Journal of Mathematical Sciences, Vol. 190, No. 1, April, 2013 SMOOTHNESS OF GENERALIZED SOLUTIONS OF ELLIPTIC DIFFERENTIAL-DIFFERENCE EQUATIONS WITH DEGENERATIONS V. A. Popov and A. L. Skubachevskii UDC 517.9 CONTENTS 1. Introduction ............................................ 135 2. Difference Operators ....................................... 136 3. A Priori Estimates and Friedrichs Extension .......................... 140 4. Inner Smoothness of Generalized Solutions in Subdomains .................. 143 References ................................................ 145 1. Introduction Elliptic differential-difference equations with degenerations were considered by many authors (see, e.g., [2, 4, 9] and bibliography therein). In papers [12, 13], elliptic differential-difference operators L R of order 2m with degeneration of the form L R u = LRu were considered. Here L is a strongly elliptic differential operator and R is a difference operator whose Hermitian part is a non-negative degenerated operator. In these papers, energy inequalities were obtained, the Friedrichs extension of the considered operator was constructed, and spectral properties and smoothness of generalized solutions were studied. In particular, it was shown that a solution may not belong to the Sobolev space W 1 2 (Q) even if its right-hand part is f ∈ ˙ C ∞ ( Q). However, the projection of a solution onto the image of the difference operator possesses a certain smoothness, but this is not smoothness in the whole domain but smoothness in its subdomains. The interest in such operators is due to certain fundamentally new (compared with the case of strongly elliptic differential-difference operators) properties (see [11]) and applications to some nonlocal elliptic problems arising in plasma theory (see [1]). In this paper, we consider the equation - n  i,j =1 ∂ 2 ∂x i ∂x j R ij u = f (x) (x ∈ Q ⊂ R n ) (1) with the boundary condition u(x)=0 (x/ ∈ Q) , (2) where R ij are differential operators acting in the space L 2 (Q) and defined by the formula R ij u(x)=  h∈M a ijh u(x + h), M is a finite set of vectors h from R n with integer coordinates, and a ijh ∈ C. Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 39, Partial Differential Equations, 2011. 1072–3374/13/1901–0135 c  2013 Springer Science+Business Media New York 135