Elliptic PDE
Prerequisiti
Nozioni base di teoria della misura, equazioni alla derivate parziali e analisi funzionale
Per Master e PhD
Programma
1 Some basic facts regarding Sobolev spaces 2 Variational formulation of some PDEs
2.1 Elliptic operators
2.2 Inhomogeneous boundary conditions
2.3 Elliptic systems
2.4 Other variational aspects
3 Lower semicontinuity of integral functionals
4 Regularity Theory
4.1 Nirenberg method
5 Decay estimates for systems with constant coefficients
6 Regularity up to the boundary
7 Interior regularity for nonlinear problems
8 Holder, Morrey and Campanato spaces
9 XIX Hilbert problem and its solution in the two- dimensional case
10 Schauder theory
11 Regularity in L^p spaces
12 Some classical interpolation theorems 13 Lebesgue differentiation theorem
14 Calderòn-Zygmund decomposition
15 The BMO space
16 Stampacchia Interpolation Theorem
17 De Giorgi’s solution of Hilbert’s
XIX problem 17.1 The basic estimates
17.2 Some useful tools
17.3 Proof of Holder continuity
18 Regularity for systems
18.1 De Giorgi’s counterexample to regularity for systems
19 Partial regularity for systems
19.1 The first partial regularity result for systems 19.2 Hausdorff measures
19.3 The second partial regularity result for systems 20 Some tools from convex and non-smooth analysis 20.1 Subdifferential of a convex function
20.2 Convex functions and Measure Theory
21 Viscosity solutions
21.1 Basic definitions
21.2 Viscosity solution versus classical solutions 21.3 The distance function
21.4 Maximum principle for semiconvex functions 21.5 Existence and uniqueness results
21.6 H ̈older regularity
22 Regularity theory for viscosity solutions
22.1 The Alexandrov-Bakelman-Pucci estimate
22.2 The Harnack Inequality.
Obiettivi formativi
Nel corso vengono presentate le tecniche di base della teoria delle equazioni differenziali parziali ellittiche. Partendo dai metodi più classici basati sulle disuguaglianze energetiche e sui quozienti differenziali, vengono introdotti strumenti, risultati e concetti più recenti, presentando anche la teoria delle soluzioni di viscosità.
Riferimenti bibliografici
L.C.EVANS. Partial Differential Equations, American Mathematical Society, 1998.
M.GIAQUINTA, L.MARTINAZZI. An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. Edizioni della Normale, 2005.
L.AMBROSIO, A.CARLOTTO, A.MASSACCESI, Lecture Notes in PDE, Edizioni della Normale
L.C.EVANS. Partial Differential Equations, American Mathematical Society, 1998.