Abstract
We mainly deal with the boundary value problem for triharmonic function with value in a universal Clifford algebra: Δ3[u](x) = 0, x ∈ R n∖∂Ω, u +(x) = u −(x)G(x) + g(x), x ∈ ∂Ω, (D j u)+(x) = (D j u)−(x)A j + f j(x), x ∈ ∂Ω, u(∞) = 0, where (j = 1,…, 5) ∂Ω is a Lyapunov surface in R n, D = ∑k=1 n e k(∂/∂x k) is the Dirac operator, and u(x) = ∑A e A u A(x) are unknown functions with values in a universal Clifford algebra Cl(V n,n). Under some hypotheses, it is proved that the boundary value problem has a unique solution.
Highlights
R4 2)
Using the condition (D3u)+(x) = (D3u)−(x)A3 + f3(x), x ∈ ∂Ω, we obtain that φ2+ (x) = φ2− (x) A3 + f3 (x), x ∈ ∂Ω, (37)
Letting T denote the integral operator defined by the right hand side of (49), we get (Tφ6)
Summary
The theory of Riemann boundary value problems in complex plane has been systematically developed in [1, 2]. It is an interesting topic to generalize the classical Riemann boundary value problems theory to Clifford analysis. The aim of this paper is to study the Riemann boundary value problem for triharmonic functions. On the basis of the above results, we consider the following Riemann boundary value problems: Δ3 [u] (x) = 0, x ∈ Rn \ ∂Ω, u+ (x) = u− (x) A + g (x) , x ∈ ∂Ω, (1). Let Ω be an open nonempty subset of Rn with a Lyapunov boundary, u(x) = ∑A eAuA(x), where uA(x) are real functions; u(x) is called a Holder continuous function on Ω if the following condition is satisfied:. Let Hα(∂Ω, Cl(Vn,n)) denote the set of Holder continuous functions with values in Cl(Vn,n is α, 0 < α < 1).
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