A family of fundamental solutions for elliptic quaternion coefficient differential operators and application to perturbation results for single layer potentials M. Dalla Riva, J. Morais,

P. Musolino

Abstract In this note we announce some of the results that will be presented in a forthcoming paper by the authors, and which are concerned about the construction of a family of fundamental solutions for elliptic partial differential operators with quaternion constant coefficients. The elements of such a family are functions which depend jointly real analytically on the coefficients of the operators and on the spatial variable. A detailed description of such fundamental solutions has been deduced in order to study regularity and stability properties in the frame of Schauder spaces for the corresponding layer potentials.

Keywords: Fundamental solutions, quaternion analysis, elliptic partial differential operators with quaternion constant coefficients, layer potentials. PACS: 02.30.Em, 02.30.Fn, 02.30.Jr, 02.60.Lj

1

Introduction

ful tools in the analysis of perturbed boundary value problems. This note announces the construction of a particular family of fundamental solutions for the quaternion constant coefficients elliptic partial differential operators of [16] and shows an extension of the results of [17] within the non-commutative structure of quaternions. The elements of such a family are quaternion valued functions which depend jointly real analytically on the (quaternion) coefficients of the operators and on the spatial variable in Rn \ {0}. A detailed description of such functions has been exploited to study regularity and stability properties in the frame of Schauder spaces for the corresponding layer potentials. The principal goal that is central in our approach is the treatment of perturbed elliptic boundary value problems by means of layer potentials. In this sense, the construction announced here can be considered as a first step toward the generalization of the potential theoretic approach of Lanza de Cristoforis et

The study of fundamental solutions is a recurring theme as it constitutes an important tool for the analysis of boundary value problems of elliptic systems of differential equations by means of potential theory (cf. Fichera [1], Miranda [2], Kupradze et al. [3]). More recently, a potential theoretic approach has been adopted in order to investigate perturbed boundary value problems and in particular domain perturbations. We mention, as an example, the works of Potthast [4, 5, 6], Costabel and Le Lou¨er [7, 8], and Lanza de Cristoforis and collaborators [9, 10, 11, 12, 13, 14, 15]. In view of such applications, it is important to understand the dependence of the layer potentials corresponding to a fundamental solution of a partial differential operator upon perturbations of the support of integration and of data such as the coefficients of the operator and the density function. When regarded in such a way, the study of fundamental solutions provides use1

al. to the case of general elliptic partial differential index α ≡ (α1 , . . . , αn ) ∈ Nn . If m ∈ N, we deoperators with quaternion constant coefficients. note the space of the m times continuously differentiable quaternion functions on Ω by C m (Ω, H), and by C m (clΩ, H) ⊆ C m (Ω, H) the subspace of those 2 Quaternion analysis functions f whose derivatives ∂xα f of order |α| ≡ α1 +· · ·+αn ≤ m can be extended to continuous funcQuaternion analysis is a powerful tool for treating 3D tions on clΩ. As usual, the definitions of C m (Ω, H) and 4D boundary value problems of elliptic partial and C m (clΩ, H) are understood componentwise. In differential equations. The rich structure of this thecase Ω is bounded, then C m P (clΩ, H) endowed with ory involves the study of functions defined on subsets α m the norm kf k ≡ C (clΩ,H) |α|≤m supclΩ |∂x f | is of Rn and with values in the quaternions. A thorough well known to be a Banach space. If λ ∈]0, 1[, treatment of the subject is listed in the bibliography, then C 0,λ (clΩ, H) denotes the space of the funce.g., G¨ urlebeck and Spr¨ oßig [18, 19], Kravchenko and tions from clΩ to H which are uniformly H¨ older Shapiro [20], Kravchenko [21], Shapiro and Vasilevski continuous with exponent λ. If f ∈ C 0,λ (clΩ, H), [22, 23], and Sudbery [24]. then its H¨older constant |f : Ω|λ is defined as Let −λ sup |f (x) − f (y)||x − y| : x, y ∈ clΩ, x = 6 y . We n o m,λ (clΩ, H) the subspace of C m (clΩ, H) H ≡ z ≡ z0 + z1 i + z2 j + z3 k : zi ∈ R, i ∈ {0, 1, 2, 3} denote by C of the quaternion functions with m-th order deriva0,λ (clΩ, H). If Ω is bounded, then be the real quaternion algebra, where the imaginary tives in C m,λ the space C (clΩ, H) equipped norm units i, j and k obey the following laws of multiplicaP with the α 2 2 2 m kf k = kf k + |∂ f : Ω|λ , m,λ C (clΩ,H) C (clΩ,H) |α|=m x tion: i = j = k = −1; ij = k = −ji, jk = i = −kj, is well known to be a Banach space. We retain a ki = j = −ik. m,λ n m,λ similar notation for C (clΩ, R ) and C (clΩ, C). Then each element z = z0 + z1 i + z2 j + z3 k of H m,λ if its clocan be identified with the vector z = (z0 , z1 , z2 , z3 ) Also, we say that Ω is a set of class C sure is a manifold with boundary embedded in Rn 4 of p R . The norm |z| of z is defined by |z| ≡ of class C m,λ . If Ω is an open bounded subset of z02 + z12 + z22 + z32 , and coincides with the corren m,λ and l ∈ {0, . . . , m} then we define sponding Euclidean norm as a real vector. The scalar R of classl,λC the sets C (∂Ω, H), C l,λ (∂Ω, Rn ), and C l,λ (∂Ω, C) and vector parts of z = z0 + z1 i + z2 j + z3 k ∈ H are defined by Sc(z) ≡ z0 and Vec(z) ≡ z1 i + z2 j + z3 k, by exploiting the local parametrizations of ∂Ω. For respectively. A function with values in H is called standard properties of functions in Schauder spaces, quaternion function or H-valued function. Proper- we refer the reader to Gilbarg and Trudinger [25]. ties (like integrability, continuity or differentiability) For standard definitions of real analytic functions between real Banach spaces we refer, e.g., to Deimling are defined componentwise. [26, p. 150] (see also Prodi and Ambrosetti [27]).

