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 $u$ to a partial differential equation on a domain $Ω$ with boundary $\partialΩ$ , it is said to satisfy a mixed boundary condition if, consisting $\partialΩ$ of two disjoint parts, $Γ 1$ and $Γ 2$ , such that $\partialΩ = Γ 1 \cup Γ 2$ , $u$ verifies the following equations:

\left.u\right|_{\Gamma _{1}}=u_{0}

and

\left.{\frac {\partial u}{\partial n}}\right|_{\Gamma _{2}}=g,

where $u 0$ and $g$ 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.

^ Obviously, it is not at all necessary to require $u 0$ and $g$ being functions: they can be distributions or any other kind of generalized functions.

[1] Obviously, it is not at all necessary to require $u 0$ and $g$ being functions: they can be distributions or any other kind of generalized functions.

[1]