Basic notions of Functional Analysis, Measure Theory and Sobolev Spaces.
Master or PhD level
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 H¨older, 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 H¨older continuity
18 Regularity for systems
18.1 De Giorgi’s counterexample to regularity for systems
19 Partial regularity for systems 93
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.
Educational goals: In the course the basic techniques of the theory of elliptic partial differential equations are presented. Starting from the very classical methods based on energy inequalities and difference quotients, more recent tools, results and concepts are introduced, touching also the theory of viscosity solutions.
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, Lectures on Elliptic Partial Differential Equations, pu blished by Edizioni SNS in 2019.