3

Notation

4

The symbol Bn denotes the unit ball in Rn , namely Bn ≡ {x ∈ Rn : |x| < 1}. If D is a subset of Rn , then clD denotes its closure and ∂D denotes is boundary. Let Ω be an open subset of Rn . Let f be an H-valued function on Ω. For any x in Ω, ∂xj f (x) denotes the partial derivative of f at x with respect to xj for all j ∈ {1, . . . , n}, and ∂x f (x) ≡ (∂x1 f (x), . . . , ∂xn f (x))T , where T stands for transpose, and ∂xα f (x) ≡ ∂xα11 . . . ∂xαnn f (x) for any multi-

A family of fundamental solutions for quaternion coefficient operators

In this section, we state our main Theorem 1 concerning the construction of a family of fundamental solutions of elliptic partial differential operators with quaternion constant coefficients. The elements of such a family are expressed by means of jointly 2

B and C from EH (k, n) × Rn to H such that   x S(a, x) = |x|k−n A a, , |x| + log |x| B(a, x) + C(a, x) |x| ∀(a, x) ∈ EH (k, n) × Rn \ {0} .

analytic functions of the coefficients of the operators and of the spatial variable. Moreover, in Theorem 1 we give a detailed description of such a family in order to deduce regularity and jump properties of the corresponding layer potentials (cf. [28], [17]). Before doing so, we need to introduce some notation. If m, n ∈ N, m ≥ 1, n ≥ 2, then N (m, n) denotes the set of all multi-indexes α ≡ (α1 , . . . , αn ) ∈ Nn with |α| ≡ α1 + · · · + αn ≤ m. Similarly, H(m, n) denotes the set of quaternion functions a ≡ (aα )α∈N (m,n) defined in N (m, n). We identify H(m, n) with a finite dimensional vector space q on H and we endow H(m, n) with the norm P 2 |a| ≡ α∈N (m,n) |aα | . Then we set

The functions B and C are identically 0 if n is odd and there exist a sequence {fj }j∈N of real analytic functions from EH (k, n) × ∂Bn to H, and a family {bα }|α|≥sup{k−n,0} of real analytic functions from EH (k, n) to H, such that fj (a, −θ) = (−1)j+k fj (a, θ) ∀(a, θ) ∈ EH (k, n) × ∂Bn , and

EH (m, n)

 ≡

a ∈ H(m, n) : X

A(a, θ, r) =

aα ξ α 6= 0 ∀ξ ∈ ∂Bn .

aα ∂xα ,

(1)

∀ (a, θ, r) ∈ EH (k, n) × ∂Bn × R , X bα (a)xα (2)

B(a, x) =

|α|≥sup{k−n,0}

The set EH (m, n) is open in H(m, n). Finally, we set L[a] ≡

fj (a, θ)rj

j=0



α∈N (m,n) , |α|=m

X

∞ X

∀(a, x) ∈ EH (k, n) × Rn ,

∀a ∈ H(m, n) .

where the series in equalities (1) and (2) converge absolutely and uniformly in all compact subsets of EH (k, n) × ∂Bn × R and of EH (k, n) × Rn , respectively.

