Homogenization, stochastic homogenization, singularly perturbed operators, Gamma-convergence, behaviour of solutions at infinity.
Main publications:
Kozlov, S.M.; Piatnitski, A.L. Effective diffusion for a parabolic operator with periodic potential. SIAM J. Appl. Math. 53, No. 2, 401–418 (1993).
Kozlov, S.M.; Piatnitski, A.L. Degeneration of effective diffusion in the presence of periodic potential. Ann. Inst. Henri Poincare, Probab. Stat. 32, No. 5, 571–587 (1997).
Piatnitski, A.L. Asymptotic behavior of the ground state of singularly perturbed elliptic equations. Commun. Math. Phys. 197, No. 3, 527–551 (1998).
Chechkin, G.A.; Friedman, A. ; Piatnitski, A.L. The boundary-value problem in domains with very rapidly oscillating boundary. J. Math. Anal. Appl. 231, No. 1, 213–234 (1999).
Piatnitski, A.; Remy, E. Homogenization of Elliptic Difference Operators. SIAM J. Math. Analysis 33, No. 1, 53–83, (2001).
Allaire, G.; Piatnitski, A. Uniform spectral asymptotics for singularly perturbed locally periodic operators. Communications in Partial Differential Equations 27, No. 3-4, 705–725, (2002).
Kleptsyna, M.L.; Piatnitski, A.L. Homogenization of Random Nonstationary Convection-Diffusion Problem. Russian Mathematical Surveys, 57, No. 4, 729–751, (2002).
Bourgeat, A.; Jurak, M.; Piatnitski A.L. Averaging a transport equation with small diffusion and oscillating velocity. Mathematical Methods in the Applied Sciences, 26, No. 2, 95–117, (2003).
Pardoux, E.; Piatnitski A. Homogenization of a nonlinear random parabolic partial differential equation. Stochastic Processes and their Applications, 104, No. 1, 1–27, (2003).
Allaire, G.; Capdeboscq, Y.; Piatnitski, A.L. Homogenization and localization with an interface. Indiana Univ. Math. J. 52, No. 6, 1413–1446, (2003).
Bourgeat, A.; Mikelic, A.; Piatnitski, A. On the double porosity model of a single phase flow in random media. Asymptotic Analysis 34, No. 2, 311–332, (2003).
Cioranescu, D.; Piatnitski, A. Homogenization in perforated domains with rapidly pulsing perforations. ESAIM: Control, Optimisation and Calculus of Variations 9, 461–484, (2003).
Bourgeat, A.; Piatnitski, A. Approximations of effective coefficients in stochastic homogenization. Ann. Inst. H. Poincare, Prob. Statistics 40, No. 2, 153–165, (2004).
Piatnitski, A.; Shamaev, A. On the Asymptotic Behavior of Eigenvalues and Eigenfunctions of Non-Self-Adjoint Elliptic Operators. Journal of Mathematical Sciences 120, No. 3, 1411–1423, (2004).
Braides, A.; Chiado Piat, V.; Piatnitski, A. A variational approach to double-porosity problems. Asymptotic Analysis 39, No. 3-4, 281–308, (2004).
Allaire, G.; Capdeboscq, Y.; Piatnitski, A. Siess, V.; Vannitathan, M. Homogenization of Periodic Systems with Large Potentials. Arch. Rational Mech. Anal. 174, 179–220, (2004).
Allaire, G.; Piatnitski, A. Homogenization of the Schredinger Equation and Effective Mass Theorems. Comm. Math. Physics 258, 1–22, (2005).
Marusic-Paloka, E.; Piatnitski, A. Homogenization of a nonlinear convection-diffusion equation with rapidly oscillating coefficients and strong convection. Journal of London Math. Soc. 72, No. 2, 391–409 (2005).
Diop, M. ; Iftimie, B.; Pardoux, E.; Piatnitski, A. Singular homogenization with stationary in time and periodic in space coefficients. Journal of Functional Analysis 231, No. 1, 1–46 (2006).
Cioranescu, D.; Piatnitski, A. Homogenization of a porous medium with randomly pulsating microstructure. Multiscale Modeling & Simulation 5, No. 1, 170–183 (2006).
Zhikov, V.V.; Piatnitski, A.L. Homogenization of random singular structures and random measures. Izvestiya: Mathematics 70, No. 1, 19–67 (2006).
Piatnitski, A.L. Homogenization of random nonstationary parabolic operators. Topics on Concentration Phenomena and Problems with Multiple Scales, A. Braides and V. Chiado Piat (Eds.) Lecture Notes of the Unione Matematica Italiana, 2.
