In this talk I will discuss surface integrals of rapidly oscillating functions, and their limit behavior: $$ lim_{e to 0} int_{Gamma} g(y,frac{y}{e}) dsigma_y, $$ where $g(x,y)$, is integrable in both variables, and continuous $1$-periodic in $y$-variable, $Gamma$ is a given $C1$ surface in $R^n$ ($ngeq 2$). It turns out that the lower dimensional character of the surface $Gamma$ gives rise to unexpected and surprising effective limits. In general, the limit of the integral depends strongly on the sequence $e=e_j$ chosen.
A direct consequence of this integral averaging is homogenization of Dirichlet and related problems for operators with Poison kernels, in smooth domains, and with rapidly oscillating data. The latter, in its own turn, might give accurate information about the boundary layer phenomena for numerical approximations and fast convergence of homogenization problems that has been much in focus lately.
(This is a joint work with Ki-Ahm Lee.) See the Nonlinear Analysis Seminar web page.
Speaker: Henrik Shagholian, Royal Institute of Technology, Stockholm