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Stagnation Zones for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq1_HTML.gif -Harmonic Functions on Canonical Domains

Boundary Value Problems20102009:853607

DOI: 10.1155/2009/853607

Received: 1 July 2009

Accepted: 15 November 2009

Published: 4 February 2010

Abstract

We study stagnation zones of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq2_HTML.gif -harmonic functions on canonical domains in the Euclidean https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq3_HTML.gif -dimensional space. Phragmén-Lindelöf type theorems are proved.

1. Introduction

In this article we investigate solutions of the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq4_HTML.gif -Laplace equation on canonical domains in the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq5_HTML.gif -dimensional Euclidean space.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq6_HTML.gif is a domain in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq7_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq8_HTML.gif be a function. For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq9_HTML.gif , a subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq10_HTML.gif is called https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq11_HTML.gif -zone (stagnation zone with the deviation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq12_HTML.gif ) of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq13_HTML.gif if there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq14_HTML.gif such that the difference between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq15_HTML.gif and the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq16_HTML.gif is smaller than https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq17_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq18_HTML.gif . We may, for example, consider difference in the sense of the sup norm

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ1_HTML.gif
(1.1)

the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq19_HTML.gif -norm

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ2_HTML.gif
(1.2)

or the Sobolev norm

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq20_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq21_HTML.gif -dimensional Hausdorff measure in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq22_HTML.gif .

For discussion about the history of the question, recent results and applications the reader is referred to [1, 2].

Some estimates of stagnation zone sizes for solutions of the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq23_HTML.gif -Laplace equation on locally Lipschitz surfaces and behavior of solutions in stagnation zones were given in [3]. In this paper we consider solutions of the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq24_HTML.gif -Laplace equation in subdomains of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq25_HTML.gif of a special form, canonical domains. In two-dimensional case, such domains are sectors and strips. In higher dimensions, they are conical and cylindrical regions. The special form of domains allows us to obtain more precise results.

Below we study stagnation zones of generalized solutions of the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq26_HTML.gif -Laplace equation

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ4_HTML.gif
(1.4)

(see [4]) with boundary conditions of types (see Definitions 1.1 and 1.2 below)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ5_HTML.gif
(1.5)

on canonical domains in the Euclidean https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq27_HTML.gif -dimensional space, where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq28_HTML.gif is a closed subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq29_HTML.gif . We will prove Phragmén-Lindelöf type theorems for solutions of the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq30_HTML.gif -Laplace equation with such boundary conditions.

1.1. Canonical Domains

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq31_HTML.gif . Fix an integer https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq32_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq33_HTML.gif and set

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ6_HTML.gif
(1.6)

We call the set

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ7_HTML.gif
(1.7)

a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq34_HTML.gif -ball and

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ8_HTML.gif
(1.8)

a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq35_HTML.gif -sphere in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq36_HTML.gif . In particular, the symbol https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq37_HTML.gif denotes the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq38_HTML.gif -sphere with the radius https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq39_HTML.gif , that is, the set

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ9_HTML.gif
(1.9)

For every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq40_HTML.gif , we set

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ10_HTML.gif
(1.10)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq41_HTML.gif be fixed, and let (see Figure 1)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Fig1_HTML.jpg
Figure 1

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq42_HTML.gif (a) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq43_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq44_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ11_HTML.gif
(1.11)

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq45_HTML.gif , we also assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq46_HTML.gif . Then for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq47_HTML.gif , the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq48_HTML.gif is the a layer between two parallel hyperplanes, and for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq49_HTML.gif the boundary of the domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq50_HTML.gif consists of two coaxial cylindrical surfaces. The intersections https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq51_HTML.gif are precompact for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq52_HTML.gif . Thus, the functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq53_HTML.gif are exhaustion functions for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq54_HTML.gif .

