Elliptic and Parabolic Maximum Principles

Abstract

This chapter focuses on the weak and strong parabolic maximum principles for second-order uniformly elliptic operators on bounded domains. Maximum principles are the fundamental tool to prove the uniqueness of solutions to parabolic Cauchy problems as well as elliptic boundary value problems. The chapter deals with solutions to the inequality and considers functions which satisfy the inequality. The main results are the weak and the strong maximum principles which hold true in both the parabolic and elliptic case. A comparison principle can be proved from the weak maximum principles. The chapter proves the uniqueness of the classical solution of the Cauchy-Dirichlet problem associated with the operator, and also considers different sets of boundary conditions, such as Neumann boundary conditions. To prove the uniqueness of the classical solution to such problems, the parabolic Hopf’s lemma plays a crucial role.