Non Linear Partial Differential Equations

1. Maximum principles for linear elliptic equations and applications to nonlinear equations:
· The Hopf theory for uniformly elliptic equations: weak and strong maximum principles, the Hopf lemma, Serrin's comparison principle, the generalized maximum principle for narrow domains. A priori estimates for semilinear equations.
· The Alexandroff theory for elliptic equations: the Alexandroff estimate and the corresponding maximum principle, the comparison principle for domains with small volume. A priori estimates for quasilinear and fully nonlinear equations.

2. Differentiability of convex functions:
· Subdifferentials and the first order theory.
· Alexandroff's theoem on the second order differentiability in the Peano sense.
· Second order subdifferentials and superdifferentials, semicontinuous functions and semiconvex functions.
· The Jensen e Slodkowski lemmas.

3. Visocsity solutions of fully nonlinear elliptic equations:
· The notions of viscosity subsolutions and supersolutions for fully nonlinear second order equations.
· Existence of viscosity solutions of the Dirichlet problem via Perron's method (upper envelope of viscosity subsolutions) and Ishii's theorem.
· Validity of the comparison principle by way of the analysis of local extema of semicontinuous functions.
· Construction of suitable upper and lower solutions.

Time permitting, some or all of the following arguments will be treated:
· The Harvey-Lawson nonlinear potential theory approach and the notion of Krylov of viscosity solutions with admissibility constraints.
· Hölder regularity of viscosity solutions.
· The notion of generalized principle eigenvalue for homogeneous nonlinear operators.