Non-Uniqueness of Solutions to Boundary Value Problems with Wentzell Condition

Let Q C R n , n E N \ { 1 } be a bounded connected domain with the boundary d Q of the class C ^∞ . For the first time, the Wentzell boundary condition

∂u

A u ( x ) + adv ( x ) + ви ( х ) = 0 , x E d Q ,                      (1)

appeared in [1]. Boundary value problems with condition (1) for second-order linear elliptic equations were studied by various methods [2–6]. Over time [7], condition (1) was understood as a description of a process occurring at the boundary of the domain and influenced by processes within the domain. The work [8] was the first to represent condition (1) in the following form

A u 2 ( x ) + a^u ¹ (x) + ви 2 (х) = 0 , x E d Q .                     (2)

∂ν

Here the boundary d Q is understood as a compact smooth Riemannian manifold without boundary, A is the Laplace-Beltrami operator, and the second term characterizes the influence of the processes occurring inside the domain.

In this context, consider condition (2) together with the Laplacian

Au1(x) = 0, x E Q.

By solution to problem (2), (3) we mean the function

/ A    J ui(x), x E Q;

u(x) = 5    / \ an(4)

• [ u₂ ( x ) , x E d Q .

Perform the replacemnet dUl.(x) = ^(x), x g dfi.                                 (5)

Following [9], we can always find a pair of Banach spaces in U 1 = U 1 (fi) and F = F (dfi) such that for any function у G F there exists the unique solution u 1 G U i to problem (3), (5). Following [10], we can find the Banach space U 2 = U 2 (d fi) for the space F and find the coefficients a G R \ { 0 } and в G R such that there exists a unique solution u ₂ G U 2 to problem (2), (5) for any function у G F . Obviously, due to the arbitrary choice of ^ , solution (4) to problem (2), (3) cannot be unique.

A similar situation arises if we replace a Laplacian with the Bi-Laplacian

A ² u 1 ( x ) = 0 , x G fi .                                      (6)

For completeness, we introduce the Dirichlet condition u1(x) = 0, x G dfi.                                      (7)

Reasoning by analogy with the previous case, we find a triple of Banach spaces U ₁ , U ₂ and F such that for any у G F there exist unique solutions to problems (5)-(7) and (3), (5). However, the solution to problem (2), (6), (7) cannot be unique.

Also, note that despite all the ab ove, under boundary conditions (7) and

Au1(x) + a-dV (x) + eu1(x) = 0, x G dfi, the solution to equation (6) exists and is unique in a suitably chosen space [11].

Acknowledgements. The research was funded by RFBR and Chelyabinsk Region, project number 20-41-740010.