*Mathematical Methods in the Applied Sciences*Two Operator Boundary - Domain Integral Equations for variable
coefficient Neumann BVP in 2D

The Neumann boundary value problem (BVP) for the second order
“stationary heat transfer” elliptic partial differential equation with
variable coefficient is considered in two-dimensional bounded domain.
Using an appropriate parametrix (Levi function) and applying the
two-operator approach, this problem is reduced to some systems of
boundary-domain integral equations (BDIEs). The two-operator BDIEs in 2D
have 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 Neumann BVP, BDIEs solvability,
uniqueness/non uniqueness of the solution, as well as Fredholm property
and invertibility of the BDIE operator 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.\\
\noindent\textbf{Key words:} Partial
differential equation, Two-operator Boundary-Domain Integral Equations,
finite-dimensional perturbations , equivalence, invertibility.

30 Jan 2020

01 Feb 2020

01 Feb 2020

05 Feb 2020