Aims: To introduce students to modern PDE theory through the study of qualitative properties of solutions, principally those that derive from Maximum Principles. The treatment will be rigorous and will lead to specific nonlinear examples.
Students should be able to state definitions, and state and prove theorems, in the analysis of partial differential equations. They should be able to apply maximum principles to questions of existence, uniqueness, symmetry and boundedness/blow-up for solutions of PDE.
Analysis of structure of problems, choice and application of appropriate techniques for their solution, presentation of theory and solution of problems in a rigorous and lucid manner.
Weak maximum principles for twice continuously differentiable solutions of linear elliptic PDE; interior ball property, Hopf boundary point lemma, strong maximum principles. Applications might include; uniqueness of solutions of linear Poisson equations; symmetry of non-negative solutions of nonlinear Poisson equation in a ball; Perron approach to existence. Maximum principles for parabolic equations; comparison theorem for nonlinear case. Applications might include: bounds for diffusive Burgers' equation via upper solutions; proof of blow-up for nonlinear diffusion equations via lower solutions; connection with similarity solutions; systems of reaction-diffusion equations.