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Methods for solving boundary value problems for partial differential equations by reducing them (whenever possible) to suitably selected variational problems (i.e. to problems of finding the minimum or the maximum of a certain functional).

Variational methods are extensively used first of all in theoretical investigations (to demonstrate the theorems of existence, uniqueness and stability of solutions, to study the differentiability properties of solutions, in spectral theory, in the study of different optimization problems, etc.) and secondly in problems of finding approximate solutions of the equations. Approximate solutions of variational problems may be found by solving finite systems of algebraic equations; often algorithms for finding approximate solutions of variational problems are simpler and practically more convenient than the available algorithms for solving the same problems for partial differential equations. The variational method of studying boundary value problems was discovered in mid-19th century as the so-called [[Dirichlet principle|Dirichlet principle]] of finding, in a domain  $  G $, 
a harmonic function assuming on the boundary  $  \partial  G $
a given value  $  \phi ( x) $, 
$  x \in \partial  G $, 
which, in the class of functions being considered, yields the minimum of the [[Dirichlet integral|Dirichlet integral]]. Originally, the Dirichlet principle was used only in the theory of second-order linear elliptic equations (subsequently also in elliptic equations of higher order), after which it was applied in the theory of equations of other types, both linear and non-linear. The development of variational methods started from the work of B. Riemann, K. Weierstrass and D. Hilbert. [[Imbedding theorems|Imbedding theorems]] and their generalizations played an important role in the development of variational methods, and in particular in their foundation.

A simple typical example of the application of variational methods is the solution of the [[Dirichlet problem|Dirichlet problem]] for a second-order self-adjoint elliptic equation

$$ \tag{1 }
A u + c u  =  0 ,
$$

where  $  c = c ( x) \geq  0 $,

$$ \tag{2 }
\left . u \right | _ {\partial  G }   =  \phi ,
$$

$$ \tag{3 }
A u  =  - \sum _ {i , j = 1 } ^ { n } 

\frac \partial {\partial  x _ {i} }
 \left [ a _ {ij} ( x) 
\frac{\partial  u }{\partial  x _ {j} }
 \right ] ,\  x = ( x _ {1} \dots x _ {n} ) \in G
$$

( $  G $
is a domain in a finite-dimensional Euclidean space,  $  \partial  G $
is its boundary and  $  \phi $
is a function given on  $  \partial  G $), 
and such that there exists a constant  $  \kappa > 0 $
such that for all points  $  x \in G $
and all numbers  $  \xi _ {1} \dots \xi _ {n} $
the following inequality (ellipticity condition) is valid:

$$ 
\sum _ {i , j = 1 } ^ { n } 
a _ {ij} ( x) \xi _ {i} \xi _ {j}  \geq   \kappa \sum _ {i = 1 } ^ { n }  \xi _ {i}  ^ {2} .
$$

In this case the variational method for solving the problem (1), (2) consists in finding a function  $  u ( x) $
for which the functional

$$ 
K ( u)  =  \int\limits _ { G } \left [ \sum _ {i , j = 1 } ^ { n } 
a _ {ij} ( x) 
\frac{\partial  u }{\partial  x _ {i} }
 
\frac{\partial  u }{\partial  x _ {j} }
 + c u  ^ {2} \right ]  d x
$$

assumes its lowest value in the class of admissible functions, i.e. of functions  $  u ( x) $
for which  $  K ( u) < + \infty $
and for which the boundary condition (2) is met (equation (1) is the Euler equation for the functional  $  K ( u) $). 
The variational method is applicable only if the class of admissible functions is non-empty. The conditions to be met by a function  $  \phi $
defined on the boundary so that the class of admissible functions is non-empty are given by the imbedding theorems. The function  $  u ( x) $
for which the functional  $  K ( u) $
assumes its lowest value in the class of admissible functions is a [[Generalized solution|generalized solution]] of the problem (1), (2) (cf. [[Differential equation, partial, functional methods|Differential equation, partial, functional methods]]) and is, for example in the classical case when the coefficients  $  a _ {ij} ( x) $
of the operator (3) are continuously differentiable, an ordinary solution of the problem.

Another typical example of the use of variational methods is their application to finding eigen values and eigen functions of the operator (3).

A function giving the minimum of some functional may be obtained as the limit of a so-called minimizing sequence, i.e. a sequence of functions for which the values of the functional on these functions tend to the given minimum. Special methods (e.g. the [[Ritz method|Ritz method]]) have been developed for the construction of minimizing sequences and for the determination of their rate of convergence.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)  {{MR|0195654}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Courant,   "Dirichlet's principle, conformal mapping, and minimal surfaces" , Interscience  (1950)  (With appendix by M. Schiffer: Some recent developments in the theory of conformal mapping)  {{MR|36317}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.L. Sobolev,   "Applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)  {{MR|0165337}} {{ZBL|0123.09003}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Smirnov,   "A course of higher mathematics" , '''3–5''' , Addison-Wesley  (1964)  (Translated from Russian)  {{MR|0182690}} {{MR|0182688}} {{MR|0182687}} {{MR|0177069}} {{MR|0168707}} {{ZBL|0122.29703}} {{ZBL|0121.25904}} {{ZBL|0118.28402}} {{ZBL|0117.03404}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.G. [S.G. Mikhlin] Michlin,   "Variationsmethoden der mathematischen Physik" , Akademie Verlag  (1962)  (Translated from Russian)  {{MR|0141248}} {{ZBL|0098.36909}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  O.A. Ladyzhenskaya,   N.N. Ural'tseva,   "Linear and quasilinear elliptic equations" , Acad. Press  (1968)  (Translated from Russian)  {{MR|0244627}} {{ZBL|0177.37404}} {{ZBL|0164.13002}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.P. Mikhailov,   "Partial differential equations" , MIR  (1978)  (Translated from Russian)  {{MR|0601389}} {{MR|0511076}} {{MR|0498162}} {{ZBL|0342.35052}} {{ZBL|0111.29009}} </TD></TR></table>

====Comments====
See also [[Dirichlet principle|Dirichlet principle]], and the references cited there.