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On a class of nonlocal elliptic problems. (English. Russian original) Zbl 0798.58074
Russ. Acad. Sci., Dokl., Math. 45, No. 2, 317-321 (1992); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 323, No. 2, 245-249 (1992).
The authors define a class of nonlocal elliptic problems, motivated by several examples, consider sets \(\mathbb{A}(m,l,r)\) of operators of the form \({\mathcal A} = \widehat{A} + \widehat{\phi}_{11} + \widehat{\phi}_{12} + \widehat{\phi}_{21} + \widehat{\phi}_{22}\), \(\widehat{A}\) a \(\psi\) DO of order \(r\) and the \(\widehat{\phi}_{ij}\) certain Fourier integral operators. An element \({\mathcal A} \in {\mathbb{A}}(m,l,r)\) is said to be elliptic if \(\widehat{A}\) and \(1 + \widehat{A}^{-1} \circ \widehat{\phi}_{11} + \widehat{A}^{-1} \circ \widehat{\phi}_{12} + \widehat{A}^{-1} \circ \widehat{\phi}_{21} + \widehat{A}^{-1} \circ \widehat{\phi}_{22}\) is elliptic. The authors assert that under certain additional conditions an elliptic operator is Fredholm. Proofs are not given but several examples.
58J40 Pseudodifferential and Fourier integral operators on manifolds
58J05 Elliptic equations on manifolds, general theory
35J45 Systems of elliptic equations, general (MSC2000)