Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 67 (2006) 2006, Pages 127–197 S 0077-1554(06)00159-2 Article electronically published on December 27, 2006 FREDHOLM PROPERTY OF GENERAL ELLIPTIC PROBLEMS A. VOLPERT AND V. VOLPERT Dedicated to Ya. B. Lopatinskii on the occasion of his 100th birthday anniversary Abstract. Linear elliptic problems in bounded domains are normally solvable with a finite-dimensional kernel and a finite codimension of the image, that is, satisfy the Fredholm property, if the ellipticity condition, the condition of proper ellipticity and the Lopatinskii condition are satisfied. In the case of unbounded domains these conditions are not sufficient any more. The necessary and sufficient conditions of normal solvability with a finite-dimensional kernel are formulated in terms of limiting problems. Adjoint operators to elliptic operators in unbounded domains are studied and the conditions in order for them to be normally solvable with a finite-dimensional kernel are also formulated by means of limiting problems. The properties of the direct and of the adjoint operators are used to prove the Fredholm property of elliptic problems in unbounded domains. Some special function spaces introduced in this work play an important role in the study of elliptic problems in unbounded domains. 1. Introduction It is known that elliptic operators in bounded domains satisfy the Fredholm property, that is, the dimension of their kernel is finite, the image is closed, and the codimension of the image is also finite (see [2], [36], [42] and the references therein). If we consider unbounded domains, then the ellipticity condition, proper ellipticity and the Lopatinskii condition are not sufficient, generally speaking, in order for the operator to satisfy the Fredholm property. Some additional conditions formulated in terms of limiting problems should be imposed. The typical result says that the operator satisfies the Fredholm property if and only if all its limiting operators are invertible. The question is about the classes of operators for which this result is applicable. Limiting operators and their interrelation with solvability conditions and with the Fredholm property were first studied in [15], [20], [21] (see also [39]) for differential operators on the real axis, and later for some classes of elliptic operators in R n [8], [25], [26], in cylindrical domains [9], [43], or in some specially constructed domains [6], [7]. Some of these results are obtained for the scalar case, some others for the vector case, under the assumption that the coefficients of the operator stabilize at infinity or without this assumption. This theory is also developed for some classes of pseudodifferential operators [12], [19], [30]–[34], [37], [38] and discrete operators [5], [35]. A survey of this literature is presented in the recent monograph [35]. In spite of the considerable progress in the understanding of properties of elliptic oper- ators in unbounded domains, this question is not yet completely elucidated. The results existing in the literature are formulated for some classes of operators. For example, scalar elliptic problems in unbounded cylinders with constant coefficients at infinity are studied in the works cited above only for some classes of second-order operators. Moreover, in some cases it can be difficult to verify imposed conditions, and even simple problems 2000 Mathematics Subject Classification. Primary 35J25; Secondary 34D09, 47F05. c 2006 American Mathematical Society 127 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use