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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, Holder and Lipschitz continuity. Exercises 2. Differentiability Definitions of differentiability, differentiation rules, differentiable functions are continuous, higher order derivatives. Exercises . . . . . . . I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 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' = 7yf, 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 43 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 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-Lindel6f on the local existence and uniqueness of solutions of ODEs with a Lipschitz condition. Exercises . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . .. 6 1 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 7 . .................................................. 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 " " " ' " " ' """ * , ** . , 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 Chapter IV. The Lebesgue Integral 12. Preparations. Semicontinuous Functions Theorem of Dini, upper and lower semicontinuous functions, the characteristic function of a set. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15 7 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 ................................................... 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, Hl6der'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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4 1 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, Poincar6 inequality, compactness theorem of Rellich-Kondrachov on the LP-convergence of sequences with bounded Sobolev norm. Exercises 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 . . I ... . ................................................. 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 ....................................9. 23. Regularity of Weak Solutions Smoothness of weakly harmonic functions and of weak solutions of general elliptic PDEs, boundary regularity, classical solutions. Exercises 3 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 . . . . . . . . . . . . . . . . . . . . . . ..... . .. . . . . . . . . . . . . . . . . . . . . . ... 3 5 9