A generalization of the maximum-modulus principle for analytic functions to the case of functions that are given a priori as unbounded; it was first given in its simplest form by E. Phragmén and E. Lindelöf . Let be a regular analytic function of a complex variable in a domain of the plane with boundary . One says that does not exceed a number in modulus at a boundary point if
that is, for every there is a disc (depending on and ) with centre such that for . The main content of the result of Phragmén and Lindelöf, in a somewhat modernized form, consists in the following two propositions, which are successive extensions of the maximum-modulus principle.
1) If the regular analytic function exceeds in modulus nowhere on , then everywhere in . This proposition is sometimes called the Phragmén–Lindelöf principle. It extends the maximum-modulus principle to functions about the behaviour of which on the boundary only partial information is available.
2) Suppose that the regular analytic function does not exceed in modulus at any point of not belonging to some set . Suppose also that there is a function with the following properties: a) is regular in ; b) in ; c) in ; and d) for every the function does not exceed in modulus at any point . Under these conditions everywhere in .
The Phragmén–Lindelöf theorem has received numerous applications, also often called Phragmén–Lindelöf theorems, and associated with a concrete form of , and (see –, in particular the generalization given in ). In applications most often consists of the single point . For example, suppose that is regular in the angular domain
and does not exceed in modulus on the sides of the angle. Then the following alternative holds: Either
everywhere in , or the maximum modulus
increases faster than as for any , . This theorem is obtained from propositions 1 and 2 for , , where .
The statements of 1 and 2 remain valid for a holomorphic function , , given in a domain of the complex space , . Many papers have been devoted to obtaining results of the type of the Phragmén–Lindelöf theorem for the solutions of partial differential equations and systems of equations of elliptic type. Propositions 1 and 2 remain true for a subharmonic function defined in a domain of a Euclidean space , , or , , provided that is replaced by and the function , , is assumed to be logarithmically subharmonic (cf. Logarithmically-subharmonic function) in (see , ). For example, suppose that is a subharmonic function in the angular domain (*) and does not exceed in modulus on the sides of the angle. Then the following alternative holds: Either everywhere in , or the maximum
increases faster than for every , . There are also similar results for solutions of other elliptic equations. They may be called "weak" theorems of Phragmén–Lindelöf type, in the sense that, on account of their weak restriction only on the function itself on the boundary, one obtains a fairly weak assertion about its growth.
In other results, which may be called "strong" theorems of Phragmén–Lindelöf type, on account of the restriction on the function itself and its normal derivative on the boundary, one obtains a stronger assertion about its growth. An example is the following statement for the cylindrical domain
in . Suppose that is a harmonic function in the cylinder and on its lateral surface , with and on . Then either everywhere in , or the maximum
increases, as , faster than
|||E. Phragmén, E. Lindelöf, "Sur une extension d'un principe classique de l'analyse" Acta Math. , 31 (1908) pp. 381–406|
|||E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979)|
|||S. Stoilov, "The theory of functions of a complex variable" , 1–2 , Moscow (1962) (In Russian; translated from Rumanian)|
|||M.A. Evgrafov, "Analytic functions" , Saunders , Philadelphia (1966) (Translated from Russian)|
|||I.I. Privalov, "Subharmonic functions" , Moscow-Leningrad (1937) (In Russian)|
|||E.D. Solomentsev, "Harmonic and subharmonic functions and their generalizations" Itogi Nauk. Mat. Anal. Teor. Veryatnost. Regulirovanie. 1962 (1964) pp. 83–100 (In Russian)|
|||M.A. Evgrafov, "Generalization of the Phragmén–Lindelöf theorems for analytic functions to solutions of various elliptic systems" Izv. Akad. Nauk SSSR Ser. Mat. , 27 (1963) pp. 843–854 (In Russian)|
|||E.M. Landis, "Second-order equations of elliptic and parabolic type" , Moscow (1971) (In Russian)|
For Phragmén–Lindelöf type theorems for subharmonic functions in see [a3].
Theorems of Phragmén–Lindelöf type are known also for parabolic equations. For instance, if solves the heat equation in the half-space and is continuous for , then implies that for all in the strip , provided satisfies the growth condition for certain positive constants , uniformly with respect to . Disregarding the growth condition above, it is possible to find unbounded solutions with bounded initial values. A well-known example is due to A.N. Tikhonov [a1].
|[a1]||A.N. Tikhonov, "Uniqueness theorems for the heat equation" Mat. Sb. , 42 (1935) pp. 199–216 (In Russian)|
|[a2]||J.R. Cannon, "The one-dimensional heat equation" , Addison-Wesley (1984)|
|[a3]||L.I. Ronkin, "Inroduction to the theory of entire functions of several variables" , Transl. Math. Monogr. , 44 , Amer. Math. Soc. (1974) (Translated from Russian)|
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