1.2. Structure Conditions

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq55_HTML.gif be a subdomain of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq56_HTML.gif and let

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ12_HTML.gif
(1.12)

be a vector function such that for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq57_HTML.gif the function

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ13_HTML.gif
(1.13)

is defined and is continuous with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq58_HTML.gif . We assume that the function

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ14_HTML.gif
(1.14)

is measurable in the Lebesgue sense for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq59_HTML.gif and

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ15_HTML.gif
(1.15)

Suppose that for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq60_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq61_HTML.gif the following properties hold:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ16_HTML.gif
(1.16)

with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq62_HTML.gif and some constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq63_HTML.gif . We consider the equation

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ17_HTML.gif
(1.17)

An important special case of (1.17) is the Laplace equation

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ18_HTML.gif
(1.18)

As in [4, Chapter 6], we call continuous weak solutions of (1.17) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq64_HTML.gif -harmonic functions. However we should note that our definition of generalized solutions is slightly different from the definition given in [4, page 56].

1.3. Frequencies

Fix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq66_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq67_HTML.gif be an open subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq68_HTML.gif (with respect to the relative topology of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq69_HTML.gif ), and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq70_HTML.gif be a nonempty closed subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq71_HTML.gif . We set

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ19_HTML.gif
(1.19)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq72_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq73_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq74_HTML.gif , then we call https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq75_HTML.gif the first frequency of the order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq76_HTML.gif of the set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq77_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq78_HTML.gif , then the quantity https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq79_HTML.gif is thethird frequency.

The second frequency is the following quantity:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ20_HTML.gif
(1.20)

where the supremum is taken over all constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq80_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq81_HTML.gif . See also Pólya and Szegö [5] as well as Lax [6].

1.4. Generalized Boundary Conditions

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq82_HTML.gif is a proper subdomain of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq83_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq84_HTML.gif be a locally Lipschitz function. We denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq85_HTML.gif the set of all points https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq86_HTML.gif at which https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq87_HTML.gif does not have the differential. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq88_HTML.gif be a subset and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq89_HTML.gif be its boundary with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq90_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq91_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq92_HTML.gif -rectifiable, then it has locally finite perimeter in the sense of De Giorgi, and therefore a unit normal vector https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq93_HTML.gif exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq94_HTML.gif -almost everywhere on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq95_HTML.gif [7, Sections 3.2.14, 3.2.15].

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq96_HTML.gif be a domain and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq97_HTML.gif be a subset of the boundary of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq98_HTML.gif . Define the concept of a generalized solution of (1.17) with zero boundary conditions on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq99_HTML.gif . A subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq100_HTML.gif is called admissible, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq101_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq102_HTML.gif have a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq103_HTML.gif -rectifiable boundary with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq104_HTML.gif .

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq105_HTML.gif is unbounded. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq106_HTML.gif be a set closed in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq107_HTML.gif . We denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq108_HTML.gif the collection of all subdomains https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq109_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq110_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq111_HTML.gif -rectifiable boundaries https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq112_HTML.gif .

Definition 1.1.

We say that a locally Lipschitz function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq113_HTML.gif is a generalized solution of (1.17) with the boundary condition
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ21_HTML.gif
(1.21)
if for every subdomain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq114_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ22_HTML.gif
(1.22)
and for every locally Lipschitz function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq115_HTML.gif the following property holds:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ23_HTML.gif
(1.23)

Here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq116_HTML.gif is the unit normal vector of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq117_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq118_HTML.gif is the volume element on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq119_HTML.gif .

Definition 1.2.

We say that a locally Lipschitz function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq120_HTML.gif is a generalized solution of (1.17) with the boundary condition
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ24_HTML.gif
(1.24)
if for every subdomain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq121_HTML.gif with (1.22) and for every locally Lipschitz function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq122_HTML.gif the following property holds:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ25_HTML.gif
(1.25)

In the case of a smooth boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq123_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq124_HTML.gif , the relation (1.23) implies (1.17) with (1.21) everywhere on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq125_HTML.gif . This requirement (1.25) implies (1.17) with (1.24) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq126_HTML.gif . See [8, Section 9.2.1].

