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Introduction to Elliptic Equations

Lecturer: 
Gianni Dal Maso
Course Type: 
PhD Course
Academic Year: 
2013-2014
Period: 
October-January
Duration: 
50 h
Description: 

Laplace equation:

  • harmonic functions, mean value properties,
  • maximum principle,
  • Green's function,
  • Poisson kernel,
  • Harnack inequality,
  • subharmonic functions,
  • Perron-Wiener-Brelot method for the Dirichlet problem,
  • regular boundary points.

Variational theory of elliptic equations:

  • existence and uniqueness of weak solutions in Sobolev Spaces,
  • weak maximum principle,
  • eigenvalues and eigenfunctions,
  • regularity theory in Sobolev spaces and in spaces of Hölder continuous functions.

Some remarks on nonlinear elliptic equations:

  • Euler equations for minimum problems of the calculus of variations,
  • direct methods for the existence of a minimum point,
  • monotonicity methods for existence and uniqueness of solutions to some nonlinear problems,
  • variational inequalities,
  • use of fixed points theorems for the solution of some nonlinear partial differential equations.
Location: 
A-133

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