Abstract
Ellipticity condition, proper ellipticity and Lopatinskii condition imply the Fredholm property of elliptic problems in bounded domains. In addition, invertibility of limiting problems determines the Fredholm property and solvability conditions of elliptic problems in unbounded domains. If this property is satisfied, then the index of the operator is defined. There is an extensive literature devoted to the index of elliptic operators in bounded domains and for some classes of operators in unbounded domains (see the bibliographical comments.)
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