The principal theme of this book is “the existence and differentiability of the solutions of variational problems involving multiple integrals.” We shall discuss the corresponding questions for single integrals only very briefly since these have been discussed adequately in every other book on the calculus of variations.Moreover， applications to engineering，physics，etc.，are not discussed at all；however，we do discuss mathematical applications to such subjects as the theory of harmonic integrals and the so-called “d-Neumann” problem （see Chapters 7 and 8）.Since the plan of the book is described in Section 1.2 below we shall merely make a few observations here.
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 differentiabilitv theory. Introduction
1.11. Differentiability; reduction to linear equations
Chapter 2 Semi-classical results
2.1. Introduction
2.2. Elementary properties of harmonic functions
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 differentiabilitv 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'Sfunctions
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 thesecond order
……
Chapter 3 The spaces Hmp and Hmpo
Chapter 4 Existence theorems
Chapter 5 Differentiability of weak solutions
Chapter 6 Regularity theorems for the solutions of general ellipticsystems and boundary value problems
Chapter 7 A variational method in the theory of harmonicintegrals
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