a new concept of conjugate Laplace series given in [4, 7].

It is well known that if u is a harmonic function in the unit ball B = {x ∈ Rn : |x| < 1} , it can be expanded by means of harmonic polynomials u(x) = 1∑ |x|h pnk∑ h=0 k=1 ahkYhk ( ) , x |x| where pnh = (2h + n − 2) (h+n3) harmonic functions.

We suppose {Yhk} is orthonormal, i.e.

(n2)!h!
(n2)!h!

and {Yhk} is a complete system of spherical ∫ Σ YhkYrsdσ = { 1, 0, if h = r and k = s, otherwise.

The trace of u on Σ = {x ∈ Rn : |x| = 1} is given by the expansion 1∑ pnk∑ h=0 k=1 ahkYhk (x) (|x| = 1).

(6) If the coeﬃcients ahk are ∫ ( ∫ ) ahk = f Yhkdσ resp.

ahk = Yhkdµ Σ Σ we say that (6) is the Laplace series of the function f (resp.

of the measure µ).
of the measure µ).

Let us consider the 2-form v = 1∑ pnh∑ h=0 k=1 ahk (h + 2)(n + h − 2) dYhk ) ( x |x| ( ) ∧ d |x|h+2 and its adjoint ∗v = 1∑ pnh∑ h=0 k=1 ahk (h + 2)(n + h − 2) ∗ ( ( dYhk x |x| ) ( )) ∧ d |x|h+2 .

(7) (8) It is possible to show that dv = 0, δv = du in B, i.e.

u + v is self-conjugate.

We say that (8), with |x| = 1, is the Laplace conjugate series of (6).
It represents 28 Bulletin of TICMI the restriction of ∗v on Σ, while the restriction of v, provided it does exist, is equal to 0, because of (7).

In [4] the following expansion is introduced as the series conjugate to (6): √ 1∑ pnh∑ h=0 k=1 h n + h − 2 ahkψhk, (9) with ψhk = √ 1 h(n + h − 2) (h + 2) ∗ (dYhk ∧ dρh+2) = = 1√ h(n + h − 2) ρh+1 n1∑ (−1)n1j j=1 √ ggjj ∂Yhk ∂ϕj dϕ1 .

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bj .

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dϕn1, where ϕ1, .

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, ϕn1 are the usual polar coordinates and gij is the inverse matrix of the relevant metric tensor gij on Σ.

The next result provides a Brother Riesz theorem for Laplace series.

Theorem 5.2 ([4]) : Let (6) be a Laplace series of a measure µ ∈ M0(Σ).

If its conjugate series (9) is a Fourier series of a (n − 2)-measure, i.e.

there exists β ∈ Mn2(Σ) such that √ ∫ n + h − 2 h ahk = β ∧ ∗ Σ ψhk +Σ (h = 1, 2, .

.

.

; k = 1, .

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pnh) ψ denotes the adjoint of ψ on Σ with respect to the usual metric on Σ) and (∗ Σ ∫ β ∧ ∗ Σ γ = 0, +Σ ∀γ ∈ C1 n2(Rn) : dγ = 0 on Σ, then µ and β are absolutely continuous with respect to the Lebesgue (n − 1)- dimensional measure on Σ.

6.

The Brother Riesz theorem for conjugate dierential forms In this section we lay out the results obtained in [5].

The Brother Riesz theorem 5.1 can be reformulated in the following way.
If u(x, y) and v(x, y) are two conjugate real functions in a planar Theorem 6.1 : domain Ω and both of them have traces on ∂Ω in the sense of measures, then these measures have to be absolutely continuous with respect to the arc-length measure on ∂Ω.

In order to obtain a generalization of this statement in Rn for conjugate dif- ferential forms, we introduce the concept of the trace in the sense of k-measures (see [5]).

To this aim let us construct a family of approximating domains Ω.
Let λ(x) be a unit vector deﬁned and continuously diﬀerentiable on Σ such that Vol.

18, No.

2, 2014 29 λ(x) · ν(x) ≥ p0 > 0, ∀x ∈ Σ.

By Σ we denote the hypersurface x = x + ρλ(x), x ∈ Σ, where |ρ| ≤ ρ0 (ρ0 small enough).

Ω is the bounded domain whose bound- ary is Σ.

The domains Ω with 0 < ρ ≤ ρ0 are contained in Ω, while Ω is contained in Ω with −ρ0 ≤ ρ < 0.

