FUNDAMENTALS OF
APPLIED FUNCTIONAL ANALYSIS

Dragisa Mitrovic and Darko Zubrinic
University of Zagreb

Pitman Monograph and Surveys in Pure and Applied Mathematics 91, Longman, 1998 (400 pages)

Contents

1. Algebra of sets...1
2. Definition and properties of measure...4
3. Outer measure, measurable sets...7
4. Construction of measure and outer measure...8
5. Definition of a measure by extension...10
6. Measurable functions...14
7. Arithmetic operations with measurable functions...16
8. Sequences of measurable functions...18
9. Convergence almost everywhere...19
10. Convergence in measure...21

2. LEBESGUE INTEGRAL

1. Integral of simple nonnegative measurable functions...23
2. Integral of a nonnegative measurable function...25
3. Integral of measurable functions of arbitrary sign...27
4. Dominated convergence...29
5. Product of measure spaces...31
6. Fubini's theorem...33
7. Relation between the Lebesgue and Riemann integrals...37

3. L^p--SPACES

1. Basic properties of $L^p$--spaces...40
2. Reflexivity and separability of $L^p$--spaces...49
3. Convolution and regularization of functions...50
4. Regularizing sequences (mollifiers)...53
5. Kolmogorov's theorem...55
6. Nemytzki operators...57

4. SPACES OF DISTRIBUTIONS

1. Introduction...59
2. Definition of distribution...62
3. Operations with distributions...75
4. Support of a distribution...94
5. Fourier transformation...100
6. Tensor products of distributions...109
7. Convolution of distributions...116
8. Fundamental solutions...121
9. Schwartz spaces...131
10. Tempered distributions...141
11. Plemelj's distributional formulas...154

5. INTRODUCTION TO SOBOLEV SPACES

1. Weak derivatives...160
2. The definition of Sobolev spaces...166
3. Lipschitz and smooth domains...176
4. Aproximation by smooth functions...179
5. Poincar\'e's inequality...184
6. Imbedding theorems...186
7. Compact imbeddings...192
8. Dual spaces and scales of Sobolev spaces...195
9. Trace spaces...202
10. Additional remarks...208

6. ELLIPTIC BOUNDARY VALUE PROBLEMS

1. The Lax--Milgram theorem...214
2. Examples of linear elliptic problems...216
3. Regularity of weak solutions...226
4. Maximum principles...233
5. Spectrum of the Laplace operator...238
6. Monotone iterations...246
7. Galerkin's method...250
8. Poho\v zaev's identity...253

7. SOLVABILITY OF NONLINEAR ELLIPTIC EQUATIONS

1. Classification of nonlinear elliptic problems...258
2. Nonresonant problems...260
3. Lower semicontinuity and convexity...265
4. Energy functionals...271
5. Monotone operators...275
6. Problems of Landesman--Lazer type...286
7. Problems of Ambrosetti--Prodi type...291
8. Deformation lemma...303
9. Mountain Pass Theorem...306

8. TOPOLOGICAL DEGREE AND ITS APPLICATIONS

1. Basic properties...312
2. Construction of the degree...315
3. Borsuk's theorem about antipodes...322
4. Infinite-dimensional case...324
5. Degree of isolated solutions...330
6. Topological theory of bifurcations...332

Appendix...339

Solutions and hints...357

Notation...385

References...388

Index...394

Errata (PDF, 144 KB)