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

$$ \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 $, 
in a bounded domain  $  \Omega $, 
with a normal at each point of the boundary  $  \Gamma $. 
The second boundary value problem for equation (*) in  $  \Omega $
is the following problem: Out of the set of all solutions of equation (*), 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,   "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,   "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Miranda,   "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.G. Petrovskii,   "Partial differential equations" , Saunders  (1967)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Garabedian,   "Partial differential equations" , Wiley  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR></table>