Zhongwei Shen, lecture 4.2, Boundary Value Problems with Oscillating Boundary Data,

Lectures on Elliptic Homogenization
Lecture IV Boundary value problems with oscillating boundary data
Zhongwei Shen, University of Kentucky

Boundary Layers in Periodic Homogenization

In this short course we will present some of the recent development on homogenization of second-order elliptic equations with rapidly oscillating periodic coefficients. We will start with an introduction of the classical homogenization theory and then study the quantitative properties of solutions: convergence rates and uniform regularity estimates. Finally, we describe recent work on the analysis of boundary layers for solutions of boundary value problems with oscillating boundary data.

Lecture I Introduction to homogenization of elliptic equations
Lecture II Convergence rates
Lecture III Uniform regularity estimates
Lecture IV Boundary value problems with oscillating boundary data

Prerequisites: A year-long graduate course on PDEs and a year-long graduate course on real analysis.

Presented at the Presented at the 28th Annual PCMI Summer Session will be held July 1–21, 2018, at the Prospector Conference Center in Park City, Utah. The residential, three-week Summer Session is the flagship activity of the IAS/Park City Mathematics Institute (PCMI), an outreach program of the Institute for Advanced Study. The 2018 research theme is Harmonic Analysis.

The PCMI Summer Session is an intensive program that includes several parallel sets of activities aimed at different groups of participants across the entire mathematics community. These activities include a program for mathematics researchers, eight mini-courses for graduate students on topics related to harmonic analysis, two lecture series for undergraduate students, a program for faculty from predominantly undergraduate institutions, a workshop on equity and mathematics education for undergraduate faculty, and a professional development program for K–12 school teachers. All of PCMI’s groups meet simultaneously, pursuing individual courses of study and a meaningful amount of interaction both scientifically and socially.