The surface integrals exist by (1.22). Indeed, this assumption guarantees that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq127_HTML.gif exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq128_HTML.gif a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq129_HTML.gif . The assumption that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq130_HTML.gif implies existence of a normal vector https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq131_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq132_HTML.gif a.e. points on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq133_HTML.gif [7, Chapter 2, Section 3.2]. Thus, the scalar product https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq134_HTML.gif is defined and is finite a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq135_HTML.gif .

2. Saint-Venant's Principle

In this section, we will prove the Saint-Venant principle for solutions of the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq136_HTML.gif -Laplace equation. The Saint-Venant principle states that strains in a body produced by application of a force onto a small part of its surface are of negligible magnitude at distances that are large compared to the diameter of the part where the force is applied. This well known result in elasticity theory is often stated and used in a loose form. For mathematical investigation of the results of this type, see, for example, [9].

In this paper the inequalities of the form (2.5), (2.4) are called the Saint-Venant principle (see also [9, 10]). Here we consider only the case of canonical domains. We plan to consider the general case in another article.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq137_HTML.gif . Fix a domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq138_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq139_HTML.gif with compact and smooth boundary, and write

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ26_HTML.gif
(2.1)

We write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq140_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq141_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq142_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq143_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq144_HTML.gif and

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ27_HTML.gif
(2.2)

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq145_HTML.gif , we set

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ28_HTML.gif
(2.3)

Theorem 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq146_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq147_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq148_HTML.gif is a generalized solution of (1.17) with the generalized boundary condition (1.21) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq149_HTML.gif , then the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ29_HTML.gif
(2.4)

holds for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq150_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq151_HTML.gif is a generalized solution of (1.17) with the generalized boundary condition (1.24), then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ30_HTML.gif
(2.5)
holds for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq152_HTML.gif . Here
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ31_HTML.gif
(2.6)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ32_HTML.gif
(2.7)

Proof.

Case A.

At first we consider the case in which https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq153_HTML.gif is a generalized solution of (1.17) with the generalized boundary condition (1.24) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq154_HTML.gif . It is easy to see that a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq155_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ33_HTML.gif
(2.8)
The domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq156_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq157_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq158_HTML.gif be a locally Lipschitz function. By (1.25) we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ34_HTML.gif
(2.9)
But
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ35_HTML.gif
(2.10)
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq159_HTML.gif , we have by (1.16) and (1.25)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ36_HTML.gif
(2.11)
since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq160_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq161_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq162_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq163_HTML.gif . We obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ37_HTML.gif
(2.12)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ38_HTML.gif
(2.13)
Note that we may also choose
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ39_HTML.gif
(2.14)

to obtain an inequality similar to (2.12).

Next we will estimate the right side of (2.12). By (1.16) and the Hölder inequality,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ40_HTML.gif
(2.15)
By using (1.19), we may write
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ41_HTML.gif
(2.16)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ42_HTML.gif
(2.17)
By (2.12) and the Fubini theorem,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ43_HTML.gif
(2.18)
By integrating this differential inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ44_HTML.gif
(2.19)
for arbitrary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq164_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq165_HTML.gif . We have shown that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ45_HTML.gif
(2.20)

Case B.

Now we assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq166_HTML.gif is a generalized solution of (1.17) with the boundary condition (1.21) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq167_HTML.gif . Fix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq168_HTML.gif . By choosing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq169_HTML.gif in (1.23), we see that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ46_HTML.gif
(2.21)
For an arbitrary constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq170_HTML.gif , we get from this and (1.23)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ47_HTML.gif
(2.22)
Thus
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ48_HTML.gif
(2.23)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ49_HTML.gif
(2.24)
or
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ50_HTML.gif
(2.25)
As above, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ51_HTML.gif
(2.26)
By using (1.20), we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ52_HTML.gif
(2.27)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq171_HTML.gif is the constant from (1.20). Then by (2.26) and (2.27),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ53_HTML.gif
(2.28)
and by (2.25) we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ54_HTML.gif
(2.29)
or
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ55_HTML.gif
(2.30)
By integrating this inequality, we have shown that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ56_HTML.gif
(2.31)

3. Stagnation Zones

Next we apply the Saint-Venant principle to obtain information about stagnation zones of generalized solutions of (1.17). We first consider zones with respect to the Sobolev norm. Other results of this type follow immediately from well-known imbedding theorems.

