Mathematics

1. 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.

2. 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.

3. 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.