Periodic Homogenization for Fully Nonlinear Equations in Half-Space Type Domains with Neumann Boundary Conditions

Francesca Da Lio

University of Padova
ABSTRACT:

The theory of viscosity solutions, initiated in the early 80 an
extremely convenient PDE framework for dealing with the lack of
smoothness of the solutions to fully nonlinear first and second order
equations. This theory provides a body of simple and effective
techniques for ascertaining the existence, uniqueness, and stability of
solutions for certain nonlinear equations via maximum principle type
arguments entailing smooth test functions. The scope of the theory is
quite broad and one of the main applications is the homogenization of
fully nonlinear equations. The approach to homogenization of nonlinear
equations by viscosity solutions methods begins with the pioneering
unpublished paper by Lions, Papanicolau & Varadhan[5] that
introduced the effective Hamiltonian and gave the first convergence
result for Hamilton-Jacobi equations. The convergence proof was then
simplified and extended to second-order equations by Evans[3, 4]. The
theory of homogenization was then continued by many authors to cover a
number of different issues. In this talk we will describe some
viscosity methods and techniques to studyperiodic homogenization of
fully nonlinear PDEs in Rn . We will also present some recent results
obtained in a joint work with G. Barles. P.L. Lions and P. Souganidis
[1] about periodic homogenization of elliptic and parabolic boundary
value problems in half-space type domains with Neumann boundary
conditions.

References
[1]G.Barles, F. Da Lio, Francesca, P.L. Lions, P. Souganidis. Ergodic problems 
and periodic homogenization for fully nonlinear equations in half-space type 
domains with Neumann boundary conditions. to appear in Indiana Univ. Math. J.
[2] M.G. Crandall, P.L. Lions. Viscosity solutions of Hamilton-Jacobi equations. 
Trans. Amer. Math. Soc. 277 (1983), no. 1, 1-42.
[3] 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
[4] L.C. Evans.Periodic homogenization of certain fully nonlinear partial differential 
equations. Proc. Roy.
Soc. Edinburgh Sect. A 120 (1992), no. 3-4, 245
[5] P.L Lions, G. Papanicolau, S. Varadhan. Homogenization of Hamilton-Jacobi 
equations. Unpublished.