3.1. Stagnation Zones with Respect to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq172_HTML.gif -Norm

We rewrite (2.4) and (2.5) in another form. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq173_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq174_HTML.gif . Fix a domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq175_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq176_HTML.gif with compact and smooth boundary, and write

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ57_HTML.gif
(3.1)

We write

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ58_HTML.gif
(3.2)

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq177_HTML.gif and

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ59_HTML.gif
(3.3)

we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ60_HTML.gif
(3.4)

and we denote

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ61_HTML.gif
(3.5)

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq178_HTML.gif . We write

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ62_HTML.gif
(3.6)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ63_HTML.gif
(3.7)

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq179_HTML.gif By (2.5) we have, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq180_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ64_HTML.gif
(3.8)

where

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ65_HTML.gif
(3.9)

By choosing the estimate as in (2.14), we also have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ66_HTML.gif
(3.10)

where

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ67_HTML.gif
(3.11)

By adding these inequalities and noting that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq181_HTML.gif , we obtain

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ68_HTML.gif
(3.12)

Thus we have the estimate

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ69_HTML.gif
(3.13)

Similarly, from (2.4) we obtain

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ70_HTML.gif
(3.14)

From this we obtain the following theorem on stagnation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq182_HTML.gif -zones.

Theorem 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq183_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq184_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq185_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq186_HTML.gif is as in (3.3). If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq187_HTML.gif is a solution of (1.17) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq188_HTML.gif with the generalized boundary condition (1.21) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq189_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq190_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ71_HTML.gif
(3.15)
or a solution of (1.17) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq191_HTML.gif with the generalized boundary condition (1.24) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq192_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ72_HTML.gif
(3.16)
then the subdomain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq193_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq194_HTML.gif -zone with respect to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq195_HTML.gif -norm, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ73_HTML.gif
(3.17)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq196_HTML.gif is as in (3.6).

3.2. Stagnation Zones with Respect to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq197_HTML.gif -Norm

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq198_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq199_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq200_HTML.gif is as in (3.3).

Denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq201_HTML.gif the best constant of the imbedding theorem from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq202_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq203_HTML.gif that is in the inequality

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ74_HTML.gif
(3.18)

if such constant exists (see Maz'ya [11] or [12]). Then we obtain from (3.13), (3.14)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ75_HTML.gif
(3.19)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ76_HTML.gif
(3.20)

These relations can be used to obtain information about stagnation zones with respect to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq204_HTML.gif -norm. Namely, we have the following.

Theorem 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq205_HTML.gif , and let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ77_HTML.gif
(3.21)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq206_HTML.gif is a domain in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq207_HTML.gif with compact and smooth boundary. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq208_HTML.gif is a solution of (1.17) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq209_HTML.gif , with the generalized boundary condition, (1.21) or (1.24), on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq210_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq211_HTML.gif , and the right side of, (3.19) or (3.20), is smaller than https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq212_HTML.gif , then the domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq213_HTML.gif is a stagnation zone with the deviation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq214_HTML.gif in the sense of the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq215_HTML.gif -norm on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq216_HTML.gif .

3.3. Stagnation Zones for Bounded, Uniformly Continuous Functions

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq217_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq218_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq219_HTML.gif is as in (3.3).

