List of Differential Operators and Equations

Introduction

This article lists the differential operators and the differential equations the corpus introduces. A differential operator is a map on functions assembled from the partial derivatives, and a differential equation is the equation obtained by setting such an operator, possibly with nonlinear terms, equal to a datum; a row below names one, records its order, its symbol and the solution theory that accompanies it, and points to the article that introduces it. Every row points to an article; this article introduces nothing and proves nothing. The objects are grouped by the layer that introduces them: the operators of the vector and complex calculus, the linear equations of first and second order, the classification by symbol into the elliptic, parabolic and hyperbolic types, the nonlinear and nonlocal equations, and the solution theories that the corpus attaches to them. The rows that record a failure — an equation with no solution, an ill-posed problem, a breakdown of uniqueness — stand beside the equations as non-examples, so that the article records examples and non-examples side by side.

The Operators of the Vector, Complex and Harmonic Calculus

Object	Its order, symbol and setting	Introduced in
the derivative and the Fréchet derivative	order $1$; symbol $\sigma(D)(\xi) = i\xi$; the differential of a map between normed spaces	Differential Calculus on Normed Spaces
the gradient, divergence and curl	order $1$; the symbols $\xi$, $\langle\xi,\cdot\rangle$ and $\xi\times\cdot$; the operators of the vector calculus, and the Laplacian as $\operatorname{div}\nabla$	Differential Forms and Stokes' Theorem
the exterior derivative $d$	order $1$; symbol the exterior multiplication by $i\xi$; the operator of the de Rham complex, with $d^2=0$	Differential Forms and Stokes' Theorem
the Laplacian $\Delta$	order $2$; symbol $-\lvert\xi\rvert^2$; the elliptic operator of the corpus, with $\Delta = \operatorname{div}\nabla$	Partial Differential Equations; Harmonic Maps
the d'Alembertian $\Box = \partial_t^2 - \Delta$	order $2$; symbol $-\xi_0^2 + \lvert\xi\rvert^2$; the hyperbolic operator of the wave equation	Partial Differential Equations; Biquaternion Analysis
the Cauchy–Riemann operator $\bar\partial$ and its conjugate	order $1$; symbol $i(\xi_1 + i\xi_2)/2$; the operator of the complex and hypercomplex analysis	Regularity and the Cauchy–Riemann Operator; Complex Analysis
the Dirac operator $D$ and the twisted Cauchy–Riemann operator	order $1$; symbol the Clifford multiplication by $\xi$; the square is the Laplacian with a curvature term	Dirac Differential Operators; Clifford Modules and the Twisted Cauchy–Riemann Operator
a Fourier multiplier	order and symbol free; defined by $\widehat{m(D)f} = m(\xi)\hat f$, and the general constant-coefficient operator	Fourier Analysis on Euclidean Spaces; Pseudodifferential Operators
a pseudodifferential operator	order $m$; a symbol $a(x,\xi)$ with an asymptotic expansion; the calculus that closes under composition and adjoints	Pseudodifferential Operators; Microlocal Analysis

The Linear Equations of First and Second Order

Object	Its order, symbol and solution theory	Introduced in
a linear ordinary differential equation $y' = A(t)y$	order $1$; symbol $-i\lambda$; solved by the matrix exponential, with the fundamental matrix	Ordinary Differential Equations
the Sturm–Liouville problem	order $2$; symbol $-(p\xi^2) + q$; the eigenvalue problem with the Riccati and oscillation theory	Ordinary Differential Equations; The Calculus of Variations
the Laplace equation $\Delta u = 0$	order $2$; symbol $-\lvert\xi\rvert^2$; harmonic functions, the mean value property and the maximum principle	Partial Differential Equations; Harmonic Maps
the Poisson equation $\Delta u = f$	order $2$; symbol $-\lvert\xi\rvert^2$; solved by the Newtonian potential, the fundamental solution of $\Delta$	Distributions and Fundamental Solutions; Partial Differential Equations
the heat equation $u_t = \Delta u$	order $2$; symbol $i\tau + \lvert\xi\rvert^2$; solved by the Gaussian kernel and the heat semigroup	Semigroups and Evolution Equations; Partial Differential Equations
the wave equation $u_{tt} = \Delta u$	order $2$; symbol $-\tau^2 + \lvert\xi\rvert^2$; solved by the Kirchhoff and d'Alembert formulas, finite propagation speed	Partial Differential Equations
the Schrödinger equation $iu_t = -\Delta u + Vu$	order $2$; symbol $\tau - \lvert\xi\rvert^2 + V$; solved by the unitary group $e^{-itH}$ of a self-adjoint operator, by Stone's theorem	Semigroups and Evolution Equations; Integrable Systems
the transport equation $u_t + b\,u_x = 0$	order $1$, hyperbolic; symbol $i\tau + ib\xi$; solved along the characteristics by translation of the datum, with no smoothing	Partial Differential Equations
a delay differential equation	order $1$ with memory; the symbol is entire, $i\tau - A - Be^{-\tau h}$; solutions by the method of steps	Delay and Functional Differential Equations
a fractional differential equation	order $\alpha \in (0,1)$; symbol $\lvert\xi\rvert^\alpha$; solved by the Mittag-Leffler functions and the fractional semigroup	Fractional Differential Equations

