Lectures on Elliptic Partial Differential Equations

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Main Notation
Chapter 1 Variational aspects of some classes of elliptic problems
	1.1. Weak solvability results
	1.2. Inhomogeneous boundary conditions
	1.3. Elliptic systems
	1.4. Necessary minimality conditions
	1.5. Lower semicontinuity of integral functionals
Chapter 2 Classical regularity theory for linear problems
	2.1. Caccioppoli-Leray inequality
	2.2. Sobolev embeddings
	2.3. A priori estimates and the Nirenberg method
	2.4. Decay estimates for systems with constant coefficients
	2.5. Regularity up to the boundary
Chapter 3 Interior regularity for nonlinear equations
	3.1. Hölder, Morrey and Campanato spaces
	3.2. Hilbert\'s XIX problem and its solution in the two-dimensional case
	3.3. Back to linear problems: Schauder theory
	3.4. The space of BMO functions
	3.5. Regularity in Lp spaces
	3.6. De Giorgi\'s solution of Hilbert’s XIX problem
Chapter 4 Regularity for systems
	4.1. De Giorgi\'s counterexample to regularity for systems
	4.2. Partial regularity for systems: basic definitions and ancillary results
	4.3. Partial regularity for systems: Ln (Σ(u)) = 0
	4.4. Partial regularity for systems: Hn−2+ε (Σ(u)) = 0
Chapter 5 Viscosity solutions
	5.1. Basic definitions
	5.2. Viscosity versus classical solutions
	5.3. The distance function
	5.4. Maximum principle for semiconvex functions
	5.5. Existence and uniqueness results
	5.6. The ABP maximum principle
	5.7. A regularity result for viscosity solutions
Appendix A Some basic facts concerning Sobolev spaces
	A.1. Two definitions and their comparison
	A.2. Poincaré inequalities
	A.3. Sobolev inequalities
Appendix B Miscellaneous results in real and harmonic analysis
	B.1. Weak Lp spaces and the maximal operator
	B.2. Some classical interpolation theorems
	B.3. Lebesgue differentiation theorem
	B.4. Calderón-Zygmund decomposition
Appendix C Hausdorff measures
	C.1. Basic definitions
	C.2. An ad hoc covering theorem
	C.3. A comparison theorem for Hausdorff measures
Appendix D Some tools from convex and nonsmooth analysis
	D.1. Subdifferential of a convex function
	D.2. Convex functions and Measure Theory
References
Index
Lecture Notes