1 First Order PDEs
1.1 Homogeneous Transport Equation
1.2 Inhomogeneous Transport Equation
1.3 Scalar Conservation Laws
1.4 Noncharacteristic Hypersurfaces
1.5 Method of Characteristics
1.6 Weak Solutions
2 General Concepts
2.1 Types of Second Order PDEs
2.2 The Questions
2.3 Divergence Theorem
2.4 Distributions
3 Laplace Equation
3.1 Fundamental Solution
3.2 Mean Value Property
3.3 Maximum Principle
3.4 Green’s Function
3.5 A PDE with no solutions
4 Heat Equation
4.1 Spectral Theory and the Fourier Transform
4.2 Fundamental Solution
4.3 Maximum Principle
4.4 Heat Kernels
4.5 Heat Kernel of $(0,1)$
5 Wave Equation
5.1 D’Alembert’s Formula
5.2 Solution on the half-line
5.3 Spherical Means of the Wave Equation
5.4 Solution in Dimension 3
5.5 Solution in Dimension 2
5.6 Inhomogeneous Wave Equation
5.7 Energy Methods
Appendix