Analysis of Two-Operator Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems in 2D
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Date
2020-10-24
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Addis Ababa University
Abstract
In this dissertation, the boundary value problems (BVPs) for the second order
elliptic partial differential equation with variable coefficient in two-dimensional
bounded domain is considered. Using an appropriate parametrix (Levi function)
and applying the two-operator approach (where the one operator approach
fails ), the problems are reduced to some systems of boundary-domain
integral equations (BDIEs). The two-operator BDIEs in 2D need a special consideration
due to their different equivalence properties as compared to higher
dimensional case due to the logarithmic term in the parametrix for the associated
partial differential equation. Consequently, we need to set conditions
on the domain or function spaces to insure the invertibility of the corresponding
layer potentials, and hence the unique solvability of BDIEs. Equivalence of
the two-operator BDIE systems to the original BVPs, BDIEs solvability, uniqueness/
non uniqueness of the solution, as well as Fredholm property and invertibility
of the BDIEs are analysed. Moreover, the two-operator boundary domain
integral operators for the Neumann BVP are not invertible, and appropriate
finite-dimensional perturbations are constructed leading to invertibility of the
perturbed operators.
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Keywords
Analysis of Two-Operator, Boundary-Domain, Integral Equations, Variable-Coefficient, Boundary Value, Problems in 2D