Alkhutov, Yuriy Alexandrovich

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Total publications: 29
Scientific articles: 27
Presentations: 2

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Doctor of physico-mathematical sciences (1992)
Speciality: 01.01.02 (Differential equations, dynamical systems, and optimal control)
Keywords: elliptic and parabolic equations; solvability of a boundary value problem; a priori estimate; boundary properties of solutions; embedding theorem; capacity; removable singularities of solutions; maximal function.


The class of nondivergent elliptic equations of the second order with Wiener test regularity of a boundary point in terms of introduced function of ellipticity was described. This class ņontain equations with dicontinuous coefficients. The parabolic analog of Cordes condition guaranteeing unique solvability of the first boundary value problem for nondivergent parabolic equations of the second order in the Sobolev space $W^{2,1}_{2,0}$ was found (with I. T. Mamedov). Necessary and sufficient condition on a boundary for unique $L_p$–solvability of the Dirichlet problem together with the corresponding coercive estimate for divergent elliptic equations of the second order was obtained. The smoothness at a point for solutions of parabolic equations of the second order under minimal assumptions on coefficients was investigated. Inner and boundary properties for solutions of quasilinear elliptic equations for integrands $|\xi|^{p(x)}$ were studied. The Holder property for solutions of degenerate elliptic equations of the second order with a weight that is not satisfying neither Muckenhoupt condition nor double condition was proved (with V. V. Zhikov). Interesting feature of these equations is absent of Harnack inequality for positive solutions.


Graduated from department of applied mathematics of Azerbaijan Institute of Oil and Chemistry in 1979. Ph.D. thesis was defended in 1982. D.Sci thesis was defended in 1992.

Main publications:
  • Alkhutov Yu. A., Mamedov I. T. Pervaya kraevaya zadacha dlya nedivergentnykh parabolicheskikh uravnenii vtorogo poryadka s razryvnymi koeffitsientami // Matem. cbornik, 1986, 173(4), 477–500.
  • Alkhutov Yu. A. Ustranimye osobennosti reshenii parabolicheskikh uravnenii vtorogo poryadka // Matem. zametki, 1991, 50(5), 9–17.
  • Alkhutov Yu. A. Neravenstvo Kharnaka i gelderovost reshenii nelineinykh ellipticheskikh uravnenii s nestandartnym usloviem rosta // Differents. uravneniya, 1997, 33(12), 1651–1660.
  • Alkhutov Yu. A. $L_p$&-otsenki resheniya zadachi Dirikhle dlya ellipticheskikh uravnenii vtorogo poryadka // Matem. cbornik, 1998, 189(1), 3–20.
  • Alkhutov Yu. A., Zhikov V. V. O gelderovosti reshenii vyrozhdayuschikhsya ellipticheskikh uravnenii // Doklady RAN, 2001, 378(5), 583–588.
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List of publications on ZentralBlatt