α∈N (2k,n)

If a ∈ H(m, n), then L[a] is a partial differential operator of order ≤ m with quaternion constant coefficients. If we further assume that a ∈ EH (m, n), then L[a] is a quaternion elliptic operator of order m. We are now in the position to state the following Theorem 1. For a proof, we refer to [16]. Here we just say that it is based on the corresponding result of [17] for real partial differential operators, which in turn exploits the construction of a fundamental solution for a real partial differential operator with analytic coefficients provided by John in [29, Chapter III].

5

The corresponding layer potential

single

This section is devoted to the presentation of some regularity properties for the single layer potential corresponding to the fundamental solution S(a, ·) of Theorem 1. We introduce some notation. Let m, n, k ∈ N, n ≥ 2, m, k ≥ 1. Let λ ∈]0, 1[. Let Ω be an open bounded subset of Rn of class C m,λ . Let a ∈ EH (k, n). Let µ ∈ C m−1,λ (∂Ω, H). Let β ∈ Nn and |β| ≤ k − 1. We introduce the function vβ [a, µ] from Rn to H by setting Z vβ [a, µ](x) ≡ ∂xβ S(a, x−y)µ(y) dσy ∀x ∈ Rn .

Theorem 1 Let k, n ∈ N, k ≥ 1, n ≥ 2. Then there exists a real analytic function S from EH (k, n)×(Rn \ {0}) to H such that S(a, ·) is a fundamental solution of the operator L[a] for all fixed a ∈ EH (k, n). Moreover, there exist a real analytic function A from EH (k, n) × ∂Bn × R to H, and real analytic functions

∂Ω

3

Here, the integral is understood in the sense of singular integrals if x ∈ ∂Ω and |β| = k − 1, and dσ denotes the area element. If β = (0, . . . , 0), we find convenient to set

where νΩ denotes the outward unit normal to the boundary of Ω.

v[a, µ] ≡ v(0,...,0) [a, µ] .

6

As a consequence, vβ [a, µ](x) =∂xβ v[a, µ](x) ∀x ∈ Rn \ ∂Ω , β ∈ Nn , |β| ≤ k − 1 . In the following Theorem 2, we state some regularity properties for the single layer potentials v[a, µ] and for the functions vβ [a, µ]. For a proof we refer to [16]. Here we say that the validity of the theorem follows by the results in [17], by the construction of S(a, ·), and by standard theorems of differentiation under the integral sign.

In this section, we consider the single layer potential corresponding to the fundamental solution of Theorem 1 in the case of complex partial differential operators of order two. We state a real analyticity result for the dependence of such a layer potential upon perturbation of the support of integration, of the density, and of the coefficients of the corresponding operator, which can be proved by exploiting the results by M. Lanza de Cristoforis and the first named author in [13].

Theorem 2 Let m, n, k ∈ N, n ≥ 2, m, k ≥ 1. Let λ ∈]0, 1[, and β ∈ Nn . Let Ω be a bounded open subset of Rn of class C m,λ . Let a ∈ EH (k, n). Let µ ∈ C m−1,λ (∂Ω, H). Then the following statements hold:

We introduce some notation. Let Ω be a bounded open connected subset of Rn of class C m,λ , for some integer m ≥ 1 and λ ∈]0, 1[, such that Rn \clΩ is connected. We consider Ω as a “base domain”. We denote by A∂Ω the set of functions of class C 1 (∂Ω, Rn ) which are injective and whose differential is injective at all points x ∈ ∂Ω. The set A∂Ω is open in C 1 (∂Ω, Rn ) (cf. Lanza de Cristoforis and Rossi [30, Cor. 4.24, Prop. 4.29], [11, Lem. 2.5]). Moreover, if φ ∈ A∂Ω , by the Jordan-Leray Separation Theorem one verifies that Rn \ φ(∂Ω) has exactly two open connected components and we denote by I[φ] the bounded connected component. Furthermore, φ(∂Ω) ≡ ∂I[φ]. If we further assume that φ ∈ A∂Ω ∩C m,λ (∂Ω, Rn ) then I[φ] is an open bounded subset of Rn of class C m,λ (cf. Lanza de Cristoforis and Rossi [12, §2]). In the sequel, φ(∂Ω) plays the role of the support of integration of our layer potentials.

