Potential theory

Potential theory studies scalar functions whose derivatives generate fields, especially harmonic functions and solutions of Laplace’s equation. It connects boundary data, field behavior, and properties such as mean-value and maximum principles.

Laplace's equation: The partial differential equation stating that a function’s Laplacian is zero. Its solutions are the central functions studied in classical potential theory.
Poisson's equation: A partial differential equation relating a function's Laplacian to a specified source term. It models potentials produced by distributed sources rather than source-free regions.
Electrostatics: The study of electric charges at rest and the electric fields and potentials they produce. Electrostatic potential is a physical instance of the scalar fields studied here.
Hodge theory: A framework relating differential forms, cohomology, and harmonic representatives on suitable spaces. It generalizes harmonic analysis beyond scalar potentials to differential forms and topology.
Isaac Newton: An English mathematician and physicist whose laws of motion and universal gravitation shaped classical mechanics. His inverse-square law led to the gravitational potentials later treated by the subject.
Harmonic function: A twice-differentiable function whose Laplacian vanishes at every point in its domain. Harmonic functions are the basic local objects characterized by potential theory.
Poisson kernel: A boundary integral kernel that reconstructs harmonic functions inside a domain from boundary values. It gives an explicit method for solving the Dirichlet problem in standard domains.
Gravitational potential: A scalar field whose negative gradient gives gravitational acceleration in a conservative gravitational field. Mass distributions generate Newtonian potentials governed by Poisson's equation.
Spectral theory: The study of eigenvalues and spectral decompositions of operators, especially linear operators on function spaces. It studies Laplacian eigenfunctions and global spectra rather than potentials alone.
Joseph-Louis Lagrange: A mathematician and mechanician whose work helped establish analytical mechanics and mathematical physics. He developed early mathematical formulations of gravitational potential.

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