Alphabeta MathMaximum Principles Harnack and Liouville in Rn
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Norms and Realification
- Complexification, Realification and Real Structures
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Darboux, L'Hôpital, and Taylor's Theorem
- Density Separability and Convolution in Lᵖ
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Euclidean Surface Measure, Divergence, and Green Identities
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Harmonic Functions and Mean Values in Rn
- Hausdorff via the Diagonal
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Lebesgue Measure on Euclidean Space
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Norming and Separation under Hahn–Banach
- Order, Zorn's Lemma, and the Axiom of Choice
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Radon Measures and the Riesz Markov Kakutani Theorem
- Regular Surfaces and Surface Integrals
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Simple Field Extensions and the Construction of the Complex Numbers
- Smooth Partitions of Unity and Exhaustions
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Divergence Theorem and Classical Stokes
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Maximal Function and Lebesgue Differentiation
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Definition
Subharmonic and superharmonic functions in rn
Lemma
Strict subharmonic perturbation
Theorem
Weak maximum principle for the laplacian
Corollary
Weak minimum principle for the laplacian
Theorem
Comparison principle for classical subharmonic functions
Lemma
Classical subharmonic mean value inequalities
Theorem
Strong maximum principle for classical subharmonic functions
Theorem
Strong maximum principle for harmonic functions
Corollary
Nonnegative harmonic function with an interior zero vanishes
Corollary
Uniqueness for the classical dirichlet problem
Corollary
Poisson supremum estimate from a quadratic barrier
Definition
Interior sphere condition and sphere normal
Lemma
Interior sphere barrier for the laplacian
Theorem
Hopf boundary point lemma for the laplacian
Theorem
Harnack inequality on a ball
Lemma
Finite harnack chain on a compact connected subset
Corollary
Harnack inequality on compact subsets
Theorem
Harnack convergence principle
Lemma
Smooth sphere data have a harmonic replacement under Countable Choice
Lemma
Derivative estimate proof of one sided harmonic liouville
Theorem
Liouville theorem for bounded harmonic functions
Corollary
Positive entire harmonic functions are constant
Corollary
Entire harmonic functions with bounded gradient are affine
Corollary
Removable singularity for bounded harmonic functions under Countable Choice
Theorem
Maximum principle with limsup control at infinity
Remark