# 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 $\Gamma$.

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.