Traces of generalized solutions of elliptic differential-difference equations with degeneration

V. A. Popov

Contemporary Mathematics. Fundamental Directions, Proceedings of the Seminar on Differential and Functional Differential Equations supervised by A. L. Skubachevskii (Peoples' Friendship University of Russia), Tome 62 (2016), pp. 124-139

Résumé

The paper is devoted to differential-difference equations with degeneration in a bounded domain $Q\subset\mathbb R^n$. We consider differential-difference operators that cannot be expressed as a composition of a strongly elliptic differential operator and a degenerated difference operator. Instead of this, operators under consideration contain several degenerated difference operators corresponding to differentiation operators. Generalized solutions of such equations may not belong even to the Sobolev space $W^1_2(Q)$. Earlier, under certain conditions on difference and differentiation operators, we had obtained a priori estimates and proved that the orthogonal projection of the generalized solution onto the image of the difference operator preserves certain smoothness inside some subdomains $Q_r\subset Q$ ($\bigcup_r\overline Q_r=\overline Q)$ instead of the whole domain. In this paper, we prove necessary and sufficient conditions in algebraic form for existence of traces on some parts of boundaries of subdomains $Q_r$.