We call {Σ} a family of approximating hypersurfaces.
Definition 6.2: We say that the k-form u ∈ Ck(Ω) admits a trace α ∈ Mk(Σ) in the sense of k-measures with respect to the approximating family {Σ} if ∫ lim !0+ ∫ p ∧ u = p ∧ α, +Σ +Σ ∀p ∈ C1 n1k(Ω).

Concerning conjugate diﬀerential forms admitting traces in this sense, we have: k (Ω) and v ∈ C1 Theorem 6.3 : Let u ∈ C1 k+2(Ω) be two conjugate forms.
Let us suppose that they and their adjoint forms admit traces in the sense of h-measures with respect to the approximating family {Σ} .

Namely let α ∈ Mk(Σ), eα ∈ Mnk(Σ), β ∈ Mk+2(Σ), eβ ∈ Mnk2(Σ), the traces of u, ∗u, v, ∗v, respectively.

Then the following formulas hold ∫ [ αy ∧ ∗ y +Σ dysk(y, x) − δysk(y, x) ∧ eαy + dysk(y, x) ∧ eβy = ] { u(x), 0, x ∈ Ω, x /∈ Ω, ∫ [ βy ∧ ∗ y +Σ sk+2(y, x)−δysk+2(y, x) ∧ eβy −αy ∧ ∗ y δysk+2(y, x) = ] { v(x), x ∈ Ω, x /∈ Ω.
0, Moreover u, ∗u, v, ∗v admit the same traces α, eα, β, eβ with respect to any approx- imating family {Σ} .

We can now consider the generalization of the Brother Riesz Theorem for con- jugate diﬀerential forms.

k (Ω) and v ∈ C1 Theorem 6.4 : Let u ∈ C1 k+2(Ω) be two conjugate forms.

Let us suppose that u, ∗u, v, ∗v admit traces in the sense of h-measures with respect to one approximating family (and therefore to any approximating family).
Namely let α ∈ Mk(Σ), eα ∈ Mnk(Σ), β ∈ Mk+2(Σ), eβ ∈ Mnk2(Σ), be the traces of u, ∗u, v, ∗v, respectively.

Then all these h-measures are absolutely continuous with respect to the Lebesgue (n − 1)-dimensional measure on Σ.

It is possible to extend this theorem to non homogeneous diﬀerential forms.

We denote by M0 ⊕ · · · ⊕ Mn1 the space of non homogeneous k-measures Θ = ∑ n1 h=0 θh, where θh ∈ Mh.
We say that a non homogeneous diﬀerential form U = ∑ n h=0 uh admits trace Θ in the sense of k-measures if uh(h = 0, .

.

.

, n − 1) admits a trace θh ∈ Mh in the sense introduced before.

If the self-conjugate form U ∈ C1 n(Ω) is such that Theorem 6.5 : U and ∗U admit traces on Σ in the sense of k-measures with respect to one ap- proximating family (and therefore to any approximating family), then these traces are absolutely continuous with respect to the Lebesgue (n − 1)-dimensional measure on Σ.
0 (Ω) ⊕ .

.

.

⊕ C1 30 Bulletin of TICMI 7.

The Dirichlet problem for the equation dU U = F We say that U ∈ L1 loc(Ω) is a weak solution of dU − δU = F ∈ L1 loc(Ω) if ∫ Ω ∫ (dϕ − δϕ) ∧ ∗U = − F ∧ ∗ϕ, ∀ϕ ∈ ˚C1(Ω).

(10) Ω We shall be interested in seeing whether the Dirichlet problem for the equation dU −δU = F admits a solution.

Next results give necessary and suﬃcient conditions for the existence of this solution.

For the proofs we refer to [3].

0(Σ) ⊕ .

.

.


⊕ L1 In what follows, the symbol L1(Σ) stands for L1 Let us denote by wi1;:::;ik the k-form whdxi1 .

.

.

dxik, where {wh} is a complete system of homogeneous harmonic polynomials.

Such a system can be obtained by ordering in one sequence the polynomials: n1(Σ).

h |x|kY k s ( ) , x |x| k = 0, 1, 2, .

.

.

; s = 1, .

.

.

, pnk; pnk = (2k + n − 2) (k + n − 3)!

(n − 2)!k!

, where Y k degree k. 1 (ω), .

.

.


, Y k pnk (ω) is a complete system of (surface) spherical harmonics of Theorem 7.1 : ϕk, eϕ = ϕ = n1∑ n1∑ Let Ω be a regular domain such that Rn \ Ω is connected.