Publications in Math-Net.Ru
1. Yu. A. Alkhutov, M. D. Surnachev, “Hölder Continuity and Harnack's Inequality for $p(x)$-Harmonic Functions”, Tr. Mat. Inst. Steklova, 308 (2020),  7–27  mathnet  elib; Proc. Steklov Inst. Math., 308 (2020), 1–21  isi  scopus
2. Yu. A. Alkhutov, M. D. Surnachev, “Estimates of the fundamental solution for an elliptic equation in divergence form with drift”, Zap. Nauchn. Sem. POMI, 489 (2020),  7–35  mathnet
3. Yu. A. Alkhutov, M. D. Surnachev, “Behavior of solutions of the Dirichlet problem for the $p(x)$-Laplacian at a boundary point”, Algebra i Analiz, 31:2 (2019),  88–117  mathnet
4. Yu. A. Alkhutov, M. D. Surnachev, “Harnack's inequality for the $p(x)$-Laplacian with a two-phase exponent $p(x)$”, Tr. Semim. im. I. G. Petrovskogo, 32 (2019),  8–56  mathnet  elib; J. Math. Sci. (N. Y.), 244:2 (2020), 116–147  scopus
5. Yu. A. Alkhutov, V. N. Denisov, “Necessary and sufficient condition for the stabilization of the solution of a mixed problem for nondivergence parabolic equations to zero”, Tr. Mosk. Mat. Obs., 75:2 (2014),  277–308  mathnet  elib; Trans. Moscow Math. Soc., 75 (2014), 233–258  scopus
6. Yu. A. Alkhutov, V. V. Zhikov, “Existence and uniqueness theorems for solutions of parabolic equations with a variable nonlinearity exponent”, Mat. Sb., 205:3 (2014),  3–14  mathnet  mathscinet  zmath  elib; Sb. Math., 205:3 (2014), 307–318  isi  scopus
7. Yu. A. Alkhutov, “Hölder continuity of solutions of nondivergent degenerate second-order elliptic equations”, Tr. Semim. im. I. G. Petrovskogo, 29 (2013),  5–42  mathnet  elib; J. Math. Sci. (N. Y.), 197:2 (2014), 151–174  scopus
8. Yu. A. Alkhutov, E. A. Khrenova, “Harnack inequality for a class of second-order degenerate elliptic equations”, Tr. Mat. Inst. Steklova, 278 (2012),  7–15  mathnet  mathscinet  elib; Proc. Steklov Inst. Math., 278 (2012), 1–9  isi  elib  scopus
9. Yu. A. Alkhutov, V. V. Zhikov, “Hölder continuity of solutions of parabolic equations with variable nonlinearity exponent”, Tr. Semim. im. I. G. Petrovskogo, 28 (2011),  8–74  mathnet  zmath  elib; J. Math. Sci. (N. Y.), 179:3 (2011), 347–389  scopus
10. Yu. A. Alkhutov, V. V. Zhikov, “Existence theorems for solutions of parabolic equations with variable order of nonlinearity”, Tr. Mat. Inst. Steklova, 270 (2010),  21–32  mathnet  mathscinet  zmath  elib; Proc. Steklov Inst. Math., 270 (2010), 15–26  isi  scopus
11. Yu. A. Alkhutov, A. N. Gordeev, “$L_p$-solubility of the Dirichlet problem for the heat operator”, Uspekhi Mat. Nauk, 64:1(385) (2009),  137–138  mathnet  mathscinet  zmath  elib; Russian Math. Surveys, 64:1 (2009), 131–133  isi  elib  scopus
12. Yu. A. Alkhutov, O. V. Krasheninnikova, “On the Continuity of Solutions to Elliptic Equations with Variable Order of Nonlinearity”, Tr. Mat. Inst. Steklova, 261 (2008),  7–15  mathnet  mathscinet  zmath  elib; Proc. Steklov Inst. Math., 261 (2008), 1–10  isi  elib  scopus
13. Yu. A. Alkhutov, “Hölder continuity of $p(x)$-harmonic functions”, Mat. Sb., 196:2 (2005),  3–28  mathnet  mathscinet  zmath  elib; Sb. Math., 196:2 (2005), 147–171  isi  elib  scopus
14. Yu. A. Alkhutov, O. V. Krasheninnikova, “Continuity at boundary points of solutions of quasilinear elliptic equations with a non-standard growth condition”, Izv. RAN. Ser. Mat., 68:6 (2004),  3–60  mathnet  mathscinet  zmath  elib; Izv. Math., 68:6 (2004), 1063–1117  isi  scopus
15. Yu. A. Alkhutov, “$L_p$-solubility of the Dirichlet problem for the heat equation in non-cylindrical domains”, Mat. Sb., 193:9 (2002),  3–40  mathnet  mathscinet  zmath; Sb. Math., 193:9 (2002), 1243–1279  isi  scopus
16. Yu. A. Alkhutov, V. V. Zhikov, “The leading term of the spectral asymptotics for the Kohn–Laplace operator in a bounded domain”, Mat. Zametki, 64:4 (1998),  493–505  mathnet  mathscinet  zmath; Math. Notes, 64:4 (1998), 429–439  isi
17. Yu. A. Alkhutov, “$L_p$-estimates of the solution of the Dirichlet problem for second-order elliptic equations”, Mat. Sb., 189:1 (1998),  3–20  mathnet  mathscinet  zmath; Sb. Math., 189:1 (1998), 1–17  isi  scopus
18. Yu. A. Alkhutov, “The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth condition”, Differ. Uravn., 33:12 (1997),  1651–1660  mathnet  mathscinet; Differ. Equ., 33:12 (1997), 1653–1663
19. Yu. A. Alkhutov, “The behavior of solutions of parabolic second-order equations in noncylindrical domains”, Dokl. Akad. Nauk, 345:5 (1995),  583–585  mathnet  mathscinet  zmath
20. Yu. A. Alkhutov, V. A. Kondratiev, “Solvability of the Dirichlet problem for second-order elliptic equations in a convex domain”, Differ. Uravn., 28:5 (1992),  806–818  mathnet  mathscinet; Differ. Equ., 28:5 (1992), 650–662
21. Yu. A. Alkhutov, “Removable singularities of solutions of second-order parabolic equations”, Mat. Zametki, 50:5 (1991),  9–17  mathnet  mathscinet  zmath; Math. Notes, 50:5 (1991), 1097–1103  isi
22. Yu. A. Alkhutov, “Smoothness and limiting properties of solutions of a second-order parabolic equation”, Mat. Zametki, 50:4 (1991),  150–152  mathnet  mathscinet  zmath; Math. Notes, 50:4 (1991), 1085–1087  isi
23. Yu. A. Alkhutov, “Local properties of solutions of non-divergent parabolic equations of second order”, Uspekhi Mat. Nauk, 45:5(275) (1990),  175–176  mathnet  mathscinet  zmath; Russian Math. Surveys, 45:5 (1990), 221–222  isi
24. Yu. A. Alkhutov, “Removable singularities of solutions of parabolic equations”, Uspekhi Mat. Nauk, 43:1(259) (1988),  189–190  mathnet  mathscinet  zmath; Russian Math. Surveys, 43:1 (1988), 229–230  isi
25. Yu. A. Alkhutov, I. T. Mamedov, “The first boundary value problem for nondivergence second order parabolic equations with discontinuous coefficients”, Mat. Sb. (N.S.), 131(173):4(12) (1986),  477–500  mathnet  mathscinet  zmath; Math. USSR-Sb., 59:2 (1988), 471–495
26. Yu. A. Alkhutov, I. T. Mamedov, “Some properties of the solutions of the first boundary value problem for parabolic equations with discontinuous coefficients”, Dokl. Akad. Nauk SSSR, 284:1 (1985),  11–16  mathnet  mathscinet  zmath
27. Yu. A. Alkhutov, “Regularity of boundary points relative to the Dirichlet problem for second-order elliptic equations”, Mat. Zametki, 30:3 (1981),  333–342  mathnet  mathscinet  zmath; Math. Notes, 30:3 (1981), 655–660  isi

