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|j| j ∂ ∂ = j1 j2 jm , ∂x1 ∂x2 ...∂xm m where j = (j1, j2,...,jm) ∈ (N ∪ {0}) is a m-dimensional multi-indices and |j| = j1 + · · · + jm. k,α • C (Ω, R0,m), α ∈ (0, 1] the right module of all R0,m-valued functions, k-times α-H¨older continuously differentiable in Ω.

• Lp(Ω, R0,m), (1 ≤ p < ∞) the right module of all equivalence classes of p- Lebesgue measurable R0,m-functions over Ω. 1 m i m 1 Let an ordered set ψ := {ψ ,...,ψ }, with ψ ∈ R ⊂ R0,m. On the set C (Ω, R0,m) we define the generalized Dirac operator by: ∂ ∂ ∂ ψ∂ := ψ1 + ψ2 + · · · ψm . (1) ∂x1 ∂x2 ∂xm

For the particular case of the standard R0,m-basic vector set ψst := {e1,e2 ...,em}, operator ψ∂ becomes the Dirac operator. Let ∆m be the (m)-dimensional Laplace operator. It is easy to prove that the equality ψ ψ ∂ ∂ = −∆m (2) in Rm hold, if and only if

i j j i ψ ψ + ψ ψ = −2δij (i, j = 1, 2,...m).

Note that last equality yields

i j j i i j 2δi,j = ψ · ψ¯ + ψ · ψ¯ = 2 ψ , ψ , (3) Rm D E where h, iRm denotes the scalar product, hence factorization (2) holds if and only if ψ represents an orthonormal basis of Rm. A set ψ with the property (3) is called structural set. Notion of structural sets goes back to [14,15]. ψ The R0,m-valued solutions of ∂u = 0 are the so-called ψ-hyperholomorphic func- tions. Next we mention basic facts of this theory to be used in the paper, thus making our exposition self-contained. Deeper discussions can be found in [1–3] and the refer- ences given there.

2 The fundamental solution of the operator ψ∂ is given by

−xψ Kψ(x)= m , σm|x| where m m i xψ = xiψ if x = xiei i=1 i=1 X Xm and σm stands for the area of the unit sphere in R . This particularly important function, referred to as Cauchy kernel, comes from 2−m ψ |x| acting ∂ to the fundamental solution of the Laplacian ∆m given by , i.e. σm(2 − m)

2−m ψ |x| Kψ(x)= ∂[ ]. σm(2 − m)

m The Cauchy kernel Kψ is ψ-hyperholomorphic in R \ {0} and plays a decisive role in our context.

1 Theorem 1 (Borel-Pompeiu formula). Let u ∈ C (Ω ∪ Γ, R0,m). Then it holds that

ψ u(x) if x ∈ Ω+ Kψ(y − x)nψ(y)u(y)dy − Kψ(y − x) ∂u(y)dy = (4) 0 if x ∈ Ω , ZΓ ZΩ ( − m i where nψ(y) = i=1 ni(y)ψ , being ni(y) the i-th component of the outward unit normal vector at y ∈ Γ. From (4) oneP finds two important integral operators: the Cauchy transform

Cψu(x) := Kψ(y − x)nψ(y)u(y)dy, ZΓ which represents a ψ-hyperholomorphic function in Rm \Γ and the Teodorescu operator

Tψv(x)= − Kψ(y − x)v(y)dy, ZΩ which runs as the right-handed inverse of ψ∂, i.e.

ψ v(x), if x ∈ Ω+ ∂Tψv(x)= (5) (0 if x ∈ Ω−. 2 Preliminaries 2.1 Lipschitz classes and Whitney extension theorem The higher order Lipschitz class Lip(k + α, Γ) consists of the collections of real-valued continuous functions f := {f (j), |j|≤ k} (6) defined on Γ and satisfying the compatibility conditions

f (j+l)(y) |f (j)(x) − (x − y)l| = O(|x − y|k+α−|j|), x,y ∈ Γ, |j|≤ k. (7) l! |j+Xl|≤k

3 In 1934 the American mathematician Hassler Whitney, one of the most prominent figures from the field in the 20th century [13], proved in [17] that such a collection can be extended as a Ck,α-smooth function on Rm. For an excellent reference alone more classical lines we refer the reader to the well-known book of E. M. Stein [16, Chapter VI, p. 176]. Because the R0,m-valued Lipschitz classes are component-wise defined, by abuse of notation we continue to write Lip(k + α, Γ) for the Cliffordian situation. We shall confine ourselves to discussing the case k = 1, which will become clear shortly.

