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

This article is cited in 14 scientific papers (total in 14 papers)

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.


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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
Received: 31.10.2006 and 02.04.2007

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

Citation in format AMSBIB
\by S.~E.~Pastukhova
\paper Degenerate equations of monotone type: Lavrent'ev phenomenon and attainability problems
\jour Mat. Sb.
\yr 2007
\vol 198
\issue 10
\pages 89--118
\jour Sb. Math.
\yr 2007
\vol 198
\issue 10
\pages 1465--1494

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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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Buttazzo G., Kogut P.I., “Weak optimal controls in coefficients for linear elliptic problems”, Rev. Mat. Complut., 24:1 (2010), 83–94  crossref  mathscinet  isi  scopus
    3. Kogut P.I., Leugering G., “Optimal $L^1$-control in coefficients for Dirichlet elliptic problems: $W$-optimal solutions”, J. Optim. Theory Appl., 150:2 (2011), 205–232  crossref  mathscinet  zmath  isi  elib  scopus
    4. Kogut P.I., Leugering G., “Optimal $L^1$-control in coefficients for Dirichlet elliptic problems: $H$-optimal solutions”, Z. Anal. Anwend., 31:1 (2012), 31–53  crossref  mathscinet  zmath  isi  elib  scopus
    5. D'Apice C., De Maio U., Kogut O.P., “Optimal control problems in coefficients for degenerate equations of monotone type: shape stability and attainability problems”, SIAM J. Control Optim., 50:3 (2012), 1174–1199  crossref  mathscinet  zmath  isi  scopus
    6. Tölle J.M., “Uniqueness of weighted Sobolev spaces with weakly differentiable weights”, J. Funct. Anal., 263:10 (2012), 3195–3223  crossref  mathscinet  zmath  isi  elib  scopus
    7. V. Zh. Sakbaev, “Cauchy problem for degenerating linear differential equations and averaging of approximating regularizations”, Journal of Mathematical Sciences, 213:3 (2016), 287–459  mathnet  crossref  mathscinet
    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  crossref  mathscinet  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
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