zbMATH — the first resource for mathematics

Homogenization of periodic linear degenerate PDEs. (English) Zbl 1167.60015
Consider a second-order lindar partial differential operator \[ L_\epsilon := \frac 12 \sum_{i,j=1}^d a_{ij}(x/\epsilon)\partial_i\partial_j + \sum_{i=1}^d \left[\epsilon^{-1}b_i(x/\epsilon)+c_i(x/\epsilon)\right]\partial_i \] where the matrix \((a_{ij})_{ij}\) may degenerate (even vanish) in some open subset \(D\subset\mathbb{R}^d\). The authors study the behaviour as \(\epsilon\to 0\) of the following elliptic and parabolic PDEs: \[ L_\epsilon u^\epsilon(x) + f(x,x/\epsilon)u^\epsilon(x) = 0,\; x\in D, \quad u^\epsilon(x)=g(x),\;x\in\partial D \] resp. \[ \partial_t u^\epsilon(t,x) = L_\epsilon u^\epsilon(t,x) + \left(\epsilon^{-1} e(x(\epsilon)+ f(x,x/\epsilon)\right)u^\epsilon(t,x), \quad u^\epsilon(0,x)=g(x),\;x\in\mathbb{R}^d. \] It is assumed that \(f\) is bounded from above and periodic in the second argument and that \(g\) is continuous and polynomially bounded; moreover \(e\) is periodic. Using the fact that the solution of such systems can be expressed as functionals of certain stochastic processes arising as solution processes of stochastic (Itô) differential equations (the coefficients can be derived from the coefficients of the PDEs under investigation), the authors prove that \(u^\epsilon \to u\) for each \(x\) resp. \((t,x)\) as \(\epsilon\to 0\) where \(u\) is the solution of a suitably ‘homogenized’ PDE. The main innovation in these results is the high degree of degeneracy which is allowed in the authors’ approach. The paper also contains a study of the image of the homogenised diffusion matrix and, in the end, some worked examples.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60J65 Brownian motion
60H07 Stochastic calculus of variations and the Malliavin calculus
35J25 Boundary value problems for second-order elliptic equations
35K15 Initial value problems for second-order parabolic equations
Full Text: DOI arXiv
[1] Acerbi, E.; Chiadò Piat, V.; Dal Maso, G.; Percivale, D., An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear anal., 18, 5, 481-496, (1992) · Zbl 0779.35011
[2] Bellieud, M.; Bouchitté, G., Homogénéisation de problèmes elliptiques dégénérés, C. R. acad. sci. Paris Sér. I math., 327, 8, 787-792, (1998) · Zbl 0920.35024
[3] Biroli, M.; Mosco, U.; Tchou, N.A., Homogenization for degenerate operators with periodical coefficients with respect to the Heisenberg group, C. R. acad. sci. Paris Sér. I math., 322, 5, 439-444, (1996) · Zbl 0851.47046
[4] Bismut, J.-M., Martingales, the Malliavin calculus and hypoellipticity under general Hörmander’s conditions, Z. wahrsch. verw. geb., 56, 4, 469-505, (1981) · Zbl 0445.60049
[5] Cioranescu, D.; Paulin, J.S.J., Homogenization in open sets with holes, J. math. anal. appl., 71, 2, 590-607, (1979) · Zbl 0427.35073
[6] De Arcangelis, R.; Serra Cassano, F., On the homogenization of degenerate elliptic equations in divergence form, J. math. pures appl. (9), 71, 2, 119-138, (1992) · Zbl 0678.35036
[7] Engström, J.; Persson, L.-E.; Piatnitski, A.; Wall, P., Homogenization of random degenerated nonlinear monotone operators, Glasg. mat. ser. III, 41(61), 1, 101-114, (2006) · Zbl 1118.35003
[8] Ethier, S.N.; Kurtz, T.G., Markov processes, Wiley series in probab. math. statist., (1986), Wiley New York · Zbl 0592.60049
[9] M. Hairer, J.C. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations, Ann. Probab., in press · Zbl 1173.37005
[10] Jikov, V.V.; Kozlov, S.M.; Oleĭnik, O.A., Homogenization of differential operators and integral functionals, (1994), Springer Berlin, translated from the Russian by G.A. Yosifian [G.A. Iosif’yan] · Zbl 0801.35001
[11] Kunita, H., Stochastic flows and stochastic differential equations, Cambridge stud. adv. math., vol. 24, (1990), Cambridge Univ. Press Cambridge · Zbl 0743.60052
[12] Kusuoka, S.; Stroock, D., Applications of the Malliavin calculus. I, (), 271-306 · Zbl 0568.60059
[13] Kusuoka, S.; Stroock, D., Applications of the Malliavin calculus. II, J. fac. sci. univ. Tokyo sect. IA math., 32, 1, 1-76, (1985) · Zbl 0568.60059
[14] Malliavin, P., Stochastic calculus of variation and hypoelliptic operators, (), 195-263
[15] Norris, J., Simplified Malliavin calculus, (), 101-130
[16] Nualart, D., The Malliavin calculus and related topics, Probab. appl. (N.Y.), (1995), Springer New York · Zbl 0837.60050
[17] Pardoux, É., Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients: A probabilistic approach, J. funct. anal., 167, 2, 498-520, (1999) · Zbl 0935.35010
[18] Pardoux, E.; Veretennikov, A.Y., On the Poisson equation and diffusion approximation. III, Ann. probab., 33, 3, 1111-1133, (2005) · Zbl 1071.60022
[19] Paronetto, F., Homogenization of degenerate elliptic – parabolic equations, Asymptot. anal., 37, 1, 21-56, (2004) · Zbl 1052.35025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.