Maximum Principle in Elliptic Problems

This lecture covers the maximum principle in elliptic problems, focusing on the symmetric, positive definite properties of the solutions and their behavior on the boundary. It discusses the application of the maximum principle to ensure the minimum and maximum values of the solutions. The lecture also explores the comparison function concept and its role in determining the extremal values of the solutions. Additionally, it delves into the implications of the maximum principle, such as inverse monotonicity and continuous dependence on the data. The lecture concludes with a detailed explanation of finite difference approximation methods for elliptic problems.