monogenic initial function u0 in Banach spaces of monogenic functions equipped with an Lp-norm. The present paper is aimed at solving the same initial value.
Contained in the collection of papers: Proceedings of the 20th international conference on finite and infinite dimensional complex analysis and applications (ICFIDCAA), Hanoi University of Science and Technology, Hanoi, Vietnam, July 29 – August 3, 2012, Edts. L. H. Son and W. Tutschke, Science and Technics Publishing House, Hanoi, 2012.
Solution of Initial Value Problems for Generalized Monogenic Functions in Banach Spaces with a Weighted Sup-Norm U˘gur Y¨ uksel Atilim University, Ankara Abstract Consider the initial value problem ∑ (A) ∂u(t, x) ∂uB (t, x) = Lu(t, x) := CB,i (t, x) eA , ∂t ∂xi (1) A,B,i u(0, x) = u0 (x) ∑ where the desired function u(t, x) = uB (t, x)eB defined in [0, T ] × Ω ⊂ B n+1 R+ is a Clifford-Algebra-valued function with real-valued com0 ×R ponents uB (t, x). This paper is aimed at solving the initial value problem (1) by the contraction mapping principle in Banach Spaces of monogenic functions equipped with a weighted sup-norm.
1
Introduction
It has been recently proved by the method of associated spaces [4] in the paper [8] that the initial value problem (1) has a unique solution for an arbitrary monogenic initial function u0 in Banach spaces of monogenic functions equipped with an Lp -norm. The present paper is aimed at solving the same initial value problem in Banach spaces of monogenic functions equipped with a weighted sup-norm by the contraction mapping principle in conical domains.
2
Preliminaries
Suppose that A is the Clifford-Algebra generated by e0 , e1 , ..., en , where e0 = 1, e2j = −1, ei ej + ej ei = −2δij , for i, j = 1, ..., n. Suppose, further, that Ω is a bounded domain in Rn+1 whose points are denoted by x = (x0 , x1 , ..., xn ). Regard an A-valued function ∑ u(t, x) = uA (t, x)eA A
1
defined in [0, T ] × Ω with 2n real-valued components uA (t, x), where, as usual A runs in the set S = {0, 1, ..., n, 12, ..., 12...n}. It is called (left-)monogenic if it satisfies the equation Du = 0, where
∑ ∂ ∂ + ej ∂x0 j=1 ∂xj n
D=
is the Cauchy-Riemann operator of Clifford-Analysis (cf. [1]). Similarly the conjugate Cauchy-Riemann operator of Clifford-Analysis is defined as n ∑ ∂ ∂ D= − . ej ∂x0 j=1 ∂xj Obviously, since DD = ∆, real-valued components of a monogenic function are harmonic functions. For further definitions concerning with monogenic functions we refer the reader to [1, 2].
3
Sufficient conditions for associated pairs
In order to investigate for which operators the initial value problem (1) is soluble with arbitrary monogenic initial functions, one has to determine operators L which are associated with the operator D. Definition 1 (see [4, 5]) Let F be a first order differential operator depending on t, x, u and on the first order partial derivatives ∂xi u, while G is a differential operator with respect to the space variables xi with coefficients not depending on time t. Then F is said to be ’associated’ with G if F transforms the set of all solutions to the differential equation Gu = 0 into solutions of the same equation for fixedly chosen t, i.e. Gu = 0 ⇒ G (Fu) = 0. The function space containing all solutions to the differential equation Gu = 0 is called an associated space of F. Theorem 2 The operator L is associated with the Cauchy-Riemann operator D provided the coefficients satisfy the following systems [ ( )] ∑ ∑ (A) (A) kl,F,D CD,j − kj,D,B CB,0 D
[
B
( )] ∑ ∑ (E) (E) kl,E,A CF,j − kj,F,B CB,0 − =0 E
B
2
for all A, F ∈ S and j, l = 1, 2, ..., n with j ≤ l, ( ) ∑ ∂ (A) (A) CD,j − kj,D,B CB,0 + ∂x0 B [ ( )] n ∑ ∑ ∑ ∂ (E) (E) km,E,A CD,j − kj,D,B CB,0 =0 ∂xm E m=1 B for all A, D ∈ S and j = 1, 2, ..., n. Proof. See [8].
