Generalized double-layer potentials in anisotropic elasticity on the plane (Q498184)

The author studies the vector \(u=(u_{1},u_{2})\), solution of a elliptic Lamé system: \[ a_{11} u_{xx} +\left\{ a_{12} + a_{21}\right\} u_{xy} + a_{22} u_{yy} = 0 \] which represents an anisotropic elasticity problem on the plane. In this system the coefficients \(a_{ij},(i,j=1,2)\) are matrix of order two; the elements \(\alpha_{j}\) of this matrix are called elastic moduli and they form a third-order matrix \(\alpha\) that is positive definite. Setting the vector function \(v= (v_{1},v_{2})\) such that \[ \begin{aligned} v_{x} &= - ( a_{21} u_{x} + a_{22} u_{y} )\\ v_{y}& = ( a_{11} u_{x} + a_{12} u_{y} )\end{aligned} \] this vector function is called the adjoint of the solution u to the Lamé system. Firstly, the author proves that the space \( h^{p} (D) \) (the Hardy class of the harmonic functions), \(p>1\) of solutions to the Lamé system and the corresponding space of their adjoints are Banach spaces. Then, he proves that the Dirichlet problem \(u^{+} = f\) for the Lamé system is uniquely solvable in the class \(h^{p} (D),p>1\). In the following part of this paper, approaching to the study of boundary value problems in anisotropic elasticity on the plane with the classical potential method, developed by Kupradze and Basheleishvili, representing the general solution to the Lamé system in terms of functions analytic and introducing double-layer potentials, the author presents final results explicitly describing these potentials only in terms of elastic moduli and simple symmetric combinations of the roots of the characteristic equation in the general anisotropic case. Finally, he proves some regularity results of the solution u of the Lamé system and of the adjoint of the solution \(u\).