(i) if k ≥ 2 and |β| ≤ k − 2, then vβ [a, µ] ∈ C k−2−|β| (Rn , H) and we have ∂xβ v[a, µ](x) = vβ [a, µ](x) for all x ∈ Rn ; (ii) if |β| = k − 1, then the restriction vβ [a, µ]|Ω has a unique continuous extension to a function vβ+ [a, µ] on clΩ and the map which takes µ to vβ+ [a, µ] is linear and continuous from C m−1,λ (∂Ω, H) to C m−1,λ (clΩ, H); (iii) if |β| = k − 1, then the restriction vβ [a, µ]|Rn \clΩ has a unique continuous extension to a function vβ− [a, µ] on Rn \ Ω and if R > 0 and clΩ ⊆ RBn , then the map which take µ to vβ− [a, µ]|cl(RBn )\Ω is linear and continuous from C m−1,λ (∂Ω, H) to C m−1,λ (cl(RBn ) \ Ω, H); (iv) if |β| = k − 1, then vβ± [a, µ](x) = ∓

2

We identify C with the subalgebra of H consisting of the quaternions z = z0 +iz1 , with z0 , z1 ∈ R. Then for each k, n ∈ N, k ≥ 1, n ≥ 2, we set C(k, n) ≡ {a = (aα )α∈N (k,n) ∈ H(k, n) : aα ∈ C ∀α ∈ N (k, n)}

νΩ (x)β µ(x) α α∈N (k,n),|α|=k aα (νΩ (x))

P

+ vβ [a, µ](x)

An application to complex elliptic partial differential operators of order two

∀x ∈ ∂Ω , 4

and

Technology (FCT) via the post-doctoral grant SFRH/BPD/64437/2009. This work was supE˜C (k, n) ≡ a = (aα )α∈N (k,n) ∈ C(k, n) : ported by FEDER funds through COMPETE– Operational Programme Factors of Competitiveness X   α (“Programa Operacional Factores de CompetitiviSc aα ξ > 0 ∀ξ ∈ ∂Bn . dade”) and by Portuguese funds through the Cen|α|=k ter for Research and Development in MathematOne verifies that C(k, n) is a finite dimensional com- ics and Applications (University of Aveiro) and plex vector space and E˜C (k, n) is an open subset of the Portuguese Foundation for Science and TechC(k, n). Also, E˜C (k, n) is non-empty if and only if k nology (“FCT–Funda¸c˜ao para a Ciˆencia e a Tecis even and L[a] is an elliptic partial differential op- nologia”), within project PEst-C/MAT/UI4106/2011 erator of order k with complex constant coefficients with COMPETE number FCOMP-01-0124-FEDERfor all a ∈ E˜C (k, n). 022690. Partial support from the Foundation for If µ is a function from ∂Ω to C and φ ∈ A∂Ω ∩ Science and Technology (FCT) via the post-doctoral C m,λ (∂Ω, Rn ), one can consider the function µ◦φ(−1) grant SFRH/BPD/66342/2009 is also acknowledged defined on φ(∂Ω). As a consequence, it makes sense by the second named author. The third named auto define the single layer potential thor acknowledges the financial support of the “FonZ dazione Ing. Aldo Gini”. v[a, φ, µ](x) ≡ S(a, x − y)µ ◦ φ(−1) (y) dσy 

φ(∂Ω)

∀x ∈ Rn , and the function V [a, φ, µ](ξ) ≡ v[a, φ, µ] ◦ φ(ξ)

References [1] G. Fichera, Rend. Mat. e Appl. (5) 17, 82–191 (1958).

∀ξ ∈ ∂Ω

for all (a, φ, µ) ∈ E˜C (2, n) × (A∂Ω ∩ C (∂Ω, Rn )) × m−1,λ C (∂Ω, C). We now confine ourselves to the case k = 2. Then we have the following Theorem 3 (see [16]).

[2] C. Miranda, Partial differential equations of elliptic type, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Springer-Verlag, New York, 1970.

Theorem 3 Let m, n ∈ N, m ≥ 1, n ≥ 2. Let λ ∈]0, 1[, and Ω be a bounded open subset of Rn of class C m,λ such that both Ω and Rn \ clΩ are connected. Then the map V [·, ·, ·] from E˜C (2, n)×(A∂Ω ∩ C m,λ (∂Ω, Rn )) × C m−1,λ (∂Ω, C) to C m,λ (∂Ω, C) is real analytic.

[3] V. D. Kupradze, T. G. Gegelia, M. O. Bashele˘ıshvili, and T. V. Burchuladze, Threedimensional problems of the mathematical theory of elasticity and thermoelasticity, vol. 25 of North-Holland Series in Applied Mathematics and Mechanics, North-Holland Publishing Co., Amsterdam, 1979.

m,λ

We note that Theorem 3 is an immediate consequence of [13, Thm. 5.6]. Moreover, analogous results can be proved for the double layer potential and for other integral operators related to the single layer potential (see [16]).

[4] R. Potthast, Inverse Problems 10, 431–447 (1994). [5] R. Potthast, Math. Methods Appl. Sci. 19, 1157– 1175 (1996a).

Acknowledgments

[6] R. Potthast, J. Inverse Ill-Posed Probl. 4, 67–84 (1996b).

The first named author acknowledges financial support from the Foundation for Science and

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