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 Mat. Sb., 2007, Volume 198, Number 10, Pages 89–118 (Mi msb3793)

Degenerate equations of monotone type: Lavrent'ev phenomenon and attainability problems

S. E. Pastukhova

Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University)

Abstract: A non-linear monotone equation with degenerate weight function is considered. In the general case the smooth functions are not dense in the corresponding weighted Sobolev space $W$, which leads to a non-uniqueness of a particular kind. Taking for the energy space either $W$ itself or its subspace $H$ equal to the closure of the smooth functions one obtains at least two uniquely soluble problems. In addition, there exist infinitely many weak solutions distinct from the $W$- and $H$-solutions. The problem of approximability or attainability is considered: which solutions of the original equation can be obtained as limits of solutions of the equations with suitable non-degenerate weights? It is shown that the $W$- and the $H$-solutions are attainable; in both cases a regular approximation algorithm is described.
Bibliography: 14 titles.

DOI: https://doi.org/10.4213/sm3793

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English version:
Sbornik: Mathematics, 2007, 198:10, 1465–1494

Bibliographic databases:

UDC: 517.956.226+517.956.8+517.957.95
MSC: 35J60, 47J05

Citation: S. E. Pastukhova, “Degenerate equations of monotone type: Lavrent'ev phenomenon and attainability problems”, Mat. Sb., 198:10 (2007), 89–118; Sb. Math., 198:10 (2007), 1465–1494

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• https://doi.org/10.4213/sm3793
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This publication is cited in the following articles:
1. V. V. Zhikov, “On the Technique for Passing to the Limit in Nonlinear Elliptic Equations”, Funct. Anal. Appl., 43:2 (2009), 96–112
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8. Kupenko O.P., Manzo R., “On an Optimal l-1-Control Problem in Coefficients for Linear Elliptic Variational Inequality”, Abstract Appl. Anal., 2013, 821964
9. Kogut P.I., Leugering G., “Matrix-Valued l-1-Optimal Controls in the Coefficients of Linear Elliptic Problems”, Z. Anal. ihre. Anwend., 32:4 (2013), 433–456
10. P.I.. Kogut, Günter Leugering, “Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage”, Math. Meth. Appl. Sci, 2014, n/a
11. Kupenko O.P., Leugering G., “On the Existence of Weak Optimal Controls in the Coefficients for a Degenerate Anisotropic p-Laplacian”, Continuous and Distributed Systems II, Studies in Systems, Decision and Control, 30, eds. Sadovnichiy V., Zgurovsky M., Springer Int Publishing Ag, 2015, 315–337
12. Kogut P.I., Kupenko O.P., “Optimality Conditions For l-1-Control in Coefficients of a Degenerate Nonlinear Elliptic Equation”, Advances in Dynamical Systems and Control, Studies in Systems Decision and Control, 69, eds. Sadovnichiy V., Zgurovsky M., Springer Int Publishing Ag, 2016, 429–471
13. Durante T., Kupenko O.P., Manzo R., “on Attainability of Optimal Controls in Coefficients For System of Hammerstein Type With Anisotropic P-Laplacian”, Ric. Mat., 66:2 (2017), 259–292
14. Kupenko O.P., Manzo R., “On Optimal Controls in Coefficients For Ill-Posed Non-Linear Elliptic Dirichlet Boundary Value Problems”, Discrete Contin. Dyn. Syst.-Ser. B, 23:4 (2018), 1363–1393
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