4
Solution of the initial value problem by the contraction mapping principle
Consider the initial value problem (1), where the initial function u0 is a monogenic function. Suppose the operator L is associated with the operator D (see [8] for some special cases and examples of associated pairs). Since the desired solution is a fixed point of the operator ∫t Lu(τ, x)dτ,
U (t, x) = u0 (x) +
(2)
0
the initial value problem under consideration can be solved by the fixed-point methods applied to suitable function spaces. In the classical case of the CauchyKovalevsky problem this was done by Walter [7] by using a weighted sup-norm. In the case under hand one has to replace the space of holomorphic functions by the space of monogenic functions as will be sketched in the following: ∑ uA (t, x)eA defined in The space of all monogenic functions u(t, x) = A
ΩT = [0, T ] × Ω and equipped with the norm ( ) ∥u∥ = max sup |uA | A
ΩT
turns out to be a Banach space. Denote this space by B. Since L is a linear differential operator it satisfies a Lipschitz type condition provided the coefficients (A) CB,i (t, x) of L are bounded. Note that for the components uA of a monogenic function u in the space B the interior estimate |∂xi uA (t, x)| ≤
n+1 ∥u∥ dist(x, ∂Ω) 3
(3)
holds (see Theorem 1 of [6]). Next introduce the conical domain Mη = {(t, x) : x ∈ Ω, 0 ≤ t ≤ ηd(x)} , where d(x) = dist(x, ∂Ω) and η will be fixed later. Then d(t, x) = d(x) −
t η
is some kind of pseudo-distance measuring the distance of a point (t, x) ∈ Mη from ∑ the lateral surface of Mη . Consider the monogenic functions u(t, x) = uA (t, x)eA defined in Mη . Let B(Mη ) be the Banach space of such functions A
(
whose norm
)
∥u∥∗ = max sup |uA (t, x)| d (t, x) p
A
Mη
is finite, where p > 1 is fixedly chosen. Since L is associated with D, the operator (2) maps B(Mη ) into itself. Using the interior estimate (3), one can show that the operator (2) is contractive provided η is small enough. Applying the contraction-mapping principle one has thus proved the following theorem (cf. Section 4 and the Theorem 2 of [6]): Theorem 3 Suppose the operator L is associated with the Cauchy-Riemann operator D and the initial function u0 is a monogenic function. If the height η of the conical domain Mη is small enough then there exists a uniquely determined solution u(t, x) of the initial value problem (1) being monogenic for each t.
References [1] F. Brackx, R. Delanghe and F. Sommen, Clifford analysis, Pitman, London, 1982. [2] E. Obolashvili, Partial differential equations in Clifford analysis, Addison Wesley Longman, Harlow, 1998. [3] L. H. Son, W. Tutschke, Eds., Function Spaces in Complex and Clifford Analysis, National University Publ., Hanoi (2008). [4] W. Tutschke, Solution of initial value problems in classes of generalized analytic functions, Teubner Leipzig and Springer Verlag, 1989. [5] W. Tutschke, Associated spaces - a new tool of real and complex analysis, Contained in [3], 253-268. 4
[6] W. Tutschke and N. Thanh Van: Interior estimates in the sup-norm for generalized monogenic functions satisfying a differential equation with an anti-monogenic right-hand side. Complex Variables and Elliptic Equations, 52 (2007), No. 5, 367–375. [7] W. Walter, An elementary proof of the Cauchy-Kovalevsky theorem, Amer. Math. Monthly, 92 (1985), 115–125. [8] U. Y¨ uksel, Solution of initial value problems with monogenic initial functions in Banach spaces with Lp -norm, Adv. appl. Clifford alg., Vol. 20, 201–209 (2010).
5
