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.
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