Fredholm property of general elliptic problems
Aizik Volpert, Vitaly A Volpert · Transactions of the Moscow Mathematical Society · 2006
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.