The Classification by Symbol

Object	The classification it carries	Introduced in
an elliptic operator	principal symbol positive definite: the Laplace type, with the elliptic regularity theory	Partial Differential Equations; Regularity and the Cauchy–Riemann Operator
a parabolic operator	symbol of the form $i\tau + a(\xi)$ with $a \geq 0$: the heat type, with smoothing for positive time	Partial Differential Equations; Semigroups and Evolution Equations
a hyperbolic operator	nondegenerate symbol of signature $(1,n)$: the wave type, with finite propagation speed and the Cauchy problem	Partial Differential Equations
the characteristic variety and the wave front set	the zeros of the principal symbol, and the directions in which a distribution fails to be smooth	Microlocal Analysis; Pseudodifferential Operators
the fundamental solution of an operator	the distribution $E$ with $PE = \delta$; the Green function and the parametrix	Distributions and Fundamental Solutions
the semigroup generated by an operator	a closed densely defined $A$ with the Hille–Yosida condition; $u' = Au$ solved by $T(t)$	Semigroups and Evolution Equations

The Nonlinear, Variational and Geometric Equations

Object	Its order, symbol and solution theory	Introduced in
the Euler–Lagrange equation	order $2$; the variational equation of a functional, symbol from the second variation	The Calculus of Variations; Lagrangian and Hamiltonian Systems
the minimal-surface equation	order $2$, quasilinear elliptic; the mean curvature equation, solved by the plateau problem	Minimal Surfaces
the harmonic-map equation	order $2$, semilinear elliptic; the harmonic map equation $\Delta u = A(u)(du,du)$	Harmonic Maps
the Hamilton–Jacobi equation	order $1$, nonlinear; the eikonal and the characteristics	Lagrangian and Hamiltonian Systems; The Calculus of Variations
the Korteweg–de Vries equation	order $3$, dispersive; solved by the inverse scattering transform, with solitons	Integrable Systems; Soliton Theory
the nonlinear Schrödinger equation	order $2$, dispersive; the integrable and the focusing cases	Integrable Systems
the Navier–Stokes equations	order $2$, parabolic nonlinear; the existence and regularity problem	Partial Differential Equations; Stochastic Partial Differential Equations
a stochastic partial differential equation	order $2$ with a noise term; solved in the Itô and the Stratonovich sense	Stochastic Partial Differential Equations
an integral equation	order $0$; the Volterra and Fredholm forms, solved by the resolvent kernel	Distributions and Fundamental Solutions

Failures of the Solution Theory

Object	The failure	Introduced in
the backward heat equation	ill-posed: the solution does not depend continuously on the data	Partial Differential Equations
a nonlinear equation with blow-up	no global solution, the norm reaching infinity in finite time	Partial Differential Equations; Harmonic Maps
the wave equation with a non-smooth characteristic datum	no classical solution; the solution must be taken in the sense of distributions or of energy	Distributions and Fundamental Solutions; Partial Differential Equations
a hyperbolic equation with a double characteristic	loses the finite propagation speed of the strictly hyperbolic case	Microlocal Analysis
the Euler–Lagrange equation of a non-convex functional	a critical point need not be a minimiser, and a minimiser need not exist	The Calculus of Variations; Nonlinear Functional Analysis
the Navier–Stokes equations in three dimensions	global regularity is unresolved; only the local and the weak theories are known	Partial Differential Equations

Summary

This list gathers the differential operators and equations of the corpus: the derivative, the vector-calculus operators, the Laplacian, the d'Alembertian, the Cauchy–Riemann and Dirac operators, the Fourier multipliers and the pseudodifferential operators; the linear equations of first and second order, from the ordinary equation to the heat, wave, Schrödinger and Sturm–Liouville problems; the elliptic, parabolic and hyperbolic classification by symbol; the variational, geometric and dispersive nonlinear equations; and the solution theories — fundamental solutions, semigroups, energy methods and microlocal parametrices. The closing table records the equations whose solution theory breaks down.

Summary of Notation

The objects are named by their standard symbols; the tables use the following.

Symbol	Meaning
$\Delta$, $\Box$	Laplacian, d'Alembertian
$\bar\partial$, $D$	Cauchy–Riemann operator, Dirac operator
$\sigma(P)$, $a(x,\xi)$	principal symbol, full symbol of an operator
$E$, $G$	fundamental solution, Green function
$T(t)$, $A$	semigroup, its generator
$\operatorname{char}(P)$	characteristic variety