Dirichlet problem
E466251
boundary value problem
mathematical problem
problem in partial differential equations
problem in potential theory
The Dirichlet problem is a fundamental boundary value problem in potential theory and partial differential equations, asking for a function that solves a specified PDE inside a domain while taking prescribed values on the domain’s boundary.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Dirichlet Laplacian on bounded domains in R^n | 1 |
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
boundary value problem
ⓘ
mathematical problem ⓘ problem in partial differential equations ⓘ problem in potential theory ⓘ |
| asksFor |
a function attaining prescribed boundary values
ⓘ
a function solving a given PDE in a domain ⓘ |
| boundaryConditionType | Dirichlet boundary condition NERFINISHED ⓘ |
| field |
complex analysis
ⓘ
geometric analysis ⓘ harmonic analysis ⓘ mathematical analysis ⓘ partial differential equations ⓘ potential theory ⓘ |
| hasAspect |
existence of solutions
ⓘ
regularity of solutions ⓘ stability of solutions ⓘ uniqueness of solutions ⓘ |
| hasSpecialCase |
Dirichlet problem for the unit disk
ⓘ
Dirichlet problem on a Riemannian manifold NERFINISHED ⓘ Dirichlet problem on a bounded domain ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| involves |
boundary conditions
ⓘ
boundary of a domain ⓘ domain in Euclidean space ⓘ partial differential equation ⓘ |
| namedAfter | Peter Gustav Lejeune Dirichlet NERFINISHED ⓘ |
| relatedTo |
Neumann problem
ⓘ
Robin boundary condition NERFINISHED ⓘ mixed boundary value problem ⓘ |
| requiresCondition |
compatibility of boundary data
ⓘ
sufficient regularity of the domain ⓘ |
| solutionType |
classical solution
ⓘ
harmonic function ⓘ weak solution ⓘ |
| solvedByMethod |
Green function methods
ⓘ
Perron method NERFINISHED ⓘ finite difference method ⓘ finite element method ⓘ method of layer potentials ⓘ subharmonic function techniques ⓘ variational methods ⓘ |
| studiedIn |
Banach spaces
NERFINISHED
ⓘ
Hilbert spaces NERFINISHED ⓘ Sobolev spaces NERFINISHED ⓘ |
| typicalPDE |
Laplace equation
NERFINISHED
ⓘ
Poisson equation ⓘ elliptic partial differential equation ⓘ |
| usedIn |
electrostatics
ⓘ
fluid mechanics ⓘ gravitational potential theory ⓘ heat conduction ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Dirichlet Laplacian on bounded domains in R^n