Alphabeta MathWeak Derivatives and Sobolev Spaces — Examples
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Differential Geometry
Functional Analysis
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Absolute Continuity and the Sharp Fundamental Theorem of Calculus
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Differentiation of Monotone Functions and the Vitali Covering Theorem
- Lebesgue Measure on Euclidean Space
- Lebesgue-Stieltjes Measures and Distribution Functions
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
Partial Differential Equations
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Bounded Variation and the Riemann–Stieltjes Integral
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Counterexample
A hypersurface jump is not W^1,p
Counterexample
Cantor function has singular distributional derivative
Counterexample
Lp and Sobolev classes do not determine point values
Counterexample
A step has no locally integrable weak derivative
Counterexample
Point evaluation is unbounded below the Sobolev continuity threshold
Example
The absolute value has a weak first derivative
Example
Matching C¹ pieces across a hyperplane have no jump derivative
Example
Sharp Sobolev threshold for a radial power
Counterexample
Subcritical W^1,p is not closed under multiplication
Example
Absolute value has a Dirac second derivative
Example
A clipped affine function keeps its zero region
Sources
- Juha Kinnunen, Sobolev Spaces (2026), Chapters 1–2
- John K. Hunter, Notes on Partial Differential Equations (2014), Chapter 3
- Juha Kinnunen, Sobolev Spaces (2026), Chapter 2 §2.6
- John K. Hunter, Notes on Partial Differential Equations, Appendix on one-dimensional weak and distributional derivatives
- John K. Hunter, Notes on Partial Differential Equations (2014), Chapter 3 §3.5
- Juha Kinnunen, Sobolev Spaces (2026), Chapter 1 §1.2
- John K. Hunter, Notes on Partial Differential Equations (2014), Chapter 3 §3.2
- Juha Kinnunen, Sobolev Spaces (Aalto University, 2026)
- Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011)
- Juha Kinnunen, Sobolev Spaces (Aalto University, 2026), Chapter 1 §1.1
- Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011), Chapter 8
- Juha Kinnunen, Sobolev Spaces (2026)
- John K. Hunter, Notes on Partial Differential Equations (2014)