28. Yu. A. Alkhutov, V. F. Butuzov, V. V. Kozlov, A. A. Kon'kov, A. V. Mikhalev, E. I. Moiseev, E. V. Radkevich, N. Kh. Rozov, V. A. Sadovnichii, I. N. Sergeev, M. D. Surnachev, R. N. Tikhomirov, V. N. Chubarikov, T. A. Shaposhnikova, A. A. Shkalikov, “Vasilii Vasilievich Zhikov”, Tr. Semim. im. I. G. Petrovskogo, 32 (2019),  5–7  mathnet  elib; J. Math. Sci. (N. Y.), 244:2 (2020), 113–115  scopus
29. Yu. A. Alkhutov, I. V. Astashova, V. I. Bogachev, V. N. Denisov, V. V. Kozlov, S. E. Pastukhova, A. L. Piatnitski, V. A. Sadovnichii, A. M. Stepin, A. S. Shamaev, A. A. Shkalikov, “Vasilii Vasil'evich Zhikov (obituary)”, Uspekhi Mat. Nauk, 73:3(441) (2018),  169–176  mathnet  elib; Russian Math. Surveys, 73:3 (2018), 533–542  isi

Presentations in Math-Net.Ru
1. Sharp estimates of solutions to the Dirichlet problem for p(x)- harmonic functions in a neighborhood of the boundary conical point
Yu. A. Alkhutov, M. V. Borsuk
International Conference on Mathematical Control Theory and Mechanics
July 6, 2015 12:40
2. Degenerate elliptic equations in the presence of Lavrent'ev effect
Yu. A. Alkhutov
International Conference on Differential Equations and Dynamical Systems
July 8, 2014 14:30

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