Variational principles (in complex function theory)

Assertions which reveal the laws governing the variations of mapping functions during certain deformations of planar domains.

The principal qualitative variational principle is the Lindelöf principle, which may be described as follows. Let $ B _ {k} $, $ 0 \in B _ {k} $, $ k = 1, 2 $, be simply-connected domains in the $ z _ {k} $- plane with more than one boundary point, and let $ L( r, B _ {k} ) $, $ 0 < r < 1 $, be the level curve of the Green function for $ B _ {k} $, i.e. the image of the circle $ C( r) = \{ \zeta : {| \zeta | = r } \} $ under a univalent conformal mapping of the disc $ \{ \zeta : {| \zeta | < 1 } \} $ onto $ B _ {k} $ which leaves the origin fixed. Further, let the function $ f( z _ {1} ) $, $ f( 0) = 0 $, realize a simple conformal mapping of $ B _ {1} $ onto $ B _ {2} $. Under these circumstances: 1) To any point $ z _ {1} ^ {0} $ on $ L( r, {B _ {1} } ) $ there corresponds a point situated either on the level curve $ L( r, {B _ {2} } ) $( this is possible only if $ f( B _ {1} ) = B _ {2} $) or inside it; and 2) $ | f ^ { \prime } ( 0) | \leq | {g ^ \prime } ( 0) | $, where $ g ( z _ {1} ) $, $ g( 0) = 0 $, is a univalent conformal mapping of $ B _ {1} $ onto $ B _ {2} $( equality holds only if $ f( B _ {1} ) = B _ {2} $). Lindelöf's principle follows from Riemann's mapping theorem (cf. Riemann theorem) and from the Schwarz lemma. Finer constructions make it possible to find pointwise deviations of the mapping functions due to a given deformation of the mapped domains.

The principal quantitative variational principle obtained by M.A. Lavrent'ev [1] (see also [2]) may be stated as follows. Let $ B _ {1} $, $ 0 \in B _ {1} $, be a simply-connected domain with analytic boundary. Let there be given a family of domains $ B _ {1} ( t) $, $ 0 \in {B _ {1} } ( t) $, $ 0 \leq t \leq T $, $ T > 0 $, $ B _ {1} ( 0) \equiv B _ {1} $, with Jordan boundaries $ \Gamma _ {1} ( t) = \{ {z _ {1} } : {z _ {1} = \Omega ( \lambda , t) } \} $, $ 0 \leq \lambda \leq 2 \pi $, $ \Omega ( 0, t) = \Omega ( 2 \pi , t) $, where $ \Omega ( \lambda , t) $ is differentiable in $ t $ at $ t = 0 $, uniformly with respect to $ \lambda $; let $ F( z _ {1} , t) $, $ F( 0, t) = 0 $, $ F _ {z _ {1} } ^ { \prime } ( 0, t) > 0 $, be the function that univalently and conformally maps $ B _ {1} ( t) $ onto $ B _ {2} = \{ { {z _ {2} } } : {| z _ {2} | < 1 } \} $, and let $ \Phi ( z _ {2} , t) $ be the function inverse to $ F( z _ {1} , t) $ for a fixed $ t $. Then

$$ F ( z _ {1} , t) = F ( z _ {1} , 0) - tK ( F ( z _ {1} , 0) ) + \gamma _ {1} ( z _ {1} , t), $$

$$ \Phi ( z _ {2} , t) = \Phi ( z _ {2} , 0) + t \Phi _ {z _ {2} } ^ \prime ( z _ {2} , 0) K ( z _ {2} ) + \gamma _ {2} ( z _ {2} , t), $$

where

$$ K ( z) = \ \lim\limits _ {r \rightarrow 1 - 0 } \ \int\limits _ {C ( r) } \left . \frac{\partial \mathop{\rm ln} | F ( \Omega ( \lambda , t), 0) | }{\partial t } \right | _ {t = 0 } \frac{\zeta + z }{\zeta - z } \frac{d \zeta } \zeta , $$

and $ {\gamma _ {k} } ( z _ {k} , t) / t $ tends to zero uniformly on compact subsets of $ B _ {k} $( $ k = 1, 2 $) as $ t \rightarrow 0 $. This result has been extended [3] to doubly-connected domains. If further restrictions are imposed, it is possible to obtain, in $ B _ {1} ( t) $, estimates (uniformly in the closed domain) of the residual terms in the expansion of the mapping function with respect to the parameters characterizing the deformation of the boundaries of the domains under consideration [4].

References

[1]	M.A. Lavrent'ev, "On the theory of conformal mapping" Trudy Fiz. Mat. Inst. Steklov. , 5 (1934) pp. 159–246 (In Russian)
[2]	P.P. Kufarev, "On one-parameter families of analytic functions" Mat. Sb. , 13 (55) : 1 (1943) pp. 87–118 (In Russian)
[3]	I.A. Aleksandrov, "Variational formulas for univalent functions in doubly connected domains" Sibirsk. Mat. Zh. , 4 : 5 (1963) pp. 961–976 (In Russian)
[4]	M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)

Comments

There are many more variational principles, cf. [a3], Chapt. 10. See also Variation-parametric method; Löwner method; Internal variations, method of.

See also Boundary variation, method of. Important contributions to variational methods for univalent functions were made by M. Schiffer, cf. [a3], Chapt. 10.

References

[a1]	M. Heins, "Selected topics in the classical theory of functions of a complex variable" , Holt, Rinehart & Winston (1962)
[a2]	E. Hille, "Analytic function theory" , 1–2 , Ginn (1962)
[a3]	P.L. Duren, "Univalent functions" , Springer (1983) pp. 258