Analysis Of Two-operator Boundary-domain Integral Equations For Variable-coefficient Boundary Value Problems In 2d

In this dissertation, the boundary value problems (BVPs) for the second orderrnelliptic partial differential equation with variable coefficient in two-dimensionalrnbounded domain is considered. Using an appropriate parametrix (Levi function)rnand applying the two-operator approach (where the one operator approachrnfails ), the problems are reduced to some systems of boundary-domainrnintegral equations (BDIEs). The two-operator BDIEs in 2D need a special considerationrndue to their different equivalence properties as compared to higherrndimensional case due to the logarithmic term in the parametrix for the associatedrnpartial differential equation. Consequently, we need to set conditionsrnon the domain or function spaces to insure the invertibility of the correspondingrnlayer potentials, and hence the unique solvability of BDIEs. Equivalence ofrnthe two-operator BDIE systems to the original BVPs, BDIEs solvability, uniqueness/rnnon uniqueness of the solution, as well as Fredholm property and invertibilityrnof the BDIEs are analysed. Moreover, the two-operator boundary domainrnintegral operators for the Neumann BVP are not invertible, and appropriaternfinite-dimensional perturbations are constructed leading to invertibility of thernperturbed operators.

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