Mixed boundary condition
In mathematics, a mixed boundary condition for a partial differential equation defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary conditions on disjoint parts of the boundary of the domain where the condition is stated. Precisely, in a mixed boundary value problem, the solution is required to satisfy a Dirichlet or a Neumann boundary condition in a mutually exclusive way on disjoint parts of the boundary.
For example, given a solution uScript error: No such module "Check for unknown parameters". to a partial differential equation on a domain ΩScript error: No such module "Check for unknown parameters". with boundary ∂ΩScript error: No such module "Check for unknown parameters"., it is said to satisfy a mixed boundary condition if, consisting ∂ΩScript error: No such module "Check for unknown parameters". of two disjoint parts, ΓScript error: No such module "Su".Script error: No such module "Check for unknown parameters". and ΓScript error: No such module "Su".Script error: No such module "Check for unknown parameters"., such that ∂Ω = ΓScript error: No such module "Su". ∪ ΓScript error: No such module "Su".Script error: No such module "Check for unknown parameters"., uScript error: No such module "Check for unknown parameters". verifies the following equations:
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where uScript error: No such module "Su".Script error: No such module "Check for unknown parameters". and gScript error: No such module "Check for unknown parameters". are given functions defined on those portions of the boundary.[1]
The mixed boundary condition differs from the Robin boundary condition in that the latter requires a linear combination, possibly with pointwise variable coefficients, of the Dirichlet and the Neumann boundary value conditions to be satisfied on the whole boundary of a given domain.
Historical note
Template:Quote The first boundary value problem satisfying a mixed boundary condition was solved by Stanisław Zaremba for the Laplace equation: according to himself, it was Wilhelm Wirtinger who suggested him to study this problem.[2]
See also
- Dirichlet boundary condition
- Neumann boundary condition
- Cauchy boundary condition
- Robin boundary condition
Notes
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- ↑ Obviously, it is not at all necessary to require uScript error: No such module "Su".Script error: No such module "Check for unknown parameters". and gScript error: No such module "Check for unknown parameters". being functions: they can be distributions or any other kind of generalized functions.
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References
- Script error: No such module "citation/CS1".. In the paper "Existential analysis of the solutions of mixed boundary value problems, related to second order elliptic equation and systems of equations, selfadjoint" (English translation of the title), Gaetano Fichera gives the first proofs of existence and uniqueness theorems for the mixed boundary value problem involving a general second order selfadjoint elliptic operators in fairly general domains.
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- Script error: No such module "citation/CS1"., translated from the Italian by Zane C. Motteler.
- Script error: No such module "citation/CS1"., translated in Russian as Script error: No such module "citation/CS1"..