Steklov operator
E910282
The Steklov operator is a boundary integral operator arising in the study of elliptic partial differential equations and spectral problems, particularly in the context of Steklov eigenvalue problems.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
boundary integral operator
ⓘ
linear operator ⓘ mathematical operator ⓘ |
| actsOn | functions on the boundary of a domain ⓘ |
| alsoCalled | Dirichlet-to-Neumann map NERFINISHED ⓘ |
| appearsIn | Calderón inverse conductivity problem NERFINISHED ⓘ |
| arisesIn |
Steklov eigenvalue problems
NERFINISHED
ⓘ
boundary value problems ⓘ elliptic partial differential equations ⓘ |
| associatedWith |
Laplace equation
NERFINISHED
ⓘ
Steklov boundary conditions NERFINISHED ⓘ harmonic functions ⓘ |
| classification | non-local boundary operator ⓘ |
| context |
Riemannian manifolds with boundary
ⓘ
bounded domains in Euclidean space ⓘ |
| dependsOn |
geometry of the domain
ⓘ
metric on the boundary ⓘ |
| domain | boundary of a domain ⓘ |
| eigenvalueProblem |
Steklov eigenfunctions
NERFINISHED
ⓘ
Steklov spectrum NERFINISHED ⓘ |
| field |
mathematical analysis
ⓘ
partial differential equations ⓘ spectral theory ⓘ |
| generalizationOf | classical Steklov boundary condition operator ⓘ |
| hasKernel | constant functions on the boundary (in many standard settings) ⓘ |
| hasProperty |
elliptic pseudodifferential operator of order 1 (on smooth boundaries)
ⓘ
positive (under suitable conditions) ⓘ self-adjoint (under suitable conditions) ⓘ |
| isDefinedFor | solutions of elliptic PDEs in a domain ⓘ |
| maps | Dirichlet boundary data to Neumann boundary data ⓘ |
| mathematicalCategory | unbounded operator on a Hilbert space (in typical formulations) ⓘ |
| namedAfter | Vladimir Andreevich Steklov NERFINISHED ⓘ |
| namedFor | Steklov eigenvalue problem NERFINISHED ⓘ |
| relatedTo |
Calderón projector
NERFINISHED
ⓘ
Neumann-to-Dirichlet map NERFINISHED ⓘ boundary integral equations ⓘ |
| requires | elliptic regularity theory for definition and analysis ⓘ |
| spectralData |
Steklov eigenvalues accumulate only at infinity
ⓘ
Steklov eigenvalues form a discrete sequence under standard assumptions ⓘ |
| typicalHilbertSpace | L^2 of the boundary measure GENERATED ⓘ |
| usedIn |
control theory for PDEs
ⓘ
inverse problems ⓘ shape optimization ⓘ spectral geometry ⓘ |
| usedToStudy |
boundary determination problems
ⓘ
relationship between boundary geometry and spectrum ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.