Linked from

The 33 pages that link to Laplace's equation, each with the reason it gives.

• Boundary conditionsRelated: Electrostatic and steady-state models use boundary data to determine its interior solution.

• Holomorphic functionRelated: The real and imaginary parts of a holomorphic function satisfy this equation.

• Boundary value problemRelated: Its solutions are determined by boundary data in electrostatics, heat flow, and potential theory.

• Spherical harmonicsRelated: Separation of variables in spherical coordinates produces spherical harmonics as its angular solutions.

• Potential flowRelated: In incompressible potential flow, the velocity potential satisfies this equation.

• Harmonic functionRelated: Harmonic functions are precisely the classical solutions of this equation.

• Conformal mapRelated: Conformal coordinate changes preserve harmonicity, simplifying two-dimensional potential problems.

• Dirichlet boundary conditionRelated: Its solutions are determined by boundary values in the classical Dirichlet problem.

• Potential theoryRelated: Its solutions are the central functions studied in classical potential theory.

• Neumann boundary conditionRelated: Neumann data commonly specify its solutions, with uniqueness determined only up to a constant.

• Separation of variablesRelated: Its boundary-value problems in simple geometries often admit separated harmonic solutions.

• Dirichlet problemRelated: Its harmonic solutions are the classical target of a Dirichlet problem.

• Equipotential surfaceRelated: Electric potential satisfies it in charge-free regions, constraining possible equipotential shapes.

• Hyperbolic functionsRelated: Separation of variables in rectangular domains often produces hyperbolic functions.

• Legendre polynomialsRelated: Separation of variables in spherical coordinates produces Legendre equations for axisymmetric solutions.

• Schwarz–Christoffel mappingRelated: Conformal maps transfer two-dimensional harmonic boundary-value problems to polygonal domains.

• Earnshaw's theoremRelated: Electrostatic potential satisfies it wherever there is no charge, forbidding a local minimum.

• EquipotentialRelated: Electrostatic and gravitational potentials satisfy it in source-free regions.

• Hele-Shaw cellRelated: For incompressible Hele-Shaw flow, pressure is harmonic away from sources and boundaries.

• Legendre equationRelated: Separation of variables in spherical coordinates produces the Legendre equation for axisymmetric solutions.

• Weyl's lemmaRelated: Weyl's lemma shows distributional solutions of its homogeneous form are smooth.