In this paper we prove a Cauchy-Kowalevski theorem for the functions satisfying the system ∂xf∂x = 0 (called inframonogenic functions)

Malonek, Helmuth R.

Peña, Dixan Peña

Sommen, Frank

Okayama University Digital Information Repository

A CAUCHY-KOWALEVSKI THEOREM FOR INFRAMONOGENIC FUNCTIONS

In this paper we prove a Cauchy-Kowalevski theorem for inframonogenic functions

Malonek, Helmuth

Peña Peña, Dixan

Sommen, Franciscus

Ghent University Academic Bibliography

A Cauchy-Kowalevski theorem for inframonogenic functions

Okayama University Scientific Achievement Repository

Math. J. Okayama Univ. 53 (2011), 167–172A CAUCHY-KOWALEVSKI THEOREM FORINFRAMONOGENIC FUNCTIONSHelmuth R. MALONEK, Dixan PEN˜A PEN˜A and Frank SOMMENAbstract. In this paper we prove a Cauchy-Kowalevski theorem forthe functions satisfying the system ∂xf∂x = 0 (called inframonogenicfunctions).1. IntroductionLet R0,m be the 2m-dimensional real Clifford algebra constructed overthe orthonormal basis (e1, . . . , em) of the Euclidean space Rm (see [3]). Themultiplication in R0,m is determined by the relations ejek + ekej = −2δjk,j, k = 1, . . . ,m, where δjk is the Kronecker delta. A general element of R0,mis of the forma =∑AaAeA, aA ∈ R,where for A = {j1, . . . , jk} ⊂ {1, . . . ,m}, j1 < · · · < jk, eA = ej1 . . . ejk . Forthe empty set ∅, we put e∅ = 1, the latter being the identity element.Notice that any a ∈ R0,m may also be written as a =∑mk=0[a]k where [a]kis the projection of a on R(k)0,m. Here R(k)0,m denotes the subspace of k-vectorsdefined byR(k)0,m ={a ∈ R0,m : a =∑|A|=kaAeA, aA ∈ R}.Observe that Rm+1 may be naturally identified with R(0)0,m ⊕R(1)0,m by associ-ating to any element (x0, x1, . . . , xm) ∈ Rm+1 the “paravector” x = x0+x =x0 +∑mj=1 xjej .Conjugation in R0,m is given bya =∑AaAeA,where eA = ejk . . . ej1 , ej = −ej , j = 1, . . . ,m. One easily checks thatab = ba for any a, b ∈ R0,m. Moreover, by means of the conjugation a normMathematics Subject Classification. 30G35, 35G10.Key words and phrases. Inframonogenic functions, Cauchy-Kowalevski theorem.The second author was supported by a Post-Doctoral Grant of Fundac¸a˜o para a Cieˆnciae a Tecnologia, Portugal (grant number: SFRH/BPD/45260/2008).167168 H. R. MALONEK, D. PEN˜A PEN˜A AND F. SOMMEN|a| may be defined for each a ∈ R0,m by putting|a|2 = [aa]0 =∑Aa2A.Let us denote by ∂x = ∂x0 + ∂x = ∂x0 +∑mj=1 ej∂xj the generalized Cauchy-Riemann operator and let Ω be an open set of Rm+1. According to [11], anR0,m-valued function f ∈ C2(Ω) is called an inframonogenic function in Ωif and only if it fulfills in Ω the “sandwich” equation ∂xf∂x = 0.It is obvious that monogenic functions (i.e. null-solutions of ∂x) are in-framonogenic. At this point it is worth mentioning that the monogenicfunctions are the central object of study in Clifford analysis (see [2, 4, 5, 7,8, 9, 10, 14]). Furthermore, the concept of monogenicity of a function maybe seen