(j) Theorem 2 (Whitney). Let u = {u , |j| ≤ 1} be an R0,m-valued collection in Lip(1 + α, Γ). Then, there exists a compact supported R0,m-valued function u˜ ∈ 1,α m C (R , R0,m) satisfying (0) (j) (j) (i) u˜|Γ = u , ∂ u˜|Γ = u , |j| = 1, (ii) u˜ ∈ C∞(Rm \ Γ), (iii) |∂ju˜(x)| 6 c dist(x, Γ)α−1, for |j| = 2 and x ∈ Rm \ Γ.

2.2 Harmonic versus (ϕ, ψ)-harmonic functions This paper is concerned with a second order partial differential equation arising in a natural way from the consideration of two different structural sets ϕ and ψ, i.e. the generalized Laplace equation ϕ∂ψ∂u = 0. (8) The solutions of (8) will be referred as (ϕ, ψ)-harmonic functions. The space of all m (ϕ, ψ)-harmonic functions in R will be denoted by Hϕ,ψ(Ω, R0,m). It is worth not- ing that for ϕ = ψ, the class Hϕ,ψ(Ω, R0,m) coincides with the space H(Ω, R0,m) of harmonic functions in Ω, which justifies the name we choose for the functions in Hϕ,ψ(Ω, R0,m). Further, it is likewise easy to obtain the fundamental solution of ϕ∂ψ∂. In fact it is sufficient to apply the corresponding operator ψ∂ϕ∂ to the fundamental solution of 2 the bi-Laplacian ∆m, which is given by

|x|4 . σm(2 − m)(4 − m) This gives rise to the fundamental solution

4 −m 2−m m ψ ϕ |x| (2 − m)|x| xψxϕ + |x| i=1 ψiϕi Kϕψ(x) := ∂ ∂ = , σ (2 − m)(4 − m) 2σ (2 − m)  m  m P which is obviously (ϕ, ψ)-harmonic in Rm \ {0}. The Laplace operator ∆m is the quintessential example of a (strongly) elliptic operator, while the operator ϕ∂ψ∂ being elliptic, is not strongly elliptic. This can be shown via the following counterexample, which violates the maximum principle. Let m be even and consider an arbitrary structural set ϕ = {ϕ1, ϕ2,...,ϕm−1, ϕm} together with ψ = {ϕ2, ϕ1,...,ϕm, ϕm−1}. Introduce the function

m 2 u(x) = 1 − xi , i=1 X which obviously does vanishes on the boundary of the unit ball B(0, 1) of Rm.

4 On the other hand we get

ϕ ψ ϕ ∂u ∂u ∂u ∂u ∂ ∂u = ∂(ϕ2 + ϕ1 + · · · + ϕm + ϕm−1 ) ∂x1 ∂x2 ∂xm−1 ∂xm ∂2u ∂2u ∂2u ∂2u = ϕ1ϕ2 2 + ϕ2ϕ1 2 + · · · + ϕm−1ϕm 2 + ϕmϕm−1 2 ∂x1 ∂x2 ∂xm−1 ∂xm = −2ϕ1ϕ2 − 2ϕ2ϕ1 −···− 2ϕm−1ϕm − 2ϕmϕm−1 = 0.

This give u ∈ Hϕ,ψ(B(0, 1), R0,m) and non identically zero. Of course, the above example mathematically leads to ill-posed formulation of the Dirichlet problem for (ϕ, ψ)-harmonic functions in the sense of Hadamard [11], which is a marked difference between Hϕ,ψ(Ω, R0,m) and H(Ω, R0,m).

3 Riemann-Hilbert boundary value problem for (ϕ, ψ)-harmonic functions

In this section a Riemann-Hilbert boundary value problem for (ϕ, ψ)-harmonic func- tions will be discussed. To do so, some basic facts are firstly introduced.

2 Theorem 3. [3] Let u ∈ C (Ω ∪ Γ, R0,m). Then for x ∈ Ω it holds that

ψ u(x)= Kψ(y − x)nψ(y)u(y)dy + Kϕψ (y − x)nϕ(y) ∂u(y)dy − ZΓ ZΓ ϕ ψ Kϕψ(y − x)[ ∂ ∂u(y)]dy. (9) ZΩ The corresponding Teodorescu operator here is given by

Tϕψv(x) := − Kϕψ(y − x)v(y)dy, ZΩ which satisfies the relations (see [3, Theorem 5])

ψ ϕ ψ v(x) if x ∈ Ω+ ∂Tϕψv(x)= Tϕv(x), ∂ ∂Tϕψv(x)= (10) (0 if x ∈ Ω−. Remark 1. By means of a more detailed analysis, it may be shown that the above Borel-Pompeiu formula will remain valid under weaker requirements, namely that u ∈ 2 1 C (Ω, R0,m) ∩ C (Ω ∪ Γ, R0,m) and

|ϕ∂ψ∂u(y)|dy < +∞, ZΩ which ensure the existence of all the integrals in (9). No consideration will be given in this paper to the problem of finding the most general conditions for the validity of (9), but instead the previous ones are completely sufficient for our purposes.

