Regularity Theory for Elliptic PDE

A review on the Laplace equation §1.3.Probabilistic interpretation of harmonic functions Chapter 2. Linear elliptic PDE §2.1.Harnack's inequality §2.2.Schauder estimates for the Laplacian §2.3.Schauder estimates for operators in non-divergence form §2.4.Schauder estimates for operators in divergence form §2.5.The case of continuous coefficients §2.6.Boundary regularity Chapter 3. Nonlinear variational PDE & Hilbert's XIXth problem §3.1.Overview §3.2.Existence and basic estimates §3.3.De Giorgi's proof §3.4.Solution to Hilbert's XIXth problem §3.5.Further results and open problems Chapter 4. Fully nonlinear elliptic PDE §4.1.What is ellipticity?§4.2.Equations in two variables v -DRAFTvi Contents §4.3.Existence of solutions §4.4.Regularity of solutions: an overview §4.5.Further results and open problems Chapter 5.The obstacle problem §5.1.Some motivations and applications §5.2.Basic properties of solutions I §5.3.Basic properties of solutions II §5.4.Regularity of free boundaries: an overview §5.5.Classification of blow-ups §5.6.Regularity of the free boundary §5.7.Singular points §5.8.On the size of the singular set Appendix A. Some properties of Hölder spaces