• integral equations;
  • integro-differential equations;
  • parametrix;
  • partial differential equations;
  • variable coefficients;
  • mixed boundary-value problem;
  • Sobolev spaces;
  • equivalence;
  • invertibility


The mixed (Dirichlet–Neumann) boundary-value problem for the ‘Laplace’ linear differential equation with variable coefficient is reduced to boundary-domain integro-differential or integral equations (BDIDEs or BDIEs) based on a specially constructed parametrix. The BDIDEs/BDIEs contain integral operators defined on the domain under consideration as well as potential-type operators defined on open sub-manifolds of the boundary and acting on the trace and/or co-normal derivative of the unknown solution or on an auxiliary function. Some of the considered BDIDEs are to be supplemented by the original boundary conditions, thus constituting boundary-domain integro-differential problems (BDIDPs). Solvability, solution uniqueness, and equivalence of the BDIEs/BDIDEs/BDIDPs to the original BVP, as well as invertibility of the associated operators are investigated in appropriate Sobolev spaces. Copyright © 2005 John Wiley & Sons, Ltd.