Non-uniqueness of solutions to boundary value problems with Wentzell condition

Recently, in the mathematical literature, the Wentzell boundary condition is considered from two points of view. In the first case, let us call it classical one, this condition is an equation containing a linear combination of the values of the function and its derivatives on the boundary of the domain. Moreover, the function itself also satisfies the equation with an elliptic operator defined in the domain. In the second case, which we call neoclassical one, the Wentzell condition is an equation with the Laplace–Beltrami operator defined on the boundary of the domain understood as a smooth compact Riemannian manifold without boundary, and the external action is represented by the normal derivative of a function defined in the domain. The paper shows the non-uniqueness of solutions to boundary value problems with the Wentzell condition in the neoclassical sense both for the equation with the Laplacian and for the equation with the Bi-Laplacian given in the domain.