Third boundary value problem

One of the boundary value problems for partial differential equations (cf. Boundary value problem, partial differential equations). Suppose, for example, that in a bounded domain $ \Omega $ with boundary $ \Gamma $, each point of $ \Gamma $ has a normal, and let the following second-order elliptic equation be given:

$$ \tag{* } Lu = $$

$$ = \ \sum _ {i, j = 1 } ^ { n } a _ {ij} ( x) \frac{\partial ^ {2} u ( x) }{\partial x _ {i} \partial x _ {j} } + \sum _ {i = 1 } ^ { n } b _ {i} ( x) \frac{\partial u ( x) }{\partial x _ {i} } + c ( x) u ( x) = f ( x), $$

where $ x = ( x _ {1} \dots x _ {n} ) $, $ n \geq 2 $. Then a third boundary value problem for (*) in $ \Omega $ is the following problem: To select from the set of all solutions $ u ( x) $ of (*) those that have, at each point of the boundary, derivatives along the interior normal $ N $ and that satisfy a condition

$$ \frac{\partial u ( x) }{\partial N } + \alpha ( x) u ( x) = \ v ( x),\ \ x \in \Gamma , $$

where $ \alpha > 0 $ and $ v $ are continuous functions defined on $ \Gamma $.

Comments

The third boundary value problem is sometimes called the Robin problem, after V.G. Robin (1855–1897), and is not to be confused with the problem from potential theory of the same name, discussed in Robin problem.

Quite often, the derivative appearing in a boundary condition of the third type is not necessarily along the interior normal (see, e.g., [a1]), but along any direction varying continuously on $ \Gamma $. If such a direction is nowhere tangent to $ \Gamma $, the problem is said to be regular.

References

[a1]	A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)