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 Izv. RAN. Ser. Mat., 2003, Volume 67, Issue 6, Pages 71–110 (Mi izv460)

Non-local elliptic problems with non-linear argument transformations near the points of conjugation

P. L. Gurevich

Abstract: We consider elliptic equations of order $2m$ in a domain $G\subset\mathbb R^n$ with non-local conditions that connect the values of the unknown function and its derivatives on $(n-1)$-dimensional submanifolds $\overline\Upsilon_i$ (where $\bigcup_i\overline\Upsilon_i=\partial G$ with the values on $\omega_{is}(\overline\Upsilon_i)\subset\overline G$. Non-local elliptic problems in dihedral angles arise as model problems near the conjugation points $g\in\overline\Upsilon_i\cap \overline\Upsilon_j\ne\varnothing$, $i\ne j$. We study the case when the transformations $\omega_{is}$ correspond to non-linear transformations in the model problems. It is proved that the operator of the problem remains Fredholm and its index does not change as we pass from linear argument transformations to non-linear ones.

DOI: https://doi.org/10.4213/im460

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English version:
Izvestiya: Mathematics, 2003, 67:6, 1149–1186

Bibliographic databases:

UDC: 517.9
MSC: 35J40, 46E35, 47F05, 47A53

Citation: P. L. Gurevich, “Non-local elliptic problems with non-linear argument transformations near the points of conjugation”, Izv. RAN. Ser. Mat., 67:6 (2003), 71–110; Izv. Math., 67:6 (2003), 1149–1186

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv460
• https://doi.org/10.4213/im460
• http://mi.mathnet.ru/eng/izv/v67/i6/p71

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This publication is cited in the following articles:
1. Soldatov A. P., “The Bitsadze-Samarskii problem for Douglis analytic functions”, Differ. Equ., 41:3 (2005), 416–428
2. P. L. Gurevich, “Generalized Solutions of Nonlocal Elliptic Problems”, Math. Notes, 77:5 (2005), 614–629
3. Soldatov A.P., “Problem of Bitsadze-Samarskii type for second-order elliptic systems in the plane”, Dokl. Math., 74:2 (2006), 736–740
4. A. L. Skubachevskii, “Nonclassical boundary value problems. I”, Journal of Mathematical Sciences, 155:2 (2008), 199–334
5. A. L. Skubachevskii, “Nonclassical boundary-value problems. II”, Journal of Mathematical Sciences, 166:4 (2010), 377–561
6. P. L. Gurevich, “Elliptic problems with nonlocal boundary conditions and Feller semigroups”, Journal of Mathematical Sciences, 182:3 (2012), 255–440
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