As before, denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq220_HTML.gif the best constant of the imbedding theorem from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq221_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq222_HTML.gif , that is in the inequality

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ78_HTML.gif
(3.22)

if such constant exists. For example, if the domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq223_HTML.gif is convex, then (3.22) holds for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq224_HTML.gif (see Maz'ya [11] or [12, page 85]).

In this case from (3.13), (3.14), we obtain

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ79_HTML.gif
(3.23)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ80_HTML.gif
(3.24)

These relations can be used to obtain theorems about stagnation zones for bounded, uniformly continuous functions.

Theorem 3.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq225_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq226_HTML.gif is a solution of (1.17), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq227_HTML.gif , on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq228_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq229_HTML.gif is as before with the generalized boundary condition, (1.21) or (1.24), on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq230_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq231_HTML.gif and the right side of, (3.23) or (3.24), is smaller than https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq232_HTML.gif , then the domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq233_HTML.gif is a stagnation zone with the deviation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq234_HTML.gif in the sense of the norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq235_HTML.gif .

4. Other Applications

Next we prove Phragmén-Lindelöf type theorems for the solutions of the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq236_HTML.gif -Laplace equation with boundary conditions (1.21) and (1.24).

4.1. Estimates for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq237_HTML.gif -Norms

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq238_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq239_HTML.gif be a domain in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq240_HTML.gif with compact and smooth boundary. Write

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ81_HTML.gif
(4.1)

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq241_HTML.gif is as in (3.3). First we will prove some estimates of the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq242_HTML.gif -norm of a solution. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq243_HTML.gif be a solution of (1.17) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq244_HTML.gif with the generalized boundary condition (1.21) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq245_HTML.gif . Fix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq246_HTML.gif and estimate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq247_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq248_HTML.gif be a Lipschitz function such that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ82_HTML.gif
(4.2)

We choose

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ83_HTML.gif
(4.3)

The function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq249_HTML.gif is admissible in Definition 1.1 for

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ84_HTML.gif
(4.4)

As in (2.22), we may by (1.23) write

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ85_HTML.gif
(4.5)

By the construction of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq250_HTML.gif , (4.2), and (4.3), the surface integral is equal to zero, and we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ86_HTML.gif
(4.6)

Thus by (1.16),

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ87_HTML.gif
(4.7)

Now we note that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ88_HTML.gif
(4.8)

and by the Hölder inequality,

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ89_HTML.gif
(4.9)
From this inequality and (4.7), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ90_HTML.gif
(4.10)

Because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq251_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq252_HTML.gif , we have the following inequality:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ91_HTML.gif
(4.11)

Next we will find that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ92_HTML.gif
(4.12)

where the minimum is taken over all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq253_HTML.gif in (4.3). We have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ93_HTML.gif
(4.13)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ94_HTML.gif
(4.14)

Because by the Hölder inequality

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ95_HTML.gif
(4.15)

we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ96_HTML.gif
(4.16)
and hence,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ97_HTML.gif
(4.17)

It is easy to see that here the equality holds for a special choice of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq254_HTML.gif . Thus

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ98_HTML.gif
(4.18)

Similarly,

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ99_HTML.gif
(4.19)

From (4.14) we obtain

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ100_HTML.gif
(4.20)

By using (4.11), we obtain the inequality

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ101_HTML.gif
(4.21)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq255_HTML.gif is an arbitrary constant. From this we obtain

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ102_HTML.gif
(4.22)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq256_HTML.gif .

Similarly, for the solutions of the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq257_HTML.gif -Laplace equation with the boundary condition (1.24), we may prove that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ103_HTML.gif
(4.23)

It follows that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ104_HTML.gif
(4.24)

4.2. Phragmén-Lindelöf Type Theorems I

We prove Phragmén-Lindelöf type theorems for cylindrical domains. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq258_HTML.gif . Fix a domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq259_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq260_HTML.gif with compact and smooth boundary. Consider the domain

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ105_HTML.gif
(4.25)

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq261_HTML.gif be a generalized solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition (1.21) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq262_HTML.gif .

