Differential equation, partial, variational methods

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 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. 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 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 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 of the problem (1), (2) (cf. 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) have been developed for the construction of minimizing sequences and for the determination of their rate of convergence.

References

[1]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654
[2]	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) MR36317
[3]	S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian) MR0165337 Zbl 0123.09003
[4]	V.I. Smirnov, "A course of higher mathematics" , 3–5 , Addison-Wesley (1964) (Translated from Russian) MR0182690 MR0182688 MR0182687 MR0177069 MR0168707 Zbl 0122.29703 Zbl 0121.25904 Zbl 0118.28402 Zbl 0117.03404
[5]	S.G. [S.G. Mikhlin] Michlin, "Variationsmethoden der mathematischen Physik" , Akademie Verlag (1962) (Translated from Russian) MR0141248 Zbl 0098.36909
[6]	O.A. Ladyzhenskaya, N.N. Ural'tseva, "Linear and quasilinear elliptic equations" , Acad. Press (1968) (Translated from Russian) MR0244627 Zbl 0177.37404 Zbl 0164.13002
[7]	V.P. Mikhailov, "Partial differential equations" , MIR (1978) (Translated from Russian) MR0601389 MR0511076 MR0498162 Zbl 0342.35052 Zbl 0111.29009

Comments

See also Dirichlet principle, and the references cited there.