Steklov eigenvalue problem

E910281

eigenvalue problem spectral boundary value problem

The Steklov eigenvalue problem is a type of spectral boundary value problem in which eigenvalues appear in the boundary conditions of a partial differential equation, playing a key role in mathematical physics and geometric analysis.

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All labels observed (10)

Label	Occurrences
Steklov eigenvalue problem canonical	2
Dirichlet-to-Neumann map Λ_γ	1
Steklov boundary conditions	1
Steklov eigenfunctions	1
Steklov eigenvalue problems	1
Steklov eigenvalues	1
Steklov problem	1
Steklov spectrum	1
biharmonic Steklov problem	1
nonlinear Steklov problem	1

How this entity was disambiguated

This entity first appeared as the object of triple T11186002 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Steklov eigenvalue problem
Context triple: [Vladimir Steklov, notableFor, Steklov eigenvalue problem]

A. Sturm–Liouville problem
The Sturm–Liouville problem is a class of second-order linear differential equations with boundary conditions that yield real eigenvalues and orthogonal eigenfunctions forming a basis for function expansions in mathematical physics and engineering.
B. Dirichlet problem
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.
C. Morawetz inequalities
Morawetz inequalities are fundamental energy and decay estimates in the study of partial differential equations, especially wave and dispersive equations, that provide control over the long-time behavior of solutions.
D. Neumann boundary conditions in potential theory
Neumann boundary conditions in potential theory specify that the normal derivative of a potential function on a boundary is prescribed, modeling situations where flux across the boundary is controlled rather than the potential itself.
E. Noether boundary value problems
Noether boundary value problems are a class of boundary value problems in the theory of partial differential equations characterized by conditions ensuring well-posedness and finite-dimensional solution spaces, developed by mathematician Fritz Noether.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Steklov eigenvalue problem
Target entity description: The Steklov eigenvalue problem is a type of spectral boundary value problem in which eigenvalues appear in the boundary conditions of a partial differential equation, playing a key role in mathematical physics and geometric analysis.

A. Sturm–Liouville problem
The Sturm–Liouville problem is a class of second-order linear differential equations with boundary conditions that yield real eigenvalues and orthogonal eigenfunctions forming a basis for function expansions in mathematical physics and engineering.
B. Dirichlet problem
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.
C. Morawetz inequalities
Morawetz inequalities are fundamental energy and decay estimates in the study of partial differential equations, especially wave and dispersive equations, that provide control over the long-time behavior of solutions.
D. Neumann boundary conditions in potential theory
Neumann boundary conditions in potential theory specify that the normal derivative of a potential function on a boundary is prescribed, modeling situations where flux across the boundary is controlled rather than the potential itself.
E. Noether boundary value problems
Noether boundary value problems are a class of boundary value problems in the theory of partial differential equations characterized by conditions ensuring well-posedness and finite-dimensional solution spaces, developed by mathematician Fritz Noether.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	eigenvalue problem ⓘ spectral boundary value problem ⓘ
definedOn	Riemannian manifold with boundary ⓘ bounded domain ⓘ
eigenvaluesOf	Dirichlet-to-Neumann operator ⓘ linked to: Steklov operator
hasApplication	determining boundary behavior of harmonic functions ⓘ spectral characterization of domain geometry ⓘ
hasBoundaryCondition	∂u/∂n = σ u on ∂Ω ⓘ
hasEquation	Δu = 0 in Ω ⓘ
hasFeature	discrete spectrum under suitable conditions ⓘ eigenvalues appear in boundary conditions ⓘ orthogonal eigenfunctions with respect to suitable inner product ⓘ real eigenvalues for self-adjoint realizations ⓘ self-adjoint operator ⓘ spectral parameter in boundary condition ⓘ
hasGeneralization	biharmonic Steklov problem ⓘ linked to: Steklov eigenvalue problem nonlinear Steklov problem ⓘ linked to: Steklov eigenvalue problem weighted Steklov problem ⓘ
hasHistoricalPeriod	early 20th century ⓘ
hasKeyConcept	boundary spectral data ⓘ normal derivative on boundary ⓘ trace of harmonic functions ⓘ
hasOperator	Dirichlet-to-Neumann operator ⓘ linked to: Steklov operator
hasProperty	eigenfunctions form a basis under suitable conditions ⓘ eigenvalues depend on domain geometry ⓘ eigenvalues scale with boundary measure under rescaling ⓘ invariant under isometries of the domain ⓘ
hasSpectrum	0 = σ₀ ≤ σ₁ ≤ σ₂ ≤ … ⓘ sequence of Steklov eigenvalues ⓘ
hasUnknown	eigenfunction u ⓘ eigenvalue σ ⓘ
involves	boundary conditions ⓘ partial differential equations ⓘ
namedAfter	Vladimir Andreevich Steklov ⓘ linked to: Vladimir Steklov
relatedTo	Dirichlet boundary value problem ⓘ linked to: Dirichlet problem Laplace eigenvalue problem ⓘ Neumann boundary value problem ⓘ Robin boundary value problem ⓘ
studiedIn	PDE theory ⓘ spectral theory of elliptic operators ⓘ
usedIn	fluid–structure interaction models ⓘ geometric analysis ⓘ inverse problems ⓘ mathematical physics ⓘ shape optimization ⓘ spectral geometry ⓘ vibration analysis with boundary mass or impedance ⓘ

How these facts were elicited