Eigenvalues and eigenfunctions of the Laplace operator in a square and in a circle with a Wentzel boundary condition

Recently, in the mathematical literature, the Wentzel boundary condition has been considered from two points of view. In the first case, let us call it a classical case, this condition is an equation containing a linear combination of the values of the function and its derivatives at the boundary of the domain. Meanwhile, the function itself also satisfies an equation with an elliptic operator given in the domain. In the second, neoclassical case, the Wentzel condition is an equation with the Laplace-Beltrami operator defined on the boundary of the domain, understood as a smooth compact Riemannian manifold without an edge; and the external effect is represented by the normal derivative of the function specified in the domain. The paper considers the properties of the Laplace operator with the Wentzel boundary condition in the neoclassical sense. In particular, eigenvalues and eigenfunctions of the Laplace operator are constructed for a system of Wentzel equations in a circle and in a square.