Dirichlet problem
Harmonic function problem
The Dirichlet problem asks for a function that satisfies a given differential equation inside a region while matching specified values on the boundary.
Named after the German mathematician Peter Gustav Lejeune Dirichlet, it arises throughout physics and engineering—finding the temperature distribution in a plate when edge temperatures are fixed, or the electrical potential in a conductor.
The problem is fundamental to partial differential equations and potential theory.
- electrostaticsStudy of stationary charges
- harmonic functionLaplacian equals zero everywhere
- heat conductionEnergy flow through matter
- fluid dynamicsScience of flowing matter
- variationalRelating to functional extremization
domain · eigenvalue · elliptic · regularity · harmonic · uniqueness · Laplace · boundary · diffusion · potential · equilibrium · existence · vibration · membrane · drum · Robin · soap film · quantum mechanics
- PDEsboundary conditions, initial conditions, Schrodinger equation, Boltzmann equation, conservation laws
- potential theoryharmonic function, balayage, capacity, barrier
- wave physicswavelength, wave function, frequency, wave speed, wave front
- differential operatorsdel, gradient, divergence, curl, partial derivative
- Green's functionspropagator