Zoran Skoda partial differential equation

• $n$Lab: differential equation, homological algebra in the finite element method, regular differential operator, variational calculus, diffiety, D-scheme, D-module, h-principle, pseudodifferential operator, polyfold
• MathOverflow: why-cant-there-be-a-general-theory-of-nonlinear-pde, recent-fundamental-new-directions-in-pdes, counterexamples-in-pde
• Sergiu Klainerman, PDE as a unified subject, GAFA, Geom. Funct. Anal. pdf
• Lars Hörmander, The analysis of linear partial differential operators, 4 vols. Springer
• J. J. Duistermaat, Fourier integral operators, Progress in Math. 130, Birkhauser 1996
• Gerald B. Folland, Introductions to PDEs, Princeton Univ. Press 1995
• L. C. Evans, Partial differential equations, Amer. Math. Soc. 1998
• David Gilbarg, Neil S. Trudinger, Elliptic differential equations of second order, Springer 1977, 1983; rus. transl, Nauka 1989
• M. E. Taylor, Pseudodifferential operators, Springer 1974
• M. Kashiwara, T. Kawai, T. Kimura, Foundations of algebraic analysis, Princeton Univ. Press 1986
• M. Kashiwara, D-modules and microlocal analysis, Transl. of math. monographs 217, Amer. Math. Soc. 2000

Local behaviour

• elliptic, hyperbolic and parabolic type equations
• globally those can mix, having different type in different regions (mixed PDEs)

Some fairly general theorems and principles

• Cauchy-Kowalewski theorem (wikipedia) – quite general local existence and uniqueness theorem, but limited only to analytic coefficients and analytic solutions
• fixed point theorems (Banach, Schauder, Leray etc.): useful for evolution equations (including for ODEs, e.g. )
• index theorems
• asymptotic methods
• variational principles
• energies, conservation laws, dissipation
• various entropies
• a priori estimates
• global estimates
• Schauder estimates
• Harnack inequality
• h-principle
• characteristics of hyperbolic PDEs
• dispersion
• ellipticity
• transport
• incompressibility
• semigroups of operators

Linear PDEs

Great success of Fourier methods (esp. for constant coefficients).

Important classes:

• heat equation
• wave equation
• Helmholtz wave equation
• Laplace equation
• Maxwell equations
• self dual Maxwell equations
• Schroedinger equation
• Dirac equation
• Klein-Gordon equation
• Proca equation

Important nonlinear PDEs

• Einstein equations
• Euler equation
• Ricci flow
• Navier-Stokes equation
• Landau-Ginzburg equation
• Yang-Mills equations
• compressible Euler equation

Partial differential relations having large families of solutions

• Monge-Ampere equation
• Cauchy-Riemann equation

Exactly solvable equations and related methods

• nonlinear Schroedinger equation (coming from quantum optics)
• Burgess equation
• KdV equation

Frameworks for defining solutions and defining their precursors

• various types of weak solutions in spaces of functions and distributions
• (sheaves of) hyperfunctions
• D-modules
• parametrix
• fundamental solution and Green functions
• formal sections
• diffieties

important: propagation of singularities

Functional spaces

• spaces of analytic functions
• $L_p$-spaces, Hoelder spaces, Sobolev spaces, Besov spaces
• distributions (and also densities and currents) of Sobolev, Schwarz, Coulombeau, hyperfunctions

Role of Fredholm theory, Sobolev and other embedding theorems, iterative schemes for improving regularity, interpolation spaces and so on.

Types of problems

• boundary value problems

Numerical methods

• difference schemes
• finite element methods
• Monte-Carlo methods

Generalized PDEs and PDOs

• stochastic PDEs
• PDEs in cohesive topoi
• noncommutative PDEs
• difference equations
• integrodifferential equations
• differential equations for functions whose arguments are in Banach, Frechet spaces and so on
• second quantized equations in QFT
• fractional PDEs
• pseudodifferential operators and Fourier integral operators