On the behaviour at infinity of the solution of a second-order elliptic equation given on a cylinder
A L Pyatnitskiĭ · Russian Mathematical Surveys · 1982
given on the cylinder Q, the product of the half-line (0, +°°) and the torus У»-. In other words, the coefficients and the data of (1) are periodic in all variables except one, and we investigate the behaviour of solutions with the same properties. Throughout what follows we assume that the matrix aij(x) is uniformly elliptic. Similar questions for divergent equations have been studied in [1] and [2]. The most complete results have been obtained in the case when the coefficients of (1) are also periodic in the variable .Vj. Let a i ;(x) and 6j(i)be sufficiently smooth functions that are periodic in all the variables X\, ..., xn. Let jt* denote the operator formally adjoint to .Ж. The auxiliary problem