Let eϕnk ∈ L1(Σ) and F = Fk ∈ L1(Ω) be given forms.
There n∑ k=0 k=0 k=0 exists a non homogeneous dierential form U ∈ L1(Ω) solution of 1) if and only if (−1)(n1)(k1)+1 ∫ [ Ω { dU − δU = F, U = ϕ, ∗U = eϕ, in Ω, on Σ, (11) ] − ∗Fk1 ∧ δwi1;:::;ik h h ∗Fk+1 ∧ dwi1;:::;ik {∫ [ ϕk ∧ ∗dwi1;:::;ik h +Σ − 1 2 − δwi1;:::;ik h ∧ eϕk + dwi1;:::;ik h ∧ eϕk+2 −ϕk2 ∧ ∗δwi1;:::;ik h = 0 (12) ]} for any 1 ≤ i1 < .

.

.

< ik ≤ n, h = 1, 2, .

.

.

, k = 0, 1, .

.

.


, n (ϕk ≡ 0, k = −2, −1; eϕk ≡ 0, k = n + 1, n + 2; Fk ≡ 0, k = −1, n + 1).

1) In the problem (11) the equation dU U = F is considered in the weak sense (10).

It is possible to generalize the previous result to multiply connected domains.

Vol.

18, No.

2, 2014 31 In this case, we have to add supplementary conditions to (12) for each hole.

m j=1 Ωj be an (m + 1)-connected domain.


Let Let Ω = Ω0 \ Theorem 7.2 : ∪ ϕ = ϕk, eϕ = eϕnk ∈ L1(Σ) and F = Fk ∈ L1(Ω) be given forms.
There n1∑ k=0 n1∑ k=0 n∑ k=0 exists a non homogeneous dierential form U ∈ L1(Ω) solution of if and only if (−1)(n1)(k1)+1 ∫ [ Ω { dU − δU = F, U = ϕ, ∗U = eϕ, in Ω, on Σ, (13) ] − ∗Fk1 ∧ δwi1;:::;ik h h ∗Fk+1 ∧ dwi1;:::;ik {∫ [ ϕk ∧ ∗dwi1;:::;ik h +Σ − 1 2 − δwi1;:::;ik h ∧ eϕk +dwi1;:::;ik h ∧ eϕk+2 − ϕk2 ∧ ∗δwi1;:::;ik h ]} = 0, ∫ [ (−1)(n1)(k1)+1 ∗Fk+1(y) ∧ dy[|y − xj|2n2kwi1;:::;ik (y − xj)] h Ω − ∗ Fk1(y) ∧ δy[|y − xj|2n2kwi1;:::;ik ] (y − xj)] h {∫ [ ϕk(y) ∧ ∗ y +Σ − 1 2 dy[|y − xj|2n2kwi1;:::;ik h (y − xj)] − δy[|y − xj|2n2kwi1;:::;ik + dy[|y − xj|2n2kwi1;:::;ik h h (y − xj)] ∧ eϕk(y) (y − xj)] ∧ eϕk+2(y) ]} −ϕk2(y) ∧ ∗ y δy[|y − xj|2n2kwi1;:::;ik h (y − xj)] = 0, j = 1, .
.

.

, m, for any 1 ≤ i1 < .

.

.

< ik ≤ n, h = 1, 2, .

.

.

, k = 0, 1, .

.

.

, n (ϕk ≡ 0, k = −2, −1; eϕk ≡ 0, k = n + 1, n + 2; Fk ≡ 0, k = −1, n + 1).

Here xj is a xed point in Ωj (j = 1, .

.

.

, m).

