Mixed Boundary Value Transmission Problems.

A common approach to the abstract boundary value problems is considered on the basis of an Abstract Green’s Identity. Some examples of areas docked configurations for interfacing problems are considered on the basis of the generalized Green’s Identity for the Laplace operator (in particular the configuration of "three sliced watermelon" is considered). The initial nonhomogeneous problem interface is divided into four auxiliary problems. Heterogeneity exists only in one place in these tasks, i.e., at the equation or in the boundary condition. We find a weak solution of every problem with the help of the corresponding Green’s Identity. Theorems on existence and uniqueness of each solution are proved. At the end of the article we get the conclusion that the solution of the original problem is the sum of the solutions of four auxiliary problems.

We consider the solution of the transmission problem

$u_1 − \Delta u_1 = f_1$ (in $Ω_1$), $γ_{11}u_1 = ϕ_1$ (on $Γ_{11}$),
$u_2 − \Delta u_2 = f_2$ (in $Ω_2$), $γ_{22}u_2 = ϕ_2$ (on $Γ_{22}$),
$u_3 − \Delta u_3 = f_3$ (in $Ω_3$), $γ_{33}u_3 = ϕ_3$ (on $Γ_{33}$).

Jumps of functions and derivatives of the external boundaries of the normals are defined on the joint boundary

$γ_{21}u_1 − γ_{12}u_2 = ϕ_{12}$, $∂_{21}u_1 + ∂_{12}u_2 = ψ_{12}$ (on $Γ_{12} = Γ_{21}$),
$γ_{32}u_2 − γ_{23}u_3 = ϕ_{23}$, $∂_{32}u_2 + ∂_{23}u_3 = ψ_{23}$ (on $Γ_{23} = Γ_{32}$),
$γ_{13}u_3 − γ_{31}u_1 = ϕ_{31}$, $∂_{13}u_3 + ∂_{31}u_1 = ψ_{31}$ (on $Γ_{31} = Γ_{13}$).

The solution to this problem is found in the form of a sum of solutions of auxiliary problems. We consider auxiliary problems of Zaremba, Steklov, the first and second Krein problems. We find a weak solution of each problem with corresponding Green’s formula.

Keywords: Green’s Identity, weak solutions, boundary value problems, transmission problem, auxiliary problem, Lipschitz domain, derivative with respect to the outer normal.