**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