Open Mathematics (Apr 2017)
Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
Abstract
We consider a commutative algebra š¹ over the field of complex numbers with a basis {e1, e2} satisfying the conditions (e12+e22)2=0,e12+e22ā 0. $ (e_{1}^{2}+e_{2}^{2})^{2}=0, e_{1}^{2}+e_{2}^{2}\neq 0. $ Let D be a bounded simply-connected domain in ā2. We consider (1-4)-problem for monogenic š¹-valued functions Ī¦(xe1 + ye2) = U1(x, y)e1 + U2(x, y)i e1 + U3(x, y)e2 + U4(x, y)i e2 having the classic derivative in the domain DĪ¶ = {xe1 + ye2 : (x, y) ā D}: to find a monogenic in DĪ¶ function Ī¦, which is continuously extended to the boundary āDĪ¶, when values of two component-functions U1, U4 are given on the boundary āD. Using a hypercomplex analog of the Cauchy type integral, we reduce the (1-4)-problem to a system of integral equations on the real axes. We establish sufficient conditions under which this system has the Fredholm property and the unique solution. We prove that a displacements-type boundary value problem of 2-D isotropic elasticity theory is reduced to (1-4)-problem with appropriate boundary conditions.
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