Elliptic Partial Differential Equations: Existence and Regularity of Solutions
Prerequisites
Basic knowledge of Functional Analysis, PDE and Measure Theory
Programme
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.
Educational aims
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.
Bibliographical references
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.
M.GIAQUINTA, L.MARTINAZZI. An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. Edizioni della Normale, 2005.