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.

• Partial differential equationBroader topic: It models steady states without internal sources and exemplifies elliptic behavior.

• Pierre-Simon LaplaceBroader topic: It expresses the mathematical structure underlying important results in Laplace’s potential theory.

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

• Heat equationCompared with: It describes steady-state heat distributions, whereas the heat equation includes time evolution.

• Wave equationCompared with: It describes static equilibrium rather than time-dependent propagation.

• Poisson's equationCompared with: It is Poisson's equation with the source function set to zero.

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

• Elliptic partial differential equationBroader topic: It is the prototype elliptic equation for source-free equilibrium fields.

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

• LaplacianBroader topic: This named equation became a defining setting for the operator in potential theory.

• 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.

• Heaviside step functionCompared with: Unlike abrupt switching models, this equation describes a source-free steady field.

• 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.

• Helmholtz equationCompared with: It is the zero-wavenumber special case of the homogeneous Helmholtz equation.

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

• Hyperbolic partial differential equationCompared with: Its boundary-value behavior contrasts with wave-like initial-value evolution.

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

• Elliptic regularityBroader topic: Its solutions are smooth in the interior, a basic instance of elliptic regularity.

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

• Elliptic operatorBroader topic: Its solutions exemplify the smoothness and mean-value behavior of elliptic equations.

• 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.