On a certain property of a regular difference operator with variable coefficients

Abstract

We consider a regular difference operator with variable coefficients in a bounded domain. It will be proved that this operator maps continuously and bijectively the Sobolev space of order k with the homogeneous Dirichlet boundary conditions to the subspace of Sobolev space of order k with nonlocal boundary conditions on the shifts of the boundary. This allows to apply the results obtained for nonlocal elliptic problems to the investigation of elliptic differential-difference equations.

COMMUNICATED BY:

• A. Soldatov

Keywords:

• Elliptic differential-difference equations
• nonlocal elliptic problems
• generalized solutions
• difference operators

AMS SUBJECT CLASSIFICATIONS:

• 39A70
• 39A14

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The publication has been prepared with the support of the ‘RUDN University Program 5-100’ and the Russian Foundation for Basic Research, grant number 16-01-00450.