Chapter I. Calculus for Functions of One Variable
0. Prerequisites
Properties of the real numbers, limits and convergence of sequences of
real numbers, exponential function and logarithm. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1. Limits and Continuity of Functions
Definitions of continuity, uniform continuity, properties of continuous
functions, intermediate value theorem, H¨older and Lipschitz continuity.
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2. Differentiability
Definitions of differentiability, differentiation rules, differentiable
functions are continuous, higher order derivatives. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3. Characteristic Properties of Differentiable Functions.
Differential Equations
Characterization of local extrema by the vanishing of the derivative,
mean value theorems, the differential equation f = γf, uniqueness of
solutions of differential equations, qualitative behavior of solutions of
differential equations and inequalities, characterization of local maxima
and minima via second derivatives, Taylor expansion. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4. The Banach Fixed Point Theorem. The Concept of
Banach Space
Banach fixed point theorem, definition of norm, metric, Cauchy
sequence, completeness. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
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5. Uniform Convergence. Interchangeability of Limiting
Processes. Examples of Banach Spaces. The Theorem
of Arzela-Ascoli
Convergence of sequences of functions, power series, convergence
theorems, uniformly convergent sequences, norms on function spaces,
theorem of Arzela-Ascoli on the uniform convergence of sequences of
uniformly bounded and equicontinuous functions. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6. Integrals and Ordinary Differential Equations
Primitives, Riemann integral, integration rules, integration by parts,
chain rule, mean value theorem, integral and area, ODEs, theorem of
Picard-Lindel¨of on the local existence and uniqueness of solutions of
ODEs with a Lipschitz condition. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Chapter II. Topological Concepts
7. Metric Spaces: Continuity, Topological Notions,
Compact Sets
Definition of a metric space, open, closed, convex, connected, compact
sets, sequential compactness, continuous mappings between metric spaces,
bounded linear operators, equivalence of norms in Rd, definition of a
topological space. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Chapter III. Calculus in Euclidean and Banach Spaces
8. Differentiation in Banach Spaces
Definition of differentiability of mappings between Banach spaces,
differentiation rules, higher derivatives, Taylor expansion. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
9. Differential Calculus in Rd
A. Scalar valued functions
Gradient, partial derivatives, Hessian, local extrema, Laplace
operator, partial differential equations
B. Vector valued functions
Jacobi matrix, vector fields, divergence, rotation. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
10. The Implicit Function Theorem. Applications
Implicit and inverse function theorems, extrema with constraints,
Lagrange multipliers. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Contents XIII
11. Curves in Rd. Systems of ODEs
Regular and singular curves, length, rectifiability, arcs, Jordan arc
theorem, higher order ODE as systems of ODEs. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Chapter IV. The Lebesgue Integral
12. Preparations. Semicontinuous Functions
Theorem of Dini, upper and lower semicontinuous functions, the
characteristic function of a set. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
13. The Lebesgue Integral for Semicontinuous Functions.
The Volume of Compact Sets
The integral of continuous and semicontinuous functions, theorem of
Fubini, volume, integrals of rotationally symmetric functions and other
examples. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
14. Lebesgue Integrable Functions and Sets
Upper and lower integral, Lebesgue integral, approximation of
Lebesgue integrals, integrability of sets. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
15. Null Functions and Null Sets. The Theorem of Fubini
Null functions, null sets, Cantor set, equivalence classes of integrable
functions, the space L1, Fubini’s theorem for integrable functions.
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
16. The Convergence Theorems of Lebesgue Integration
Theory
Monotone convergence theorem of B. Levi, Fatou’s lemma, dominated
convergence theorem of H. Lebesgue, parameter dependent integrals,
differentiation under the integral sign. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
17. Measurable Functions and Sets. Jensen’s Inequality.
The Theorem of Egorov
Measurable functions and their properties, measurable sets,
measurable functions as limits of simple functions, the composition of a
measurable function with a continuous function is measurable, Jensen’s
inequality for convex functions, theorem of Egorov on almost uniform
convergence of measurable functions, the abstract concept of a measure.
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
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18. The Transformation Formula
Transformation of multiple integrals under diffeomorphisms, integrals
in polar coordinates. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Chapter V. Lp and Sobolev Spaces
19. The Lp-Spaces
Lp-functions, H¨older’s inequality, Minkowski’s inequality,
completeness of Lp-spaces, convolutions with local kernels, Lebesgue
points, approximation of Lp-functions by smooth functions through
mollification, test functions, covering theorems, partitions of unity.
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
20. Integration by Parts. Weak Derivatives. Sobolev Spaces
Weak derivatives defined by an integration by parts formula, Sobolev
functions have weak derivatives in Lp-spaces, calculus for Sobolev
functions, Sobolev embedding theorem on the continuity of Sobolev
functions whose weak derivatives are integrable to a sufficiently high
power, Poincar´e inequality, compactness theorem of Rellich-Kondrachov
on the Lp-convergence of sequences with bounded Sobolev norm.
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Chapter VI. Introduction to the Calculus of Variations and
Elliptic Partial Differential Equations
21. Hilbert Spaces. Weak Convergence
Definition and properties of Hilbert spaces, Riesz representation
theorem, weak convergence, weak compactness of bounded sequences,
Banach-Saks lemma on the convergence of convex combinations of
bounded sequences. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
22. Variational Principles and Partial Differential Equations
Dirichlet’s principle, weakly harmonic functions, Dirichlet
problem, Euler-Lagrange equations, variational problems, weak lower
semicontinuity of variational integrals with convex integrands, examples
from physics and continuum mechanics, Hamilton’s principle, equilibrium
states, stability, the Laplace operator in polar coordinates. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
23. Regularity of Weak Solutions
Smoothness of weakly harmonic functions and of weak solutions of
general elliptic PDEs, boundary regularity, classical solutions. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
Contents XV
24. The Maximum Principle
Weak and strong maximum principle for solutions of elliptic PDEs,
boundary point lemma of E. Hopf, gradient estimates, theorem of
Liouville. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
25. The Eigenvalue Problem for the Laplace Operator
Eigenfunctions of the Laplace operator form a complete orthonormal
basis of L2 as an application of the Rellich compactness theorem.
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365