as the higher dimensional counterpart of holomorphy in the complexplane.Moreover, as∆x =m∑j=0∂2xj = ∂x∂x = ∂x∂x,every inframonogenic function f ∈ C4(Ω) satisfies in Ω the biharmonic equa-tion ∆2xf = 0 (see e.g. [1, 6, 12, 15]).This paper is intended to study the following Cauchy-type problem forthe inframonogenic functions. Given the functions A0(x) and A1(x) analyticin an open and connected set Ω ⊂ Rm, find a function F (x) inframonogenicin some open neighbourhood Ω˜ of Ω in Rm+1 which satisfiesF (x)|x0=0 = A0(x),(1.1)∂x0F (x)|x0=0 = A1(x).(1.2)2. Cauchy-type problem for inframonogenic functionsConsider the formal series(2.1) F (x) =∞∑n=0xn0An(x).It is clear that F satisfies conditions (1.1) and (1.2). We also see at oncethat∂x (xn0An) ∂x = n(n− 1)xn−20 An + nxn−10(∂xAn +An∂x)+ xn0∂xAn∂x.We thus get∂xF∂x =∞∑n=0xn0((n+2)(n+1)An+2+(n+1)(∂xAn+1+An+1∂x)+∂xAn∂x).A CAUCHY-KOWALEVSKI THEOREM FOR INFRAMONOGENIC FUNCTIONS 169From the above it follows that F is inframonogenic if and only if the functionsAn satisfy the recurrence relationAn+2 = −1(n+ 2)(n+ 1)((n+ 1)(∂xAn+1 +An+1∂x)+ ∂xAn∂x), n ≥ 0.It may be easily proved by induction that(2.2) An =(−1)n+1n!n−2∑j=0∂n−j−1x A0∂j+1x +n−1∑j=0∂n−j−1x A1∂jx , n ≥ 2.We now proceed to examine the convergence of the series (2.1) with thefunctions An (n ≥ 2) given by (2.2). Let y be an arbitrary point in Ω. Thenthere exist a ball B(y,R(y))of radius R(y) centered at y and a positiveconstant M(y), such that∣∣∂n−jx As(x)∂jx∣∣ ≤M(y) n!Rn(y) , x ∈ B(y,R(y)), j = 0, . . . , n, s = 0, 1.It follows that|An(x)| ≤M(y)n+R(y)− 1Rn(y), x ∈ B(y,R(y)),and therefore the series (2.1) converges normally inΩ˜ =⋃y∈Ω(−R(y), R(y))×B(y,R(y)).Note that Ω˜ is a x0-normal open neighbourhood of Ω in Rm+1, i.e. for eachx ∈ Ω˜ the line segment {x + t : t ∈ R} ∩ Ω˜ is connected and contains onepoint in Ω.We thus have proved the following.Theorem 2.1. The function CK[A0, A1] given by(2.3) CK[A0, A1](x) = A0(x) + x0A1(x)−∞∑n=2(−x0)nn!n−2∑j=0∂n−j−1x A0(x)∂j+1x +n−1∑j=0∂n−j−1x A1(x)∂jxis inframonogenic in a x0-normal open neighbourhood of Ω in Rm+1 andsatisfies conditions (1.1)-(1.2).It is worth noting that if in particular A1(x) = −∂xA0(x), thenCK[A0,−∂xA0](x) =∞∑n=0(−x0)nn!∂nxA0(x),170 H. R. MALONEK, D. PEN˜A PEN˜A AND F. SOMMENwhich is nothing else but the left monogenic extension (or CK-extension)of A0(x). Similarly, it is easy to see that CK[A0,−A0∂x](x) yields the rightmonogenic extension of A0(x) (see [2, 5, 13, 16, 17]).Let P(k) (k ∈ N0 fixed) denote the set of all R0,m-valued homogeneouspolynomials of degree k in Rm. Let us now take A0(x) = Pk(x) ∈ P(k) andA1(x) = Pk−1(x) ∈ P(k − 1). Clearly,CK[Pk, Pk−1](x) = Pk(x) + x0Pk−1(x)−k∑n=2(−x0)nn!n−2∑j=0∂n−j−1x Pk(x)∂j+1x +n−1∑j=0∂n−j−1x Pk−1(x)∂jx ,since the other terms in the series (2.3) vanish. Moreover, we can also claimthat CK[Pk, Pk−1](x) is a homogeneous inframonogenic polynomial of degreek in Rm+1.Conversely, if Pk(x) is