To be reviewed!

This course is an introduction to Partial Differential Equations. We will review some basic notions from the book H.L. Royden's Real Analysis with Fitzpatrick 4th ed beginning \( L^p \) spaces ch. 7. Then we proceed to the rigors and applications of PDE. This course will be team teach by Prof. Joaquim Corriea of Universidade de Évora and Prof. Jayrold Arcede. The following plan maybe applicable to this course.

The two primary goals of many pure and applied scientific disciplines can be summarized as follows:

1. Formulate/devise a collection of mathematical laws (i.e., equations) that model the phenomena of interest.
2. Analyze solutions to these equations in order to extract information and make predictions.

The end result of i) is often a system of partial differential equations (PDEs). Thus, ii) often entails the analysis of a system of PDEs. This course will provide an application-motivated introduction to some fundamental aspects of both i) and ii).

In order to provide a broad overview of PDEs, our introduction to i) will touch upon a diverse array of equations including

1. The Laplace and Poisson equations of electrostatics;
2. The diffusion equation, which models e.g. the spreading out of heat energy and chemical diffusion processes;
3. The Schrödinger equation, which governs the evolution of quantum-mechanical wave functions;
4. The wave equation, which models e.g. the propagation of sound waves in the linear acoustical approximation;
5. The Maxwell equations of electrodynamics; and other topics as time permits.

In our introduction to ii), we will study three important classes of PDEs that differ markedly in their quantitative and qualitative properties: elliptic, diffusive, and hyperbolic. In each case, we will discuss some fundamental analytical tools that will allow us to probe the nature of the corresponding solutions.