Phragmén-Lindelöf theorem
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 [1]. 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
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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 [1]–[4], in particular the generalization given in [4]). 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 [5], [6]). 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
![]() |
References
[1] | E. Phragmén, E. Lindelöf, "Sur une extension d'un principe classique de l'analyse" Acta Math. , 31 (1908) pp. 381–406 |
[2] | E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) |
[3] | S. Stoilov, "The theory of functions of a complex variable" , 1–2 , Moscow (1962) (In Russian; translated from Rumanian) |
[4] | M.A. Evgrafov, "Analytic functions" , Saunders , Philadelphia (1966) (Translated from Russian) |
[5] | I.I. Privalov, "Subharmonic functions" , Moscow-Leningrad (1937) (In Russian) |
[6] | E.D. Solomentsev, "Harmonic and subharmonic functions and their generalizations" Itogi Nauk. Mat. Anal. Teor. Veryatnost. Regulirovanie. 1962 (1964) pp. 83–100 (In Russian) |
[7] | 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) |
[8] | E.M. Landis, "Second-order equations of elliptic and parabolic type" , Moscow (1971) (In Russian) |
Comments
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].
References
[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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