In particular, for u ∈ Hϕ,ψ(Ω, R0,m), one has in Ω

ψ u(x)= Kψ(y − x)nψ(y)u(y)dy + Kϕψ(y − x)nϕ(y) ∂u(y)dy, (11) ZΓ ZΓ

5 the last formula being a sort of Cauchy representation formula for (ϕ, ψ)-harmonic functions. The Lipschitz class Lip(1+α, Γ) is well adapted to define on it a Cauchy transform arising from (11). More precisely, given a Lipschitz data u ∈ Lip(1+α, Γ), the Cauchy transform is defined by

ψ Cϕψu(x)= Kψ(y − x)nψ(y)˜u(y)dy + Kϕψ(y − x)nϕ(y) ∂u˜(y)dy, (12) ZΓ ZΓ whereu ˜ denotes the Whitney extension of u in Theorem 2. Although the Whitney extension is not unique, one should remark that the ap- pearance of ambiguity in the above definition disappears since the values ofu ˜ and ψ∂u˜ on Γ are completely determined by the collection u = {u(j), |j|≤ 1}. m Of course, the function Cϕψu is by definition (ϕ, ψ)-harmonic in R \ Γ. Also, it should be noted that the weakly singularity of the kernel Kϕψ(y − x) implies that the second integral in (12) does not experience a jump when x is crossing the boundary Γ. This together with the classical Plemelj-Sokhotski formulas applied to the first Cauchy type integral in (12) lead to

+ − (0) [Cϕψu] (x) − [Cϕψu] (x) =u ˜(x)= u (x), x ∈ Γ, (13) where ± [Cϕψu] (x) = lim Cϕψu(z). Ω±∋z→x

3.1 Riemann-Hilbert boundary value problem The Riemann-Hilbert boundary value problem (RH-problem for short) for an unknown (j) u ∈ Hϕ,ψ(Ω+ ∪Ω−, R0,m) is defined for g = {g , |j|≤ 1} ∈ Lip(1+α, Γ) by the system

u+(x) − u−(x)A =g ˜(x) if x ∈ Γ, ψ + ψ − ψ [ ∂u] (x) − [ ∂u] (x)B = ∂g˜(x) if x ∈ Γ, (14) u(∞)= ψ∂u(∞) = 0,  where A, B are two invertible R0,m-valued constants.

3.2 The smooth case Let us denote

ψ −1 g∗(x)= Kϕψ (y − x)nϕ(y) ∂g˜(y)dy(−1+ B A) +g ˜(x), x ∈ Γ. ZΓ Then, the function

ψ Cψg∗(x)+ Γ Kϕψ(y − x)nϕ(y) ∂g˜(y)dy, x ∈ Ω+, u(x) :=  R (15)  −1 ψ −1 Cψg∗(x)A + Γ Kϕψ(y − x)nϕ(y) ∂g˜(y)dyB , x ∈ Ω−  R belongs to Hϕ,ψ(Ω+ ∪ Ω−, R0,m) and satisfies the boundary conditions in (14). The uniqueness of homogeneous RH-problem (14) is reduced to prove that it has only the null-solution.

6 ψ In fact, since u ∈ Hϕ,ψ(Ω+ ∪Ω−, R0,m), the function ∂u(x) is ϕ-hyperholomorphic in Ω+ ∪ Ω− and has no jump through Γ. This, together with the vanishing condition ψ∂u(∞) = 0 yields ψ∂u ≡ 0 in Rm, which is clear from the combination of classi- cal Painleve and Liouville theorems. The proof is concluded after using the same arguments for the ψ-hyperholomorphic function u. We refer the reader also to [10, p.307] and [18], where similar problems for standard R0,m-valued harmonic functions in the smooth context are studied.

