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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.
MSC:
58J40 Pseudodifferential and Fourier integral operators on manifolds
58J05 Elliptic equations on manifolds, general theory
35J45 Systems of elliptic equations, general (MSC2000)
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