Maximum principles and Harnack’s inequality

Abstract

This thesis studies two fundamental principles in the theory of second-order elliptic and parabolic partial differential equations: the maximum principles and Harnack’s inequality. We prove the weak and strong maximum principles, using Hopf’s lemma to show that an interior maximum forces the solution to be constant. We then establish Harnack’s inequality, which provides quantitative comparison bounds for positive solutions. Starting from harmonic functions, we extend the re-sult to elliptic operators with smooth coefficients, state Moser’s results for measur-able coefficients, and prove the parabolic Harnack inequality. As an application, we use it to prove the strong maximum principle for parabolic operators. These results form a unified framework for understanding the regularity and pointwise behavior of solutions.