RobinRandwerte

RobinRandwerte is a term used in the theory of boundary value problems for partial differential equations to denote the boundary data imposed in a Robin (or generalized Robin) boundary condition. The name combines the Robin boundary condition with the German word Randwert, meaning boundary value. The term emphasizes the specific boundary data that drives the solution through the Robin relation.

In a typical setting, let Ω be a bounded domain with boundary ∂Ω. A Robin boundary condition for a function u: Ω → R takes the form α u(x) + β ∂u/∂n(x) = φ(x) for x on ∂Ω, where α and β are given coefficients and ∂u/∂n denotes the outward normal derivative. The function φ defined on ∂Ω is referred to as the RobinRandwert (or Robin Randwerte in plural). The term captures the boundary data that participates in the Robin boundary condition.

Properties: φ often belongs to a function space such as L2(∂Ω) or C(∂Ω). The choice of φ affects existence, uniqueness, and regularity of the solution to the boundary value problem. In linear problems, solutions depend continuously on φ, a property studied in operator theory and variational formulations.

Applications: RobinRandwerte arise in heat and mass transfer with convective boundary conditions, electrostatics, and fluid dynamics. They are central in numerical methods such as finite element and boundary element methods, where φ is incorporated into the weak form as a boundary integral term.

See also: Robin boundary condition, Dirichlet problem, Neumann problem, boundary value problem, partial differential equation. References: standard texts on PDEs discuss Robin conditions and related boundary data.