This text on partial differential equations is intended for readers who want to understand the theoretical underpinnings of modern PDEs in settings that are important for the applications without using extensive analytic tools required by most advanced texts. The assumed mathematical background is at the level of multivariable calculus and basic metric space material, but the latter is recalled as relevant as the text progresses. The key goal of this book is to be mathematically complete without overwhelming the reader, and to develop PDE theory in a manner that reflects how researchers would think about the material. A concrete example is that distribution theory and the concept of weak solutions are introduced early because while these ideas take some time for the students to get used to, they are fundamentally easy and, on the other hand, play a central role in the field. Then, Hilbert spaces that are quite important in the later development are introduced via completions which give essentially all the features one wants without the overhead of measure theory. There is additional material provided for readers who would like to learn more than the core material, and there are numerous exercises to help solidify one's understanding. The text should be suitable for advanced undergraduates or for beginning graduate students including those in engineering or the sciences.
Where do PDE come from First order scalar semilinear equations First order scalar quasilinear equations Distributions and weak derivatives Second order constant coefficient PDE: Types and d'Alembert's solution of the wave equation Properties of solutions of second order PDE: Propagation, energy estimates and the maximum principle The Fourier transform: Basic properties, the inversion formula and the heat equation The Fourier transform: Tempered distributions, the wave equation and Laplace's equation PDE and boundaries Duhamel's principle Separation of variables Inner product spaces, symmetric operators, orthogonality Convergence of the Fourier series and the Poisson formula on disks Bessel functions The method of stationary phase Solvability via duality Variational problems Bibliography