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 Izv. RAN. Ser. Mat., 2001, Volume 65, Issue 2, Pages 27–80 (Mi izv327)

Entropy solutions of the Dirichlet problem for a class of non-linear elliptic fourth-order equations with right-hand sides in $L^1$

A. A. Kovalevsky

Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences

Abstract: In this paper we introduce and study the notion of an entropy solution of the Dirichlet problem for a class of non-linear elliptic fourth-order equations whose right-hand sides admit arbitrary growth with respect to the variable corresponding to the unknown function and belong to the space $L^1$ for each fixed value of this variable. We prove the existence and uniqueness of an entropy solution. We establish the existence of so-called $H$-solutions and $W$-solutions of the problem and prove that the entropy solutions belong to certain Sobolev spaces.

DOI: https://doi.org/10.4213/im327

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English version:
Izvestiya: Mathematics, 2001, 65:2, 231–283

Bibliographic databases:

MSC: 35J65, 35J30, 35D05

Citation: A. A. Kovalevsky, “Entropy solutions of the Dirichlet problem for a class of non-linear elliptic fourth-order equations with right-hand sides in $L^1$”, Izv. RAN. Ser. Mat., 65:2 (2001), 27–80; Izv. Math., 65:2 (2001), 231–283

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv327
• https://doi.org/10.4213/im327
• http://mi.mathnet.ru/eng/izv/v65/i2/p27

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. A. Kovalevsky, “Integrability of Solutions of Nonlinear Elliptic Equations with Right-Hand Sides from Classes Close to $L^1$”, Math. Notes, 70:3 (2001), 337–346
2. Kovalevsky A., Nicolosi F., “Entropy solutions of Dirichlet problem for a class of degenerate anisotropic fourth-order equations with $L^1$-right-hand sides”, Nonlinear Anal., 50:5 (2002), 581–619
3. A. A. Kovalevsky, “On the summability of entropy solutions for the Dirichlet problem in a class of non-linear elliptic fourth-order equations”, Izv. Math., 67:5 (2003), 881–894
4. Alexander Kovalevsky, Francesco Nicolosi, “Summability of Solutions of Dirichlet Problem for a Class of Degenerate Nonlinear High-order Equations”, GAPA, 82:2 (2003), 93
5. Kovalevsky A., Nicolosi F., “Solvability of Dirichlet problem for a class of degenerate anisotropic equations with $L^1$-right-hand sides”, Nonlinear Analysis: Theory, Methods & Applications, 59:3 (2004), 347–370
6. Alexander Kovalevsky, Francesco Nicolosi ‡, “Summability of solutions of some degenerate nonlinear elliptic fourth-order equations”, Applicable Analysis, 84:1 (2005), 1
7. A. A. Kovalevsky, F. Nicolosi, “On the sets of boundedness of solutions for a class of degenerate nonlinear elliptic fourth-order equations with $L^1$-data”, J. Math. Sci., 150:5 (2008), 2358–2368
8. A. A. Kovalevsky, “A priori properties of solutions of nonlinear equations with degenerate coercivity and $L^1$-data”, Journal of Mathematical Sciences, 149:5 (2008), 1517–1538
9. Kovalevsky A.A., Gorban Yu.S., “Degenerate anisotropic variational inequalities with $L^1$-data”, C. R. Math. Acad. Sci. Paris, 345:8 (2007), 441–444
10. Kovalevsky A.A., Nicolosi F., “On multipliers characterizing summability of solutions for a class of degenerate nonlinear high-order equations with $L^1$-data”, Nonlinear Anal., 69:3 (2008), 931–939
11. Alexander A. Kovalevsky, Francesco Nicolosi, “On the sets of -regularity of solutions for a class of degenerate nonlinear problems with slightly regular data”, Nonlinear Analysis: Theory, Methods & Applications, 68:10 (2008), 3175
12. Kovalevsky A.A., Nicolosi F., “On limit summability of solutions for a class of degenerate nonlinear high-order equations with L-1-data”, Complex Variables and Elliptic Equations, 55:11 (2010), 1047–1058
13. A. A. Kovalevsky, Yu. S. Gorban, “On $T$-solutions of degenerate anisotropic elliptic variational inequalities with $L^1$-data”, Izv. Math., 75:1 (2011), 101–156
14. S. Bonafede, F. Nicolosi, “The local boundedness of solutions for a class of degenerate nonlinear elliptic higher order equations with data close enough toL1”, Complex Variables and Elliptic Equations, 2011, 1
15. Voitovich M.V., “Existence of Bounded Solutions for Nonlinear Fourth-Order Elliptic Equations with Strengthened Coercivity and Lower-Order Terms with Natural Growth”, Electron. J. Differ. Equ., 2013
16. Cirmi G.R., D'Asero S., Leonardi S., “Fourth-Order Nonlinear Elliptic Equations With Lower Order Term and Natural Growth Conditions”, Nonlinear Anal.-Theory Methods Appl., 108 (2014), 66–86
17. Voitovych M.V., “Holder Continuity of Bounded Generalized Solutions For Nonlinear Fourth-Order Elliptic Equations With Strengthened Coercivity and Natural Growth Terms”, Electron. J. Differ. Equ., 2017, 63
18. Gorban Yu., “Existence of Entropy Solutions For Nonlinear Elliptic Degenerate Anisotropic Equations”, Open Math., 15 (2017), 768–786
19. Voitovych M.V., “On the Existence of Continuous Solutions For Nonlinear Fourth-Order Elliptic Equations With Strongly Growing Lower-Order Terms”, Rocky Mt. J. Math., 47:2 (2017), 667–685
20. Bonafede S., Voitovych M.V., “Holder Continuity Up to the Boundary of Solutions to Nonlinear Fourth-Order Elliptic Equations With Natural Growth Terms”, Diff. Equat. Appl., 11:1 (2019), 107–127
21. Voitovych M.V., “Pointwise Estimates of Solutions to 2M-Order Quasilinear Elliptic Equations With M- (P, Q) Growth Via Wolff Potentials”, Nonlinear Anal.-Theory Methods Appl., 181 (2019), 147–179
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