Math 2.0

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Partial differential equations

457 ideas in this subject. The map below lists the main ideas of the subject by area, in teaching order. 224 of its 225 ideas have a page so far.

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Map of the subject

Basic notions and classification 23 of 23

  1. Partial differential equation definition
  2. Order of a PDE definition
  3. Linear PDE definition
  4. Semilinear PDE definition
  5. Quasilinear PDE definition
  6. Fully nonlinear PDE definition
  7. Principal symbol definition
  8. Characteristic variety definition
  9. Classification of second-order PDEs concept
  10. Elliptic partial differential equation definition
  11. Parabolic partial differential equation definition
  12. Hyperbolic partial differential equation definition
  13. Mixed-type equation definition
  14. Well-posed problem definition
  15. Initial value problem definition
  16. Boundary value problem definition
  17. Dirichlet problem problem
  18. Neumann problem problem
  19. Robin boundary condition definition
  20. Free boundary problem problem
  21. Stefan problem problem
  22. Obstacle problem problem
  23. Overdetermined system concept

Existence theory 8 of 8

  1. Cauchy–Kovalevskaya theorem statement
  2. Holmgren's uniqueness theorem statement
  3. Lewy's example counterexample
  4. Malgrange–Ehrenpreis theorem statement
  5. Fundamental solution definition
  6. Green's function definition
  7. Hadamard's example of ill-posedness counterexample
  8. Local solvability concept

Generalized solutions and function spaces 25 of 25

  1. Probability distribution definition
  2. Dirac delta function object
  3. Bump function definition
  4. Schwartz space definition
  5. Tempered distribution definition
  6. Weak derivative definition
  7. Weak solution definition
  8. Sobolev space definition
  9. Sobolev inequality statement
  10. Gagliardo–Nirenberg interpolation inequality statement
  11. Morrey's inequality statement
  12. Rellich–Kondrachov theorem statement
  13. Poincaré inequality statement
  14. Trace operator definition
  15. Hölder space definition
  16. Besov space definition
  17. Triebel–Lizorkin space definition
  18. Fractional Sobolev space definition
  19. Bessel potential definition
  20. Hardy inequality in Sobolev spaces statement
  21. Lax–Milgram theorem statement
  22. Galerkin method concept
  23. Viscosity solution definition
  24. Entropy solution definition
  25. Mild solution definition

Elliptic equations 37 of 37

  1. Laplace's equation definition
  2. Laplacian definition
  3. Harmonic function definition
  4. Mean value property statement
  5. Poisson's equation definition
  6. Poisson kernel definition
  7. Newtonian potential definition
  8. Maximum principle statement
  9. Hopf lemma statement
  10. Harnack's inequality statement
  11. Liouville's theorem for harmonic functions statement
  12. Weyl's lemma statement
  13. Dirichlet's principle statement
  14. Perron method concept
  15. Kelvin transform definition
  16. Schauder estimates statement
  17. Elliptic regularity statement
  18. Calderón–Zygmund estimates statement
  19. De Giorgi–Nash–Moser theorem statement
  20. Krylov–Safonov estimate statement
  21. Fredholm alternative statement
  22. Helmholtz equation definition
  23. Biharmonic equation definition
  24. p-Laplacian definition
  25. Monge–Ampère equation definition
  26. Minimal surface equation definition
  27. Bernstein's problem problem
  28. Plateau's problem problem
  29. Yamabe problem problem
  30. Liouville's equation definition
  31. Lane–Emden equation definition
  32. Mountain pass theorem statement
  33. Lopatinskii–Shapiro condition definition
  34. Elliptic boundary value problem definition
  35. Hypoelliptic operator definition
  36. Hörmander's condition definition
  37. Cauchy–Riemann operator definition

Parabolic equations 19 of 19

  1. Heat equation definition
  2. Heat kernel definition
  3. Parabolic maximum principle statement
  4. Tychonoff's non-uniqueness example counterexample
  5. Widder's theorem statement
  6. Reaction–diffusion system definition
  7. Fisher's equation definition
  8. Allen–Cahn equation definition
  9. Cahn–Hilliard equation definition
  10. Porous medium equation definition
  11. Fokker–Planck equation definition
  12. Black–Scholes equation definition
  13. Feynman–Kac formula statement
  14. Ricci flow definition
  15. Mean curvature flow definition
  16. Kardar–Parisi–Zhang equation definition
  17. Keller–Segel model definition
  18. Turing pattern concept
  19. Harnack inequality for parabolic equations statement

