Dirichlet problem

A boundary-value problem that seeks a function satisfying a differential equation inside a domain and prescribed values on its boundary. Its solutions are fixed by the boundary data when the problem is well posed.

Laplace's equation: The partial differential equation stating that a function’s Laplacian is zero. Its harmonic solutions are the classical target of a Dirichlet problem.
Boundary-value problem: A differential equation problem whose solution must satisfy conditions specified on a domain’s boundary. The Dirichlet problem is the version that prescribes the function’s boundary values.
Neumann problem: A boundary-value problem that prescribes the normal derivative of a solution on the boundary. It fixes boundary flux rather than the function values prescribed by a Dirichlet problem.
Electrostatics: The study of electric charges and fields that are stationary or vary slowly enough to be treated as static. Fixed conductor potentials lead to Dirichlet problems for electric potential.
Dirichlet's principle: The variational principle that harmonic functions minimize Dirichlet energy among functions with the same boundary values. It links existence of solutions to energy minimization, subject to appropriate function-space conditions.
Harmonic function: A twice-differentiable function whose Laplacian vanishes throughout its domain. For Laplace’s equation, solving the problem means finding a harmonic function with the prescribed boundary values.
Domain (mathematics): A connected open subset of a topological space, commonly an open region in Euclidean space. The differential equation is imposed inside this region.
Robin boundary condition: A boundary condition prescribing a linear combination of a function and its normal derivative. It combines the value and flux conditions that Dirichlet and Neumann problems impose separately.
Heat equation: A parabolic partial differential equation describing the diffusion of heat through a medium. Prescribed wall temperatures give Dirichlet boundary data for heat diffusion.
Wiener criterion: A criterion characterizing whether a boundary point is regular for the Laplace Dirichlet problem. It identifies which irregular-looking boundary points still attain prescribed values continuously.

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