Chapter 1 Introduction
1.1. Introductory remarks
1.2. The plan of the book: notation
1.3. Very brief historical remarks
1.4. The euler equations
1.5. Other classical necessary conditions
1.6. Classical sufficient conditions
1.7. The direct methods
1.8. Lower semicontinuity
1.9. Existence
1.10. The differentiability theory. introduction
1.11. Differentiability; reduction to linear equations
Chapter 2 Semi-classical results
2.1. Introduction
2.2. Elementary properties of harmonic functions
2.3. Weyl's lemma
2.4. Poisson's integral formula; elementary functions; GREEN'S functions
2.5. Potentials
2.6. Generalized potential theory; singular integrals
2.7. The calderon-zygmund inequalities
2.8. The maximum principle for a linear elliptic equation of the second order
Chapter 3 The spaces Hmp and Hmp0
3.1. Definitions and first theorems
3.2. General boundary values; the spaces hmp0 (g); weak convergence
3.3. The dirichlet problem
3.4. Boundary values
3.5. Examples; continuity; some sobolev lemmas
3.5. Miscellaneous additional results
3.7. Potentials and quasi-potentials; generalizations
Chapter 4 Existence theorems
4.1. The lower-semicontinuity theorems of SERRIN
4.2. Variational problems with f = f(p); the equations (i. t 0. t 3) with n = i, bi=0, aα = aα(p)
4.3. The borderline cases k = v
4.4. The general quasi-regular integral
Chapter 5 Differentiability of weak solutions
5.1. Introduction
5.2. General theory; V >2
5.3. Extensions Of The DE giorgi-nash-moser results; V >2
5.4. The case V=2
5.5. Lp and SCHAUDER estimates
5.6. The Equation a.▽2u+b.▽u+cu-λu=f
5.7. Analyticity of the solutions of analytic linear equations
5.8. Analyticity of the solutions of analytic, non-linear, elliptic equations
5.9. Properties of the extremals; regular cases
5.10. The extremals in the case 1
