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, “Behavior of solutions of the Dirichlet problem for the $p(x)$-Laplacian at a boundary point”, Algebra i Analiz, 31:2 (2019), 88–117
2.
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
3.
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
4.
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
5.
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
6.
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
7.
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
8.
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
9.
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
10.
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
11.
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
12.
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
13.
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
14.
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
15.
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
16.
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
1992
17.
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
18.
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
19.
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
20.
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
21.
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
22.
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
23.
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
24.
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
25.
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
26.
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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