Hyperbolic equations 15 of 15

  1. Wave equation definition
  2. d'Alembert's formula statement
  3. Huygens principle statement
  4. Proof by infinite descent concept
  5. Finite speed of propagation statement
  6. Energy method concept
  7. Klein–Gordon equation definition
  8. Telegrapher's equations definition
  9. Strict hyperbolicity definition
  10. Strichartz estimate statement
  11. Dispersive partial differential equation definition
  12. Geometric optics concept
  13. Propagation of singularities statement
  14. Wave front set definition
  15. Christodoulou–Klainerman stability of Minkowski space statement

First-order equations and conservation laws 18 of 18

  1. Method of characteristics concept
  2. Transport equation definition
  3. Hamilton–Jacobi equation definition
  4. Eikonal equation definition
  5. Hopf–Lax formula statement
  6. Complete integral definition
  7. Conservation law definition
  8. Hyperbolic conservation law definition
  9. Burgers' equation definition
  10. Shock wave concept
  11. Rankine–Hugoniot conditions statement
  12. Riemann problem problem
  13. Rarefaction wave concept
  14. Glimm scheme concept
  15. Cole–Hopf transformation concept
  16. Lax–Friedrichs method concept
  17. Godunov's scheme concept
  18. Courant–Friedrichs–Lewy condition statement

Solution representations and transform methods 14 of 14

  1. Separation of variables concept
  2. Fourier series method concept
  3. Fourier transform definition
  4. Laplace transform definition
  5. Duhamel's principle statement
  6. Method of images concept
  7. Spherical harmonics object
  8. Similarity solution concept
  9. Traveling wave concept
  10. Lie group method for PDEs concept
  11. Bäcklund transform concept
  12. Inverse scattering transform concept
  13. Lax pair definition
  14. Soliton concept

Spectral theory and eigenvalue problems 11 of 11

  1. Dirichlet eigenvalue definition
  2. Rayleigh quotient definition
  3. Min-max theorem statement
  4. Weyl's law statement
  5. Faber–Krahn inequality statement
  6. Hearing the shape of a drum problem
  7. Schiffer's conjecture conjecture
  8. Pompeiu problem problem
  9. Schrödinger operator definition
  10. Steklov eigenvalue problem problem
  11. Cheeger constant definition

Qualitative behavior and nonlinear analysis 14 of 14

  1. A priori estimate concept
  2. Regularity theory concept
  3. Comparison principle statement
  4. Symmetric decreasing rearrangement concept
  5. Global attractor concept
  6. Homogenization concept
  7. Calculus of variations concept
  8. Euler–Lagrange equation definition
  9. Direct method in the calculus of variations concept
  10. Hilbert's nineteenth problem problem
  11. Hilbert's twentieth problem problem
  12. Leray–Schauder degree definition
  13. Nash–Moser theorem statement
  14. Aubin–Lions lemma statement

Equations of mathematical physics 24 of 24

  1. Schrödinger equation definition
  2. Nonlinear Schrödinger equation definition
  3. Korteweg–De Vries equation definition
  4. Sine-Gordon equation definition
  5. Kadomtsev–Petviashvili equation definition
  6. Navier–Stokes equations definition
  7. Navier–Stokes existence and smoothness conjecture
  8. Existence of Leray weak solutions statement
  9. Euler equations definition
  10. Stokes flow definition
  11. Boltzmann equation definition
  12. Vlasov equation definition
  13. Maxwell's equations definition
  14. Dirac equation definition
  15. Yang–Mills equations definition
  16. Einstein field equations definition
  17. Ginzburg–Landau equation definition
  18. Gross–Pitaevskii equation definition
  19. Hamilton–Jacobi–Bellman equation definition
  20. Mean-field game concept
  21. Lamé equations of elasticity definition
  22. Shallow water equations definition
  23. Darcy's law statement
  24. Tricomi equation definition

Pseudodifferential and microlocal methods 9 of 10

  1. Pseudo-differential operator definition
  2. Symbol class definition
  3. Fourier integral operator definition
  4. Microlocal analysis concept
  5. Atiyah–Singer index theorem statement
  6. Calderón projector definition
  7. Dirichlet-to-Neumann operator definition
  8. Calderón problem problem
  9. Fractional Laplacian definition
  10. Riesz potential definition

Further topics 7 of 7

  1. Stochastic partial differential equation definition
  2. Regularity structures concept
  3. Inverse problem concept
  4. Finite element method concept
  5. Finite difference method concept
  6. Lax equivalence theorem statement
  7. Optimal transport and the Monge problem problem

More ideas in this subject

Concepts 48

Theorems and results 32

Examples 8

Counterexamples 1

Problems 5

Conjectures 1

Definitions 174

This subject is MSC class 35. See it on zbMATH Open.