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A function  $  u(x) = u(x _ {1} \dots x _ {n} ) $
of real variables, defined in a domain  $  D $
of the Euclidean space  $  \mathbf R  ^ {n} $, 
$  n \geq  2 $, 
with continuous partial derivatives up to the fourth order inclusive, that satisfies in  $  D $
the equation

$$ 
\Delta  ^ {2} u  \equiv \ 
\Delta ( \Delta u)  =  0,
$$

where  $  \Delta $
is the Laplace operator. This equation is known as the biharmonic equation. The class of biharmonic functions includes the class of harmonic functions and is a subclass of the class of poly-harmonic functions (cf. [[Harmonic function|Harmonic function]]; [[Poly-harmonic function|Poly-harmonic function]]). Each biharmonic function is an analytic function of the coordinates  $  x _ {i} $.

From the point of view of practical applications the biharmonic functions in two variables  $  u(x _ {1} , x _ {2} ) $
are the most important. Such biharmonic functions can be represented using harmonic functions  $  u _ {1} , u _ {2} $
or  $  v _ {1} , v _ {2} $, 
in the form

$$ 
u(x _ {1} , x _ {2} )  = \ 
x _ {1} u _ {1} (x _ {1} , x _ {2} ) +
u _ {2} (x _ {1} , x _ {2} )
$$

or

$$ 
u(x _ {1} , x _ {2} )  = \ 
(r  ^ {2} -r _ {0}  ^ {2} )
v _ {1} (x _ {1} , x _ {2} ) +
v _ {2} (x _ {1} , x _ {2} ),
$$

where  $  r  ^ {2} = x _ {1}  ^ {2} + x _ {2}  ^ {2} $, 
while  $  r _ {0}  ^ {2} $
is a constant. The main boundary value problem for biharmonic functions is the following: To find a biharmonic function in the domain  $  D $
that is continuous together with its first-order derivatives in the closed domain  $  \overline{D}\; = D \cup C $
and satisfies on the boundary  $  C $
the conditions

$$ \tag{* }
\left . u \right | _ {C}  = \ 
f _ {1} (s),\ \ 
\left .

\frac{\partial  u }{\partial  n }

\right | _ {C}  = \ 
f _ {2} (s),
$$

where  $  \partial  u / \partial  n $
is the derivative with respect to the normal on  $  C $, 
and  $  f _ {1} (s), f _ {2} (s) $
are given continuous functions of the arc length  $  s $
on the contour  $  C $. 
The above representations of biharmonic functions give the solution of problem (*) in an explicit form for the case of the disc  $  D $, 
starting from the Poisson integral for harmonic functions [[#References|[1]]].

Biharmonic functions in two variables may also be represented as follows:

$$ 
u (x _ {1} , x _ {2} )  = \ 
 \mathop{\rm Re} \{ \overline{z}\; \phi (z) +
\chi (z) \} =
$$

$$ 
= \ 
{
\frac{1}{2}
 } \{ \overline{z}\; \phi (z) + z {\phi (z)
 } bar + \chi (z) + \overline{ {\chi (z) }}\; \} ,\  \overline{z}\; = x _ {1} - ix _ {2} ,
$$

using two analytic functions  $  \phi (z), \chi (z) $
of the complex variable  $  z = x _ {1} + ix _ {2} $. 
This representation makes it possible to reduce the boundary value problem (*) for an arbitrary domain  $  D $
to a system of boundary value problems for analytic functions, for which a method of solving was developed in detail by G.V. Kolosov and N.I. Muskhelishvili. This method was developed while solving various planar problems of elasticity theory (cf. [[Elasticity theory, planar problem of|Elasticity theory, planar problem of]]), in which the main biharmonic function is the stress function or the Airy function [[#References|[2]]], [[#References|[3]]] (cf. [[Airy functions|Airy functions]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,   A.A. Samarskii,   "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. Muskhelishvili,   "Some basic problems of the mathematical theory of elasticity" , Noordhoff  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Lavrent'ev,   B.V. Shabat,   "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR></table>

====Comments====
An axiomatic treatment of biharmonic functions, similar to that of harmonic functions, is given in [[#References|[a1]]], [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.P. Smyrnelis,   "Axiomatique des fonctions biharmoniques"  ''Ann. Inst. Fourier (Grenoble)'' , '''25''' :  1  (1975)  pp. 35–97</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.P. Smyrnelis,   "Axiomatique des fonctions biharmoniques"  ''Ann. Inst. Fourier (Grenoble)'' , '''26''' :  3  (1976)  pp. 1–47</TD></TR></table>