Fix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq263_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq264_HTML.gif be as in (3.3). Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq265_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq266_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq267_HTML.gif th unit coordinate vector, and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq268_HTML.gif By (4.22)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ106_HTML.gif
(4.26)

By using (3.14), we obtain from this the inequality

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ107_HTML.gif
(4.27)

We observe that in this case

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ108_HTML.gif
(4.28)

and hence,

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ109_HTML.gif
(4.29)

It follows that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ110_HTML.gif
(4.30)

By letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq269_HTML.gif , we obtain the following statement.

Theorem 4.1.

Fix a domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq270_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq271_HTML.gif with compact and smooth boundary. Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ111_HTML.gif
(4.31)

and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq272_HTML.gif be a generalized solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition (1.21) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq273_HTML.gif . If for a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq274_HTML.gif the right side of (4.30) goes to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq275_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq276_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq277_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq278_HTML.gif .

Similarly for a solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq279_HTML.gif of (1.17) with (1.15) and (1.16) satisfying the boundary condition (1.24), we may write

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ112_HTML.gif
(4.32)

However here we do not have any identity similar to (4.28). We have the following.

Theorem 4.2.

Fix a domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq280_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq281_HTML.gif with compact and smooth boundary. Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ113_HTML.gif
(4.33)

and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq282_HTML.gif be a generalized solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition (1.24) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq283_HTML.gif . If the right side of (4.32) tends to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq284_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq285_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq286_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq287_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq288_HTML.gif everywhere on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq289_HTML.gif , then an identity similar to (4.28) holds in the following form:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ114_HTML.gif
(4.34)

As above, we find that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ115_HTML.gif
(4.35)

Thus we obtain the following.

Corollary 4.3.

Fix a domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq290_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq291_HTML.gif with compact and smooth boundary. Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ116_HTML.gif
(4.36)

and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq292_HTML.gif be a generalized solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq293_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq294_HTML.gif . If the right side of (4.35) tends to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq295_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq296_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq297_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq298_HTML.gif .

4.3. Phragmén-Lindelöf Type Theorems II

We prove Phragmén-Lindelöf type theorems for canonical domains of an arbitrary form. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq299_HTML.gif . We consider a domain

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ117_HTML.gif
(4.37)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq300_HTML.gif is a domain in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq301_HTML.gif with compact and smooth boundary. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq302_HTML.gif be a generalized solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition (1.21) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq303_HTML.gif .

Fix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq304_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq305_HTML.gif By (4.22) we may write

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ118_HTML.gif
(4.38)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq306_HTML.gif . As in (3.14), we obtain from (2.4) the estimate

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ119_HTML.gif
(4.39)

By combining these inequalities, we obtain

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ120_HTML.gif
(4.40)

The inequality (4.40) holds for arbitrary constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq307_HTML.gif and every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq308_HTML.gif . Thus the following statement holds.

Theorem 4.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq309_HTML.gif be a generalized solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition (1.21) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq310_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq311_HTML.gif . If for a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq312_HTML.gif the right side of (4.40) tends to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq313_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq314_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq315_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq316_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq317_HTML.gif satisfies (1.17) with (1.15), (1.16) and the boundary condition (1.24) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq318_HTML.gif , then we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ121_HTML.gif
(4.41)

We obtain the following.

Theorem 4.5.

Fix a domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq319_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq320_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq321_HTML.gif , with compact and smooth boundary. Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ122_HTML.gif
(4.42)

and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq322_HTML.gif be a generalized solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition (1.24) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq323_HTML.gif . If for a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq324_HTML.gif the right side of (4.41) tends to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq325_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq326_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq327_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq328_HTML.gif .

Authors’ Affiliations

(1)
Department of Mathematics, Volgograd State University
(2)
Department of Mathematics and Systems Analysis, Aalto University
(3)
Department of Mathematics, University of Turku

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© Vladimir M. Miklyukov et al. 2009

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