On the question of regularizability of the oblique derivative type boundary value problem for second-oder elliptic systems on the plane

A. I. Basik ; E. V. Gricuk ; T. V. Kapaitsava

Problemy fiziki, matematiki i tehniki, no. 3 (2022), pp. 67-71

Résumé

The set $\mathfrak{M}(2;2;2)$ of elliptic systems of two second-order partial differential equations on the plane with positive characteristic determinant is considered. An oblique derivative type boundary value problem for a system from $\mathfrak{M}(2;2;2)$ in a bounded domain $\Omega$ with a smooth boundary $\partial\Omega$ is to find a solution for given boundary values of the derivatives along the directions $l_1$ and $l_2$ nontangential to $\partial\Omega$. It is known that the set $\mathfrak{M}(2;2;2)$ has three homotopy connected components. It is also known that if a system from $\mathfrak{M}(2;2;2)$ is a system of orthogonal type and $l_1$, $l_2$ are vector fields that are noncollinear at each point of the boundary, then the oblique derivative boundary value problem is Fredholm in its classical formulation (regardless of the homotopy class of the system). In this paper, for each component of $\mathfrak{M}(2;2;2)$ a representative is given that has the following properties: each component of an arbitrary twice continuously differentiable solution is a biharmonic function, and an oblique derivative type boundary value problem for this representative is not regularizable. Consequently, the regularizability of a problem of oblique derivative type boundary value problem for the elliptic systems under consideration is not related to the homotopy class of the system.

Détail

Keywords: elliptic system, regularizable boundary value problem, Lopatinski condition, homotopic classification.