On a method of investigating the Steklov problem for the 3-dimensional Laplace equation with non-local boundary-value conditions

The three-dimensional Laplace equation is considered in a domain D с r , convex in the direction Ox₃: _л д u (x) д u (x) д u (x) Lu = Au(x) =-^ +-^ +-^ = 0, (1) dxj dx₂ dx₃ x = (x₁, x₂, x₃) e D, with a parameter X under nonlocal homogeneous boundary conditions: du( x)i -Дл _dx l3 =Yk') + ^ 3 j=1 3=Y j0 ' _a(.)( x,) uW + a j2)( x') dx₁ dx₂ j-= Xu(x', Y_k(x')), x' e S, k = 1,2, (2) u(x) = f₀(x), x e L = Г₁ пГ₂ =dS . (3) where Г₁ and Г₂ are the lower and upper half surfaces of the boundary Г , respectively; the equations of half surfaces Г₁ and Г₂ y_k (^'), k = 1,2, are twice differentiable with respect to both the variables ; S is the projection of the domain D on the plane Ox₁x₂ = Ox'; the coefficients а(^(x') e C(S) , i, j, k = 1,2 , satisfy Holder's condition in S; the boundary Г = dD is a Lyapunov surface, XeC is a complex-valued parameter; and L is the equator connecting the half-surfaces Г₁ and Г₂ : L = Г₁ n Г₂ . The presented work is devoted to the study and proof of the Fredholm property for the solution of the Steklov boundary value problem for the three-dimensional Laplace equation in a bounded domain with non-local boundary conditions where the spectral parameter appears only in the boundary condition. The applied method is new and relies on necessary conditions derived from basic relations. These relations are obtained from the second Green's formula and from an analogue of this formula. The proposed scheme was applied to a variety of problems for partial differential equations in the two-dimensional case. However, the singularities entering the necessary conditions for three-dimensional problems are multi-dimensional; for this reason, their regularization is a difficulty which is overcome by using the proposed method.

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