O. Chkadua

Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, II: Solution regularity and asymptotics

Mapping and invertibility properties of some parametrix-based surface and volume potentials are studied in Bessel-potential and Besov spaces. These results are then applied to derive regularity and asymptotics of the solution to a system of boundary-domain integral equations associated with a mixed BVP for a variablecoefficient PDE, in a vicinity of the curve of change of the boundary condition type.

Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed bvps in exterior domains

Direct segregated systems of boundary-domain integral equations are formulated for the mixed (Dirichlet–Neumann) boundary value problems for a scalar second-order divergent elliptic partial differential equation with a variable coefficient in an exterior three-dimensional domain. The boundary-domain integral equation system equivalence to the original boundary value problems and the Fredholm properties and invertibility of the corresponding boundary-domain integral operators are analyzed in weighted Sobolev spaces suitable for infinite domains. This analysis is based on the corresponding properties of the BVPs in weighted Sobolev spaces that are proved as well.

Crack-Type Boundary Value Problems of Electro-Elasticity

Dirichlet, Neumann and mixed crack-type boundary value problems of statics are considered in three-dimensional bounded domains filled with a homogeneous anisotropic electro-elastic medium. Applying the method of the potential theory and the theory of pseudodifferential equations, we prove the existence and uniqueness theorems in Besov and Bessel potential spaces, and derive full asymptotic expansion of solutions near the crack edge.

Localized boundary-domain integral equation formulation for mixed type problems

Abstract Some modifed direct localized boundary-domain integral equations (LBDIEs) systems associated with the mixed boundary value problem (BVP) for a scalar “Laplace” PDE with variable coefficient are formulated and analyzed. The main results established in the paper are the LBDIEs equivalence to the original variable-coefficient BVPs and the invertibility of the corresponding localized boundary-domain integral operators in appropriately chosen function spaces.

Analysis of Segregated Boundary–Domain Integral Equations for Mixed Variable-Coefficient BVPs in Exterior Domains

Some direct segregated systems of boundary–domain integral equations (LBDIEs) associated with the mixed boundary value problems for scalar PDEs with variable coefficients in exterior domains are formulated and analyzed in the paper. The LBDIE equivalence to the original boundary value problems and the invertibility of the corresponding boundary–domain integral operators are proved in weighted Sobolev spaces suitable for exterior domains. This extends the results obtained by the authors for interior domains in non-weighted Sobolev spaces.

Singular Localised Boundary-Domain Integral Equations of Acoustic Scattering by Inhomogeneous Anisotropic Obstacle

We consider the time-harmonic acoustic wave scattering by a bounded {\it anisotropic inhomogeneity} embedded in an unbounded {\it anisotropic} homogeneous medium. The material parameters may have discontinuities across the interface between the inhomogeneous interior and homogeneous exterior regions. The corresponding mathematical problem is formulated as a transmission problems for a second order elliptic partial differential equation of Helmholtz type with discontinuous variable coefficients. Using a localised quasi-parametrix based on the harmonic fundamental solution, the transmission problem for arbitrary values of the frequency parameter is reduced equivalently to a system of {\it singular localised boundary-domain integral equations}. Fredholm properties of the corresponding {\it localised boundary-domain integral operator} are studied and its invertibility is established in appropriate Sobolev-Slobodetskii and Bessel potential spaces, which implies existence and uniqueness results for the localised boundary-domain integral equations system and the corresponding acoustic scattering transmission problem.

Screen type mixed boundary value problems for anisotropic pseudo-Maxwell’s equations

Abstract We investigate screen type mixed boundary value problems for anisotropic pseudo-Maxwell’s equations. We show that the problems with tangent traces are well posed in tangent Sobolev spaces. The unique solvability results are proven based on the potential method and coercivity result of Costabel on the bilinear form associated with pseudo-Maxwell’s equations.

Potential Methods for Anisotropic Pseudo-Maxwell Equations in Screen Type Problems

We investigate the Neumann type boundary value problems for anisotropic pseudo-Maxwell equations in screen type problems. It is shown that the problem is well posed in tangent Sobolev spaces and unique solvability and regularity results are obtained via potential methods and the coercivity result of Costabel on the bilinear form associated to pseudo-Maxwell equations.

Analysis of Some Localized Boundary–Domain Integral Equations for Transmission Problems with Variable Coefficients

Some segregated systems of direct localized boundary-domain integral equations (LBDIEs) associated with several transmission problems for scalar PDEs with variable coefficients are formulated and analyzed for a bounded domain composed of two subdomains with a coefficient jump over the interface. The main results established in the paper are the LBDIE equivalence to the original transmission problems and the invertibility of the corresponding localized boundary-domain integral operators in corresponding Sobolev spaces function spaces.