SUMMARY

The mixed (Dirichlet–Neumann) boundary-value problem for the ‘Laplace ’ linear dierential equa-tion with variable coecient is reduced to boundary-domain integro-dierential or integral equations (BDIDEs or BDIEs) based on a specially constructed parametrix. The BDIDEs=BDIEs contain inte-gral operators dened on the domain under consideration as well as potential-type operators dened 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-dierential 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. KEY WORDS: integral equations; integro-dierential equations; parametrix; partial dierential equations; variable coecients; mixed boundary-value problem; Sobolev spaces; equivalence; invertibility 1