Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media
- 635 Downloads
We establish a logarithmic-type rate of convergence for the homogenization of fully nonlinear uniformly elliptic second-order pde in strongly mixing media with similar, i.e., logarithmic, decorrelation rate. The proof consists of two major steps. The first, which is actually the only place in the paper where probability plays a role, establishes the rate for special (quadratic) data using the methodology developed by the authors and Wang to study the homogenization of nonlinear uniformly elliptic pde in general stationary ergodic random media. The second is a general argument, based on the new notion of δ-viscosity solutions which is introduced in this paper, that shows that rates known for quadratic can be extended to general data. As an application of this we also obtain here rates of convergence for the homogenization in periodic and almost periodic environments. The former is algebraic while the latter depends on the particular equation.
Unable to display preview. Download preview PDF.
- 14.Ishii, H.: Almost periodic homogenization of Hamilton-Jacobi equations. In: International Conference on Differential Equations, vols. 1, 2, Berlin, 1999, pp. 600–605. World Sci. Publ., River Edge (2000) Google Scholar
- 16.Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994) Google Scholar
- 23.Lions, P.-L., Papanicolaou, G., Varadhan, S.R.S.: Homogenization of Hamilton-Jacobi equations. Preprint Google Scholar
- 25.Papanicolaou, G., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. In: Fritz, J., Lebaritz, J.L., Szasz, D. (eds.) Proceed. Colloq. on Random Fields, Rigorous Results in Statistical Mechanics and Quantum Field Theory. Colloquia Mathematica Soc. Janos Bolyai, vol. 10, pp. 835–873. (1979) Google Scholar
- 26.Papanicolaou, G., Varadhan, S.R.S.: Diffusion with random coefficients. In: Krishnaiah, P.R.: Essays in Statistics and Probability. North Holland, Amsterdam (1981) Google Scholar