Alphabeta MathFundamental Solutions Newtonian Potentials and Green Functions
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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
Combinatorics
Differential Geometry
Functional Analysis
Linear Algebra
- Complexification, Realification and Real Structures
- 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
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Radon Measures and the Riesz Markov Kakutani Theorem
- Sigma Algebras and Borel Sets
- 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
Partial Differential Equations
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Constant Rank, Submersions, Immersions and Regular Level Sets
- 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
- Fundamental Trigonometric Identities
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- 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
- Regular Surfaces and Surface Integrals
- 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
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- 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 Logarithm and General Powers
- The Real Gamma and Beta Functions
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
- Volumes of Elementary Solids and Solids of Revolution
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Hausdorff via the Diagonal
- Metric Spaces
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Fundamental solution of a constant-coefficient operator
Definition
Fundamental solution for the positive operator minus Laplacian
Lemma
Distributional derivatives commute with test-function convolution
Lemma
Necessary compatibility for the classical Neumann Poisson problem
Lemma
Puncturing a connected open subset of ℝⁿ preserves path-connectedness for n≥2
Corollary
Classical Neumann solutions differ by componentwise constants
Lemma
Local integrability of the Laplace fundamental kernel
Lemma
The Laplace fundamental solution is harmonic off its pole
Definition
Newtonian potential of compactly supported data
Theorem
The negative Laplacian of the fundamental solution is the unit Dirac distribution
Lemma
Bounded compact data give an everywhere finite Newtonian potential
Theorem
Far-field asymptotics of compact-source Newtonian potentials
Theorem
Newtonian potentials solve the distributional Poisson equation
Theorem
Hölder data give a classical Newtonian solution
Definition
Dirichlet Green function for minus Laplacian
Lemma
A bounded-domain Dirichlet Green function is unique and positive
Theorem
Symmetry of the Dirichlet Green function
Definition
Poisson kernel from a Dirichlet Green function
Theorem
Green representation for classical Poisson data
Corollary
Zero-Dirichlet Green representation for Poisson data
Corollary
Uniqueness of classical Dirichlet and compatible Neumann solutions
Remark
The Green identities used here are Euclidean
Sources
- Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript)
- John K. Hunter, Notes on Partial Differential Equations (2014)
- Paul Bankston, Metric Topology: A First Course, Proposition 21.3
- N. P. Strickland, Algebraic Topology notes, Proposition 5.14
- Thomas Schmidt, Partial Differential Equations I (2026)
- Sung-Jin Oh, Math 222A: Partial Differential Equations (lecture notes)