Sobolev Spaces and Partial Differential Equations

Hölder's spaces. Sobolev spaces: approximation, extension, features, Sobolev inequalities, compactness, Poincaré inequality, characterization by Fourier transform, Other spaces (negative Sobolev and spaces with time). Second Order Elliptical Equations: definition and weak solution. Existence of weak solution: Lax-Milgram; Energy Estimates; Fredholm alternative. Regularity. Maximum principle. Eigenvalues and self-functions.

Basic Information

45 hours
PDE and Applications, Functional Analysis: Fundamentals, Functional Analysis: Linear Operators.


  • Lawrence C. Evans; Partial Differential Equations; Springer-Verlag.
  • John, Fritz (1982); Partial Differential Equations; Springer-Verlag.
  • Iório, Rafael & Iório, Valéria; (1988); Equações Diferenciais Parciais: Uma Introdução. IMPA.


  • H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Springer Science & Business Media, 2010
  • Figueredo, Djairo G. (1987); Análise de Fourier e EDP; IMPA.
  • Gustafson, Karl E. (1980); Partial Differential Equations and Hilbert Spaces Methods; John Wiley & Sons.
  • Smoller, Joel; Reaction Diffusion Equations; Springer-Verlag.
  • Trudinger, N.; Gilbarg, D. (1983); Elliptic PDE of Second Order; Springer-Verlag.
  • LECUN, Yann; BENGIO, Yoshua; HINTON, Geoffrey. Deep learning. nature, v. 521, n. 7553, p. 436, 2015.