a homogeneous inframonogenic polynomial of de-gree k in Rm+1, then Pk(x)|x0=0 ∈ P(k), ∂x0Pk(x)|x0=0 ∈ P(k − 1) andobviously CK[Pk|x0=0, ∂x0Pk|x0=0](x) = Pk(x).Call I(k) the set of all homogeneous inframonogenic polynomials of degreek in Rm+1. Then CK[., .] establishes a bijection between P(k)×P(k−1) andI(k).It is easy to check thatPk(x) = Pk(∂u)〈x, u〉kk!, Pk(x) ∈ P(k),where Pk(∂u) is the differential operator obtained by replacing in Pk(u)each variable uj by ∂uj . Therefore, in order to characterize I(k), it sufficesto calculate CK[〈x, u〉k eA, 0]and CK[0, 〈x, u〉k−1 eA]with u ∈ Rm.A simple computation shows thatCK[〈x, u〉k eA, 0](x) = 〈x, u〉k eA−k∑n=2(kn)(−x0)n 〈x, u〉k−nn−2∑j=0un−j−1eAuj+1 ,CK[0, 〈x, u〉k−1 eA](x) = x0 〈x, u〉k−1 eA−1kk∑n=2(kn)(−x0)n 〈x, u〉k−nn−1∑j=0un−j−1eAuj .A CAUCHY-KOWALEVSKI THEOREM FOR INFRAMONOGENIC FUNCTIONS 171References[1] S. Bock and K. Gu¨rlebeck, On a spatial generalization of the Kolosov-Muskhelishviliformulae, Math. Methods Appl. Sci. 32 (2009), no. 2, 223–240.[2] F. Brackx, R. Delanghe and F. Sommen, Clifford analysis, Research Notes in Math-ematics, 76, Pitman (Advanced Publishing Program), Boston, MA, 1982.[3] W. K. Clifford, Applications of Grassmann’s Extensive Algebra, Amer. J. 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Malonek, Selected topics in hypercomplex function theory, Clifford algebras andpotential theory, 111–150, Univ. Joensuu Dept. Math. Rep. Ser., 7, Univ. Joensuu,Joensuu, 2004.[11] H. R. Malonek, D. Pen˜a Pen˜a and F. Sommen, Fischer decomposition by inframono-genic functions, to appear in CUBO, A Mathematical Journal.[12] V. V. Meleshko, Selected topics in the history of the two-dimensional biharmonicproblem, Appl. Mech. Rev. 56 (2003), no. 1, 33–85.[13] D. Pen˜a Pen˜a, Cauchy-Kowalevski extensions, Fueter’s theorems and boundary valuesof special systems in Clifford analysis, Ph.D. Thesis, Ghent University, Ghent, 2008(available at http://hdl.handle.net/1854/11636).[14] J. Ryan, Basic Clifford analysis, Cubo Mat. Educ. 2 (2000), 226–256.[15] L. Sobrero, Theorie der ebenen Elastizita¨t unter Benutzung eines Systems hyperkom-plexer Zahlen, Hamburg. Math. Einzelschriften, Leipzig, 1934.[16] F. Sommen, Monogenic functions on surfaces, J. Reine Angew. Math. 361 (1985),145–161.[17] F. Sommen and B. Jancewicz, Explicit solutions of the inhomogeneous Dirac equation,J. Anal. Math. 71 (1997), 59–74.Helmuth R. MalonekDepartment of MathematicsAveiro University3810-193 AveiroPortugale-mail address: hrmalon@ua.pt172 H. R. MALONEK, D. PEN˜A PEN˜A AND F. SOMMENDixan Pen˜a Pen˜aDepartment of MathematicsAveiro University3810-193 AveiroPortugale-mail address: dixanpena@ua.pt; dixanpena@gmail.comFrank SommenDepartment of Mathematical AnalysisGhent University9000 GentBelgiume-mail address: fs@cage.ugent.be(Received May 9, 2009 )

A CAUCHY-KOWALEVSKI THEOREM FOR INFRAMONOGENIC FUNCTIONS

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