The following result plays a key role in obtaining Theorems 7.1 and 7.2.
It concerns the following two spaces of non homogeneous diﬀerential forms: { U = U ∈ L1(Ω) : ∃ ϕ = n1∑ k=0 ϕk, eϕ = n1∑ k=0 eϕnk ∈ L1(Σ), F = n∑ k=0 Fk ∈ L1(Ω) such that ∫ n1∑ k=0 Ω dvk ∧ ∗uk+1 − ∫ n∑ k=1 Ω δvk ∧ ∗uk1 − ∫ n∑ k=0 Ω vk ∧ ∗Fk 32 ∫ n1∑ = k=0 +Σ vk ∧ eϕk+1 + ∫ n∑ k=1 +Σ Bulletin of TICMI ϕk1 ∧ ∗vk for any V = } vk ∈ C1(Rn) , n∑ k=0 { V = U ∈ L1(Ω) : ∃ ϕ = ∫ n1∑ k=0 ϕk, eϕ = n1∑ k=0 eϕnk ∈ L1(Σ) , F = n∑ k=0 Fk ∈ L1(Ω) such that − [dysk(x, y) ∧ ∗Fk+1(y) − δysk(x, y) ∧ ∗Fk1(y)] ∫ [ Ω + +Σ ϕk(y) ∧ ∗ y dysk(x, y) − δysk(x, y) ∧ ∗ϕk(y) + dysk(x, y) ∧ ∗ϕk+2(y) ] { −ϕk2(y) ∧ ∗ y uk(x) 0 (ϕk ≡ 0, k = −2, −1; eϕk ≡ 0, k = n + 1, n + 2; Fk ≡ 0, k = −1, n + 1) k = 0, .
.

.

, n δysk(x, y) = x ∈ Ω x /∈ Ω } .

Roughly speaking the space U is given by the L1 diﬀerential forms solutions of dU − δU = F in Ω having L1 traces in a weak sense (see (10)), while V is the space of the L1 forms in Ω such that there exist L1 forms on Σ for which the Cauchy integral formula holds.

Actually we have: Theorem 7.3 : U = V. 8.


The Dirichlet problem for the Cimmino system In paper [13] Dragomir and Lanconelli studied the Cimmino system (3), that can be written in the following complex form { uz + vw = 0, uw − vz = 0.

The authors obtained, among many other results, a necessary condition for the resolubility of the Dirichlet problem for Cimmino system.

Theorem 8.1 : Let Ω ⊂ C2 be a bounded domain on which Green’s formula holds and Σ its boundary; let f, g ∈ L2(Ω), F, G ∈ L2(Σ).
If there is a solution u, v ∈ C1(Ω) ∩ C0(Ω) to the boundary value problem { uz + vw = f, u = F, uw − vz = g, v = G, in Ω, on Σ, (14) Vol.

18, No.

2, 2014 33 then (f, g, F, G) satises the compatibility relations { ∫ Re 2 (f h + gk)dV − Ω ∫ Σ { F [ (n1 + in2)h + (n3 + in4)k ] } +G [(n3 + in4)h − (n1 + in2)k] dσ } = 0 (15) for any solution h, k ∈ C1(Ω) ∩ C0(Ω) to hz + kw = 0, hw − kz = 0, in Ω, where (n1, n2, n3, n4) is the outward unit normal on Σ.
They left open the question of whether (15) is also suﬃcient for the solvability of (14).

More recently, Abreu Blaya et al.

[1] studied (14) by means of quaternionic anal- ysis.

In particular, they found some diﬀerent necessary and suﬃcient conditions involving some particular integral operators.

From this they deduce that (15) are also suﬃcient when f = g = 0 and Ω is a simply connected domain.
In [3], Theorem 7.2 is used to obtain other necessary and suﬃcient conditions for the solvability of (14), also in the case of multiply connected domains.

In what follows we illustrate these results.

Let us consider the problem (13) with the particular kind of data: ϕ = (ϕ0, 0, ϕ2, 0), eϕ = (−ϕ0, 0, ϕ2, 0) ∈ L1(Σ), F = F1 − ∗F1 ∈ L1(Ω), (16) where F1 = γkdxk.
Applying Theorem 7.2, necessary and suﬃcient conditions for the solvability of (13) with these particular data are ∫ ∗F1(y) ∧ dwh(y) − 1 2 Ω ∫ +Σ ∫ Ω ∗F1(y) ∧ dy[|y − xj|2wh(y − xj)] − 1 2 [ϕ0(y) ∧ ∗dwh(y) + dwh(y) ∧ ϕ2(y)] = 0; {∫ +Σ [ ϕ0(y) ∧ ∗ y } ] dy[|y − xj|2wh(y − xj)] +dy[|y − xj|2wh(y − xj)] ∧ ϕ2(y) = 0, j = 1, .

.

