On Cell Problems for Nonlinear PDES and Its Application to Homogenization

Abstract

In this talk, I will give an introductory talk on the homogenization for fully nonlinear PDEs. To prove “homogenization” in a periodic setting, it is well-known that the cell problem, which is a kind of nonlinear eigenvalue problems, plays an important role. In the talk, I will show some of basic arguments in Lions-Papanicolaou-Varadhan (1987), and Evans (1992) and prove the homogenization in a periodic setting by using a perturbed test function method introduced in Evans (1989) as a starting point. In the second half of the talk, I will show some of recent development in Davini-FathiIturriaga-Zavidovique (2016), Mitake-Tran (2017) on the selection problem appearing in the cell problem. These results did not currently have a clear application to homogenization, but may have potential. I finally refer to a lecture note [6] on this direction. [1] A. Davini, A. Fathi, R. Iturriaga, M. Zavidovique, Convergence of the solutions of the discounted equation, Invent. Math. 206 (1) (2016) 29-55. [2] L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 3-4, 359-375. [3] L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), no. 3-4, 245-265. [4] P.-L. Lions, G. Papanicolaou, S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished work (1987). [5] H. Mitake, H. V. Tran, Selection problems for a discount degenerate viscous Hamilton-Jacobi equation, Adv. Math. 306 (2017), 684-703. [6] N. Q. Le, H. Mitake and H. V. Tran, Dynamical and geometric aspects of Hamilton-Jacobi and linearized Monge-Ampere equations, to appear in Lecture Notes in Mathematics, Springer.