D. I. Borisov, E. A. Zhizhina, A. L. Piatnitski, “Spectrum of a convolution operator with potential”, Uspekhi Mat. Nauk, 77:3(465) (2022), 173–174; Russian Math. Surveys, 77:3 (2022), 546–548
E. A. Zhizhina, A. L. Piatnitski, “On the behaviour at infinity of solutions to nonlocal parabolic type problems”, Bulletin of Irkutsk State University. Series Mathematics, 30 (2019), 99–113
Andrey Piatnitski, Volodymyr Rybalko, “Singularly perturbed spectral problems in a thin cylinder with Fourier conditions on its bases”, Zh. Mat. Fiz. Anal. Geom., 15:2 (2019), 256–277
2018
4.
R. R. Gadyl'shin, A. L. Piatnitski, G. A. Chechkin, “On the asymptotic behaviour of eigenvalues of a boundary-value problem
in a planar domain of Steklov sieve type”, Izv. RAN. Ser. Mat., 82:6 (2018), 37–64; Izv. Math., 82:6 (2018), 1108–1135
V. V. Zhikov, A. L. Piatnitski, “Homogenization of random singular structures and random measures”, Izv. RAN. Ser. Mat., 70:1 (2006), 23–74; Izv. Math., 70:1 (2006), 19–67
M. L. Kleptsyna, A. L. Piatnitski, “Homogenization of a random non-stationary convection-diffusion problem”, Uspekhi Mat. Nauk, 57:4(346) (2002), 95–118; Russian Math. Surveys, 57:4 (2002), 729–751
S. B. Kuksin, N. S. Nadirashvili, A. L. Piatnitski, “Hölder estimates for solutions of parabolic SPDEs”, Teor. Veroyatnost. i Primenen., 47:1 (2002), 152–159; Theory Probab. Appl., 47:1 (2003), 157–163
A. G. Belyaev, A. L. Piatnitski, G. A. Chechkin, “Averaging in a perforated domain with an oscillating third boundary condition”, Mat. Sb., 192:7 (2001), 3–20; Sb. Math., 192:7 (2001), 933–949
A. G. Belyaev, A. L. Piatnitski, G. A. Chechkin, “Asymptotic behavior of the solution of a boundary value problem in a punctured domain with an oscillating boundary”, Sibirsk. Mat. Zh., 39:4 (1998), 730–754; Siberian Math. J., 39:4 (1998), 621–644
S. M. Kozlov, A. L. Piatnitski, “Averaging on a background of vanishing viscosity”, Mat. Sb., 181:6 (1990), 813–832; Math. USSR-Sb., 70:1 (1991), 241–261
A. L. Piatnitski, “Averaging a singularly perturbed equation with rapidly oscillating coefficients in a layer”, Mat. Sb. (N.S.), 121(163):1(5) (1983), 18–39; Math. USSR-Sb., 49:1 (1984), 19–40
A. L. Piatnitski, “On the behaviour at infinity of the solution of a second-order elliptic equation given on a cylinder”, Uspekhi Mat. Nauk, 37:2(224) (1982), 231–232; Russian Math. Surveys, 37:2 (1982), 249–250
A. L. Piatnitski, “Propagation of a solution of a hyperbolic equation
with rapidly oscillating coefficients”, Uspekhi Mat. Nauk, 36:1(217) (1981), 223–224; Russian Math. Surveys, 36:1 (1981), 250–251
14.
A. L. Piatnitski, “A scattering problem in laminar media”, Mat. Sb. (N.S.), 115(157):3(7) (1981), 478–492; Math. USSR-Sb., 43:3 (1982), 427–441
15.
A. L. Piatnitski, “On the limit behavior of the domain of dependence of a hyperbolic equation with rapidly oscillating coefficients”, Mat. Sb. (N.S.), 115(157):1(5) (1981), 130–145; Math. USSR-Sb., 43:1 (1982), 117–131
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
2016
17.
C. Wayne, T. Kappeler, G. Kokarev, W. Craig, A. Piatnitsky, I. Chueshov, A. Shirikyan, L. H. Eliasson, “Sergei Borisovich Kuksin (on his 60th birthday)”, Uspekhi Mat. Nauk, 71:1(427) (2016), 177–183; Russian Math. Surveys, 71:1 (2016), 167–173
1996
18.
N. S. Bakhvalov, A. Yu. Belyaev, M. I. Vishik, V. V. Zhikov, V. P. Maslov, O. A. Oleinik, G. P. Panasenko, A. L. Piatnitski, “Sergei Mikhailovich Kozlov (obituary)”, Uspekhi Mat. Nauk, 51:4(310) (1996), 145–146; Russian Math. Surveys, 51:4 (1996), 723–725
Homogenization of spin energies A. L. Piatnitski Sixth International Conference on Differential and Functional Differential Equations DFDE-2011 August 16, 2011 12:55
Contact us: