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​Evans

5 Sobolev spaces

    5.1 H\"older spaces

    5.2 Sobolev spaces

        5.2.1 Weak derivatives 

        5.2.2 Defintion of Sobolev spaces

        5.2.3 Elementary properties

Appendix

    A. Notation

        A.1. Notation for matrices [Evans A1]

        A.2. Geometric noation [Evans A2]

        A.3. Notation for functions [Evans A3]

        A.4. Vector-valued functions [Evans A4]

        A.5. Notation for estimates [Evans A5]

        A.6. Some comments about notation [Evans A6]

    B. Inequalities        

        B.1. Convex functions

        B.2. Useful inequalities

            B.2.1. Cauchy's inequality

            B.2.2 Cauchy's inequality with epsilon

            B.2.3. Young's inequality

            B.2.4. Young's inequality with epsilon

            B.2.5. H"older's inequality

            B.2.6. Minkowski's inequality

            B.2.7. General H"older inequality

            B.2.8. Interpolation inequality for -norms

            B.2.9. Cauchy-Schwarz inequality

            B.2.10. Gronwall's inequality (differential form)

            B.2.11. Gronwall's inequality (integral form)

    C. Calculus

        C.1. Boundaries

        C.2. Gauss-Green theorem

        C.3. Polar coordinates, coarea formula

        C.4. Moving regions

        C.5. Convolution and smoothing

        C.6. Inverse function theorem

        C.7. Implicit function theorem

        C.8. Uniform convergence

    D. Functional Analysis

        D.1. Banach spaces

        D.2. Hilbert spaces

        D.3. Bounded linear operaters

            D.3.1. Linear operators on Banach spaces

            D.3.2. Linear operators on Hilbert spaces

        D.4. Weak convergence

        D.5. Compact operators, Fredholm theory

            D.5.1 Definitions and basic properties

            D.5.2. Fredholm alternative

            D.5.3. The spectrum of compact operators

    E. Measure theory

            E.1. Lebesgue measure

            E.2. Measurable functions and integration

            E.3. Convergence theorems for integrals

            E.4. Differentiation

            E.5. Banach space-valued functions