.
, m; (17) ∫ [ Ω F1(y) ∧ dwi1;i2 h (y) − ∗F1(y) ∧ δwi1;i2 h ] (y) − 1 2 {∫ [ +Σ ϕ2(y) ∧ ∗dwi1;i2 h (y) −δwi1;i2 h (y) ∧ ϕ2(y) + dwi1;i2 h (y) ∧ −ϕ0(y) − ϕ0(y) ∧ ∗δwi1;i2 h (y) = 0; ]} 34 ∫ [ F1(y) ∧ dy[|y − xj|6wi1;i2 h Bulletin of TICMI (y − xj)] − ∗F1(y) ∧ δy[|y − xj|6wi1;i2 h ] (y − xj)] Ω {∫ [ ϕ2(y) ∧ ∗ y +Σ − 1 2 dy[|y − xj|6wi1;i2 h (y − xj)] − δy[|y − xj|6wi1;i2 h (y − xj)] ∧ ϕ2(y) ]} +dy[|y − xj|6wi1;i2 h (y − xj)] ∧ −ϕ0(y) − ϕ0(y) ∧ ∗ y δy[|y − xj|6wi1;i2 h (y − xj)] = 0, j = 1, .
.

.

, m, (18) for any 1 ≤ i1 < i2 ≤ 4, h = 1, 2, .

.

..

Here xj is a ﬁxed point in Ωj (j = 1, .

.

.

, m).

Moreover the solution U can be written as U = u0 + u2 + u4, u0 = f0, u2 = f1(dx1dx2 + dx3dx4) − f2(dx1dx3 + dx4dx2) + f3(dx1dx4 + dx2dx3), u4 = −f0dx1dx2dx3dx4.
By exploiting the relation between self-conjugate diﬀerential forms and solutions of the Cimmino system, described in section 3, it is possible to prove that (13), with the particular data (16), is equivalent to the Dirichlet problem for the Cimmino 2 (γ1 + iγ2), g = 1 system (14).

Indeed, if f = 1 2 (γ3 + iγ4), F = α0 + iα1 and G = β0 + iβ1 it suﬃces to take ϕ = (α0, 0, ϕ2, 0) and eϕ = (−α0, 0, ϕ2, 0), where ϕ2 = α1(dx1dx2 + dx3dx4) − β0(dx1dx3 − dx2dx4) − β1(dx1dx4 + dx2dx3).
This equivalence applies to obtain the next claim.

Theorem 8.2 : Let Ω ⊂ C2 be a regular domain and Σ its boundary.

Let f, g ∈ L1(Ω) and F, G ∈ L1(Σ).

Conditions (17) and (18) are necessary and sucient for the resolubility of the boundary value problem for Cimmino system (14).

The only point remaining concerns the relation between Dragomir and Lanconelli condition (15) and our conditions (17) and (18).
Let us consider the ﬁrst of (17); it can be written as ] dy +β1 ) ∫ [ Ω − 1 2 ( + ( ∂wh ∂y1 [( γ1 ∫ +Σ ∂wh ∂y2 α0 + α0 ( + α0 ∂wh ∂y3 ∂wh ∂y4 + γ2 ∂wh ∂y2 + γ3 ∂wh ∂y3 + γ4 ∂wh ∂y4 α0 ∂wh ∂y1 + α1 ∂wh ∂y2 − β0 ∂wh ∂y3 − α1 + α1 − α1 ∂wh ∂y1 ∂wh ∂y4 ∂wh ∂y3 + β0 + β0 − β0 ∂wh ∂y4 ∂wh ∂y1 ∂wh ∂y2 +β1 −β1 −β1 ∂wh ∂y3 ∂wh ∂y2 ∂wh ∂y1 ) n1 ∂wh ∂y4 (19) ) n2 n3 ) ] n4 dσ = 0.

If we put h0 = ∂wh ∂y3 (19).

In a similar way, it is possible to prove that (15) implies (17) and (18).
We , we have that (15) implies ∂wh ∂y1 ∂wh ∂y2 ∂wh ∂y4 , h1 = , k1 = , k0 = Vol.

18, No.

2, 2014 35 can conclude that (15) is not only necessary but also suﬃcient for the solvability of (14).

References [1] R. Abreu Blaya, Bory Reyes J., Guzm´an Ad´an A., B. Schneider, Boundary value problems for the Cimmino system via quaternionic analysis, Applied Mathematics and Computation, 219 (2012), 3872–3881.
[2] B. Bochner, Boundary values of analytic functions in several variables and of almost periodic func- tions, Ann.

of Math., 45 (1944), 708-722 [3] P. Caramuta, A. Cialdea, An application of the theory of self-conjugate dierential forms to the Dirichlet problem for Cimmino system, submitted [4] A. Cialdea, The brother Riesz theorem in Rn and Laplace series, Mem.

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