01.01.02 (Differential equations, dynamical systems, and optimal control)
E-mail:
Keywords:
nonlinear elliptic equations and variational inequalities; homogenization of boundary value problems in variable domains; G-convergence of nonlinear operators; Ã-convergence of integral functionals; existence and properties of solutions of nonlinear equations with L1-data; regularity of solutions of degenerate nonlinear high-order equations.
Subject:
Necessary and sufficient conditions for Ã-convergence of integral functionals with varying domain of definition were established and theorems on Ã-compactness for these functionals were proved. New results on $G$-compactness of sequences of nonlinear elliptic operators (including high-order operators) corresponding to Dirichlet and Neumann problems in variable domains were obtained. G-convergence of nonlinear operators of Neumann problems in domains of framework-type periodic structure with thin channels was studied and representations for coefficients of the G-limit operator were obtained. The asymptotic behaviour of solutions of Neumann problems for nonlinear elliptic equations in three-dimensional domains with periodically allocated simple and double accumulators was investigated. It was shown that these solutions converge in a certain sense to a solution of a problem for a system of some functional equations and one differential equation. An effect of double homogenization was first established in regard to Dirichlet problems for nonlinear elliptic second-order equations with coefficients depended on a parameter in variable domains of general structure. A notion of entropy solution of Dirichlet problem for some classes of nonlinear elliptic high-order equations with L1-data was introduced and results on existence, uniqueness and summability of such a kind of solutions were proved. New results on summability of solutions of nonlinear elliptic second-order equations with right-hand sides in logarithmic classes of functions were established.
Biography
Graduated from Mathematical Faculty of the Donetsk State University in 1979 (department of differential equations). Ph.D. thesis was defended in 1985. D. Sci. thesis was defended in 1995.
Main publications:
Kovalevskii A.A. G-convergence and homogenization of nonlinear elliptic operators in divergence form with variable domain // Russ. Acad.Sci. Izv. Math., 1995, 44(3), 431–460.
Kovalevsky A. An effect of double homogenization for Dirichlet problems in variable domains of general structure // Comptes Rendus Acad. Sci. Paris, Ser. I, 1999, 328(12), 1151–1156.
Kovalevskii A.A. Entropy solutions of the Dirichlet problem for a class of non-linear elliptic fourth-order equations with right-hand sides in L1 // Izv. Math., 2001, 65(2), 231–283.
Kovalevsky A.A. Integrability and boundedness of solutions to some anisotropic problems // J. Math. Anal. Appl. 2015, 432(2), 820–843.
Kovalevsky A.A., Skrypnik I.I., Shishkov A.E. Singular Solutions of Nonlinear Elliptic and Parabolic Equations. Berlin: De Gruyter, 2016. 436 p.
Kovalevsky A.A. On the convergence of solutions to bilateral problems with the zero lower constraint and an arbitrary upper constraint in variable domains // Nonlinear Anal., 2016, 147, 63–79.
Kovalevsky A.A. Variational problems with variable regular bilateral constraints in variable domains // Rev. Mat. Complut., 2019, 32(2), 327–351.
Kovalevsky A.A. On the convergence of solutions of variational problems with variable implicit pointwise constraints in variable domains // Ann. Mat. Pura Appl., 2019, 198(4), 1087–1119.
A. A. Kovalevsky, “Criteria for the existence of weak solutions of the Dirichlet problem for nonlinear degenerate elliptic equations for any right-hand side in $L^1$”, Mat. Zametki, 116:3 (2024), 482–485; Math. Notes, 116:3 (2024), 571–574
2021
2.
A. A. Kovalevsky, “On the convergence of minimizers and minimum values in variational problems with pointwise functional constraints in variable domains”, Trudy Inst. Mat. i Mekh. UrO RAN, 27:1 (2021), 246–257
2020
3.
