O. Chkadua and D. Natroshvili

Localized Boundary-Domain Integral Equations Approach for Dirichlet Problem of the Theory of Piezo-Elasticity for Inhomogeneous Solids

abstract:
The paper deals with the three-dimensional Dirichlet boundary-value problem (BVP) of piezo-elasticity theory for anisotropic inhomogeneous solids and develops the generalized potential method based on the localized parametrix method. Using Green's integral representation formula and properties of the localized layer and volume potentials we reduce the Dirichlet BVP to the localized boundary-domain integral equations (LBDIE) system. The equivalence between the Dirichlet BVP and the corresponding LBDIE system is studied. We establish that the obtained localized boundary-domain integral operator belongs to the Boutet de Monvel algebra and with the help of the Wiener-Hopf factorization method we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate function spaces.

Mathematics Subject Classification: 35J57, 31B10, 45F15, 47G30, 47G40, 74E05, 74E10, 74F15

Key words and phrases: Piezo-elasticity, strongly elliptic systems, variable coefficients, boundary value problem, localized parametrix, localized boundary-domain integral equations, pseudodifferential operators