Second boundary value problem

One of the boundary value problems (cf. Boundary value problem, partial differential equations) for partial differential equations. For example, let there be given a second-order elliptic equation

\begin{equation} \label{f:1} 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), \end{equation}

where $ x = ( x _ {1} \dots x _ {n} ) $, $ n \geq 2 $, in a bounded domain $ \Omega $, with a normal at each point of the boundary $ \Gamma $. The second boundary value problem for equation \eqref{f:1} in $ \Omega $ is the following problem: Out of the set of all solutions of equation \eqref{f:1}, isolate those solutions which have, at all boundary points, derivatives with respect to the interior normal $ N $ and which satisfy the condition

$$ \left . \frac{\partial u ( x, t) }{\partial N ( x) } \right | _ {x \in \Gamma } = \phi ( x), $$

where $ \phi ( x) $ is a given function. The second boundary value problem is also known as the Neumann problem.

References

[1]	A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian)
[2]	V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)
[3]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
[4]	I.G. Petrovskii, "Partial differential equations" , Saunders (1967) (Translated from Russian)
[a1]	P.R. Garabedian, "Partial differential equations" , Wiley (1963)
[a2]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)