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Partial Differential Equations

Year 3 · 4 sections · 44 concepts · 80 practice questions

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What's covered

free1 · Laplace and Poisson equations12 concepts

Overview 1 · The plan of this section
Definition 1.1 · Laplace's and Poisson's equations
Proposition 1.2 · Radial harmonic functions and the fundamental solution
Theorem 1.3 · The mean value property
Theorem 1.4 · The maximum principle
Corollary 1.5 · Uniqueness and stability for the Dirichlet problem
Proposition 1.6 · Dirichlet's principle
Lemma 1.7 · Green's identities
Theorem 1.8 · Green's functions and the representation formula
Example 1.9 · The Poisson kernel for the disc and half-plane
Corollary 1.10 · Smoothness and Liouville's theorem
Check 1.11 · Check: problems for Laplace's equation

2 · The heat equation10 concepts

Overview 2 · The plan of this section
Definition 2.1 · The heat kernel
Theorem 2.2 · Solving the initial value problem
Corollary 2.3 · Smoothing and infinite speed of propagation
Proposition 2.4 · Duhamel's principle
Theorem 2.5 · The parabolic maximum principle
Theorem 2.6 · Energy methods for the heat equation
Example 2.7 · The backward heat equation is ill-posed
Remark 2.8 · Uniqueness on the whole space
Check 2.9 · Check: working with the heat equation

3 · The wave equation10 concepts

Overview 3 · The plan of this section
Theorem 3.1 · D'Alembert's formula
Corollary 3.2 · Domains of dependence and influence
Theorem 3.3 · Energy conservation
Theorem 3.4 · Finite speed of propagation in higher dimensions
Lemma 3.5 · Spherical means (Euler–Poisson–Darboux)
Theorem 3.6 · Kirchhoff's formula
Corollary 3.7 · Huygens' principle
Proposition 3.8 · The method of descent (two dimensions)
Check 3.9 · Check: working with the wave equation

4 · Distributions and weak solutions12 concepts

Overview 4 · The plan of this section
Definition 4.1 · Test functions
Definition 4.2 · Distributions
Lemma 4.3 · Functions are determined by their distributions
Definition 4.4 · Distributional derivatives
Proposition 4.5 · The jump rule
Definition 4.6 · Weak solutions
Proposition 4.7 · Travelling waves with kinks are weak solutions
Theorem 4.8 · The fundamental solution of the Laplacian
Corollary 4.9 · Solving Poisson's equation on the whole space
Example 4.10 · Fundamental solutions of other operators
Check 4.11 · Check: computing with distributions

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Proposition 1.2 · Radial harmonic functions and the fundamental solution

$\Phi = -\frac1{2\pi}\ln|x|$ (2D), $\frac1{4\pi|x|}$ (3D): radial, harmonic off $0$ , unit flux.

Proof

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