01.01.02 (Differential equations, dynamical systems, and optimal control)
E-mail:
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.
Subject:
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.
Biography
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.
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; Proc. Steklov Inst. Math., 308 (2020), 1–21
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
2019
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
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; J. Math. Sci. (N. Y.), 244:2 (2020), 116–147
2014
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; Trans. Moscow Math. Soc., 75 (2014), 233–258
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; Sb. Math., 205:3 (2014), 307–318
2013
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; J. Math. Sci. (N. Y.), 197:2 (2014), 151–174
2012
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; Proc. Steklov Inst. Math., 278 (2012), 1–9
2011
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; J. Math. Sci. (N. Y.), 179:3 (2011), 347–389
2010
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; Proc. Steklov Inst. Math., 270 (2010), 15–26
2009
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; Russian Math. Surveys, 64:1 (2009), 131–133
2008
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; Proc. Steklov Inst. Math., 261 (2008), 1–10
2005
13.
Yu. A. Alkhutov, “Hölder continuity of $p(x)$-harmonic functions”, Mat. Sb., 196:2 (2005), 3–28; Sb. Math., 196:2 (2005), 147–171
2004
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; Izv. Math., 68:6 (2004), 1063–1117
2002
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; Sb. Math., 193:9 (2002), 1243–1279
1998
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; Math. Notes, 64:4 (1998), 429–439
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; Sb. Math., 189:1 (1998), 1–17
1997
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; Differ. Equ., 33:12 (1997), 1653–1663
1995
19.
Yu. A. Alkhutov, “The behavior of solutions of parabolic second-order equations in
noncylindrical domains”, Dokl. Akad. Nauk, 345:5 (1995), 583–585
1992
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; Differ. Equ., 28:5 (1992), 650–662
1991
21.
Yu. A. Alkhutov, “Removable singularities of solutions of second-order parabolic equations”, Mat. Zametki, 50:5 (1991), 9–17; Math. Notes, 50:5 (1991), 1097–1103
22.
Yu. A. Alkhutov, “Smoothness and limiting properties of solutions of a second-order parabolic equation”, Mat. Zametki, 50:4 (1991), 150–152; Math. Notes, 50:4 (1991), 1085–1087
1990
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; Russian Math. Surveys, 45:5 (1990), 221–222
1988
24.
Yu. A. Alkhutov, “Removable singularities of solutions of parabolic equations”, Uspekhi Mat. Nauk, 43:1(259) (1988), 189–190; Russian Math. Surveys, 43:1 (1988), 229–230
1986
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; Math. USSR-Sb., 59:2 (1988), 471–495
1985
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
1981
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; Math. Notes, 30:3 (1981), 655–660
2019
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; J. Math. Sci. (N. Y.), 244:2 (2020), 113–115
2018
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; Russian Math. Surveys, 73:3 (2018), 533–542
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