3.3 The fractal case There is an essential difference between the fractal case and those studied before. In fact, for a fractal boundary Γ, the function given by (15) is useless and meaningless as it stands. Throughout this subsection we follow [12] in assuming that Γ is d-summable for some m − 1

1 d−1 NΓ(τ) τ dτ Z0 converges, where NΓ(τ) stands for the minimal number of balls of radius τ needed to cover Γ. The notion of a d-summable subset was introduced by Harrison & Norton in [12], who showed that if Γ has box dimension [7] less than d, then is d-summable. The following lemma can be found in [12, Lemma 2] and reveals the specific im- portance of the notion of d-summability of the boundary Γ of a Jordan domain Ω in connection with the Whitney decomposition W of Ω by squares Q of diameter |Q|. Lemma 1. [12] If Ω is a Jordan domain of Rm and its boundary Γ is d-summable, d then the expression Q∈W |Q| , called the d-sum of the Whitney decomposition of Ω, is finite. P ϕ ψ p m − d Lemma 2. Let g ∈ Lip(1 + α, Γ), then ∂ ∂g˜ ∈ L (Ω, R ,m) for p = . 0 1 − α Proof. From Theorem 2 (iii), we have |ϕ∂ψ∂g˜(x)| 6 c dist(x, Γ)α−1 for x ∈ Ω. With this in mind the proof follows in a quite analogous way to that of [4, Lemma 4.1]. 

d Lemma 3. Let g ∈ Lip(1 + α, Γ) with α > . Then T [ψ∂g˜] and ψ∂T [ϕ∂ψ∂g˜] are m ψ ϕψ continuous functions in Rm. Proof. ψ 0 ψ Since ∂g˜ ∈ C (Ω ∪ Γ, R0,m), then ∂g˜ belongs to Lp(Ω, R0,m), with p>m. Similar analysis to that in the proof of [9, Proposition 8.1] can be applied to conclude that ψ 0 m Tψ[ ∂g˜] ∈ C (R , R0,m). ψ ϕ ψ To prove the continuity of ∂Tϕψ[ ∂ ∂g˜] we use the first identity in (10):

ψ ϕ ψ ϕ ψ ∂Tϕψ[ ∂ ∂g˜]= Tϕ[ ∂ ∂g˜]. d A fine point here is to note that under the condition α > we can applied Lemma m ϕ ψ 2 to ensure that ∂ ∂g˜ also belongs to Lp(Ω, R0,m) with p>m, and so, the reasoning as described above may be repeated. 

7 The following theorem describes a solution of the RH-problem (14) in fractal con- text. d Theorem 4. Let g ∈ Lip(1 + α, Γ)and let Γ be d-summable with α > . Then the m RH-problem (14) has a solution given by

ψ ψ ϕ ψ g∗(x) − Tψ[ ∂g∗](x)+ Tψ[ ∂g˜](x) − Tϕψ[ ∂ ∂g˜](x), x ∈ Ω+, u(x) :=  (16)  ψ −1 ψ ϕ ψ −1 −Tψ[ ∂g∗](x)A + Tψ[ ∂g˜](x) − Tϕψ [ ∂ ∂g˜](x) B , x ∈ Ω−,    where  ψ ϕ ψ −1 g∗(x) := Tψ[ ∂g˜](x) − Tϕψ [ ∂ ∂g˜](x) (−1+ B A) +g ˜(x).   Proof. 0 m Lemma 3 shows that g∗ ∈ C (R , R0,m). On the other hand

ψ ψ ϕ ψ −1 ψ ∂g∗(x)= ∂g˜(x) − Tϕ[ ∂ ∂g˜](x) (−1+ B A)+ ∂g˜(x)   0 m and clearly we will again have a function of C (R , R0,m). ψ Consequently, Tψ[ ∂g∗] together with the remaining terms in (16) all belong to 0 m C (R , R0,m). The first boundary condition in (14) then follows directly after a direct calculation. To prove the second condition we use again the identities (5) and (10) to obtain

ψ ϕ ψ ∂g˜(x) − Tϕ[ ∂ ∂g˜](x), x ∈ Ω+, ψ ∂u(x) :=  (17) ϕ ψ −1 −Tϕ[ ∂ ∂g˜](x)B , x ∈ Ω−.

Hence, the second boundary condition is directly deduced. Finally, the (ϕ, ψ)-harmonicity of u in Rm \ Γ follows immediately acting ϕ∂ in (17). 

If A = B, then g∗ =g ˜ and the solution is considerably simplified to

ϕ ψ g˜(x) − Tϕψ[ ∂ ∂g˜](x), x ∈ Ω+, u(x) :=  (18) ϕ ψ −1 −Tϕψ[ ∂ ∂g˜](x)B , x ∈ Ω−.

Remark 2. The method followed to prove the uniqueness of solution of the RH- problem breaks down when we drop the smoothness assumption over the boundary of the domain. Some partial evidence support the conjecture that the uniqueness may be ensured under similar conditions to those discussed in [4, Theorem 4.2].

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