mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a mixed boundary condition for a

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

defines a

boundary value problem In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...

in which the solution of the given equation is required to satisfy different

boundary condition In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...

s 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 Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In Mathematical analysis, analysis, h ...

or a

Neumann boundary condition In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann. When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative app ...

in a mutually exclusive way on disjoint parts of the boundary. For example, given a solution to a partial differential equation on a domain with boundary , it is said to satisfy a mixed boundary condition if, consisting of two disjoint parts, and , such that , verifies the following equations: :

\left. u \_ = u_0

and

\left. \frac\_ = g,

where and are given functions defined on those portions of the boundary. The mixed boundary condition differs from the Robin boundary condition in that the latter requires a

linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...

, possibly with

pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some Function (mathematics), function f. An important class of pointwise concepts are the ''pointwise operations'', that ...

variable coefficients, of the Dirichlet and the Neumann boundary value conditions to be satisfied on the whole boundary of a given domain.

Historical note

The first boundary value problem satisfying a mixed boundary condition was solved by Stanisław Zaremba for the

Laplace equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delt ...

: according to himself, it was Wilhelm Wirtinger who suggested him to study this problem.See .

Notes

References

*. 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 Existence is the state of having being or reality in contrast to nonexistence and nonbeing. Existence is often contrasted with essence: the essence of an entity is its essential features or qualities, which can be understood even if one does ...

and uniqueness theorems for the mixed boundary value problem involving a general second order selfadjoint

elliptic operator In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which im ...

s in fairly general domains. *. *. *, translated from the Italian by Zane C. Motteler. *, translated in Russian as . Boundary conditions Partial differential equations {{Mathanalysis-stub

Historical note

See also

Notes

References