A. A. Kovalevsky, “Summability of Solutions of the Dirichlet Problem
for Nonlinear Elliptic Equations with Right-Hand Side
in Classes Close to $L^1$”, Mat. Zametki, 107:6 (2020), 934–939; Math. Notes, 107:6 (2020), 1023–1028
A. A. Kovalevsky, “Integrability Properties of Functions with a Given Behavior of Distribution Functions and Some Applications”, Trudy Inst. Mat. i Mekh. UrO RAN, 25:1 (2019), 78–92; Proc. Steklov Inst. Math. (Suppl.), 308, suppl. 1 (2020), S112–S126
A. A. Kovalevsky, “On the Convergence of Solutions of Variational Problems with Implicit Pointwise Constraints in Variable Domains”, Funktsional. Anal. i Prilozhen., 52:2 (2018), 82–85; Funct. Anal. Appl., 52:2 (2018), 147–150
A. A. Kovalevsky, “On the convergence of solutions of variational problems with implicit constraints defined by rapidly oscillating functions”, Trudy Inst. Mat. i Mekh. UrO RAN, 24:2 (2018), 107–122; Proc. Steklov Inst. Math. (Suppl.), 305, suppl. 1 (2019), S86–S101
A. A. Kovalevsky, “On the convergence of solutions of variational problems with bilateral obstacles in variable domains”, Trudy Inst. Mat. i Mekh. UrO RAN, 22:1 (2016), 140–152; Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 151–163
A. A. Kovalevsky, “A priori properties of solutions of nonlinear equations with degenerate coercivity and $L^1$-data”, CMFD, 16 (2006), 47–67; Journal of Mathematical Sciences, 149:5 (2008), 1517–1538
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”, Fundam. Prikl. Mat., 12:4 (2006), 99–112; J. Math. Sci., 150:5 (2008), 2358–2368
A. A. Kovalevsky, “On the summability of entropy solutions for the Dirichlet problem in a class of non-linear elliptic fourth-order equations”, Izv. RAN. Ser. Mat., 67:5 (2003), 35–48; Izv. Math., 67:5 (2003), 881–894
A. A. Kovalevsky, “Integrability of Solutions of Nonlinear Elliptic Equations with Right-Hand Sides from Logarithmic Classes”, Mat. Zametki, 74:5 (2003), 676–685; Math. Notes, 74:5 (2003), 637–646
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
A. A. Kovalevsky, “Integrability of Solutions of Nonlinear Elliptic Equations with Right-Hand Sides from Classes Close to $L^1$”, Mat. Zametki, 70:3 (2001), 375–385; Math. Notes, 70:3 (2001), 337–346
A. A. Kovalevsky, “A necessary condition for the strong $G$-convergence of nonlinear operators of Dirichlet problems with
variable domain”, Differ. Uravn., 36:4 (2000), 537–541; Differ. Equ., 36:4 (2000), 599–604
1996
19.
A. A. Kovalevsky, “$G$-compactness of sequences of non-linear operators of Dirichlet problems with a variable domain of definition”, Izv. RAN. Ser. Mat., 60:1 (1996), 133–164; Izv. Math., 60:1 (1996), 137–168
A. A. Kovalevsky, “On the uniform boundedness of solutions of nonlinear elliptic variational inequalities in variable domains”, Differ. Uravn., 30:8 (1994), 1370–1373; Differ. Equ., 30:8 (1994), 1270–1273
21.
A. A. Kovalevsky, “$G$-convergence and homogenization of nonlinear elliptic operators in divergence form with variable domain”, Izv. RAN. Ser. Mat., 58:3 (1994), 3–35; Russian Acad. Sci. Izv. Math., 44:3 (1995), 431–460
Limits of constrained minimum problems in variable domains A. A. Kovalevsky III International Conference “Mathematical Physics, Dynamical Systems, Infinite-Dimensional Analysis”, dedicated to the 100th anniversary of V.S. Vladimirov, the 100th anniversary of L.D. Kudryavtsev and the 85th anniversary of O.G. Smolyanov July 6, 2023 12:35
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