Dirichlet boundary condition

A boundary condition that fixes a function’s value at every point on the boundary of its domain. It is named for the mathematician Peter Gustav Lejeune Dirichlet.

Laplace's equation: A partial differential equation stating that a function’s Laplacian is zero. Its solutions are determined by boundary values in the classical Dirichlet problem.
Boundary value: The value assigned to a function at a point on the boundary of its domain. A Dirichlet condition consists of these prescribed values across the boundary.
Neumann boundary condition: A boundary condition that specifies a function’s normal derivative on the boundary. It prescribes boundary flux rather than the function’s value.
Heat equation: A partial differential equation describing how temperature diffuses through a medium over time. Fixed boundary temperatures model surfaces held at prescribed temperatures.
Dirichlet problem: The problem of finding a harmonic function with specified values on a domain’s boundary. It is the canonical setting for solving an equation with Dirichlet data.
Domain (mathematics): A connected open set on which a mathematical function or equation is defined. Its boundary is where the specified function values are imposed.
Robin boundary condition: A boundary condition relating a function’s value to its normal derivative on the boundary. It combines value and derivative data where Dirichlet data fixes value alone.
Electrostatics: The study of stationary electric charges and their electric fields and potentials. Prescribed electric potential on conductors supplies Dirichlet data for potential equations.
Poisson's equation: A partial differential equation relating a function’s Laplacian to a prescribed source. Its solutions also use prescribed boundary values alongside an interior source.
Trace operator: An operator assigning boundary values to functions defined inside a domain, when ordinary restriction is unavailable. It formalizes boundary values for weak solutions and Sobolev functions.

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