Partial Differential Equations


A crash course on Sobolev spaces. Second order linear elliptic equations (existence of weak solutions; regularity in the interior and up to the boundary; maximum principles; Harnack inequality; De Giorgi-Nash-Moser theory). Second order linear parabolic equations (existence via Galerkin method; regularity theory and maximum principles). The Calculus of Variations (Euler-Lagrange equation; existence of minimizers; regularity; unilateral constraints: variational inequalities and free boundary problems). Nonvariational techniques (monotonicity and fixed point methods). Degenerate and singular PDEs (the p-Laplace equation; intrinsic scaling; the infinity Laplacian).

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Defended Theses

  • Statistical instability in chaotic dynamics
      Muhammad Ali Khan (March 2018)
      José Ferreira Alves
  • Profinite topologies on the free group
      Khadijeh Alibabaei (March 2018)
      Jorge Almeida
  •   Rui Manuel Tavares Pinto de Sá Pereira (January 2018)
      Evelina Shamarova
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