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

  • VladimirM Miklyukov1,

    Affiliated with

    • Antti Rasila2Email author and

      Affiliated with

      • Matti Vuorinen3

        Affiliated with

        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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq5_HTML.gif -dimensional Euclidean space.

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

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

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

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

        or the Sobolev norm

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

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq20_HTML.gif is the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq21_HTML.gif -dimensional Hausdorff measure in http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq24_HTML.gif -Laplace equation in subdomains of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq26_HTML.gif -Laplace equation

        http://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)

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq27_HTML.gif -dimensional space, where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq28_HTML.gif is a closed subset of http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq31_HTML.gif . Fix an integer http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq33_HTML.gif and set

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

        We call the set

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

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

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

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

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

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

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ10_HTML.gif
        (1.10)
        Let http://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)
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Fig1_HTML.jpg
        Figure 1

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

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

        For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq45_HTML.gif , we also assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq46_HTML.gif . Then for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq47_HTML.gif , the http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq49_HTML.gif the boundary of the domain http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq51_HTML.gif are precompact for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq52_HTML.gif . Thus, the functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq53_HTML.gif are exhaustion functions for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq54_HTML.gif .

        1.2. Structure Conditions

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

        http://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. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq57_HTML.gif the function

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq58_HTML.gif . We assume that the function

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq59_HTML.gif and

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

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

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

        http://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

        http://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) http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq66_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq67_HTML.gif be an open subset of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq69_HTML.gif ), and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq70_HTML.gif be a nonempty closed subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq71_HTML.gif . We set

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

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

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq80_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq82_HTML.gif is a proper subdomain of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq83_HTML.gif . Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq85_HTML.gif the set of all points http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq86_HTML.gif at which http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq87_HTML.gif does not have the differential. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq88_HTML.gif be a subset and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq89_HTML.gif be its boundary with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq90_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq91_HTML.gif is http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq93_HTML.gif exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq94_HTML.gif -almost everywhere on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq96_HTML.gif be a domain and let http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq99_HTML.gif . A subset http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq100_HTML.gif is called admissible, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq101_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq102_HTML.gif have a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq103_HTML.gif -rectifiable boundary with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq104_HTML.gif .

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq105_HTML.gif is unbounded. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq106_HTML.gif be a set closed in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq107_HTML.gif . We denote by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq108_HTML.gif the collection of all subdomains http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq109_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq110_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq111_HTML.gif -rectifiable boundaries http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ21_HTML.gif
        (1.21)
        if for every subdomain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq114_HTML.gif ,
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq115_HTML.gif the following property holds:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ23_HTML.gif
        (1.23)

        Here http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq116_HTML.gif is the unit normal vector of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq117_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq118_HTML.gif is the volume element on http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ24_HTML.gif
        (1.24)
        if for every subdomain http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq122_HTML.gif the following property holds:
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq123_HTML.gif , and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq127_HTML.gif exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq128_HTML.gif a.e. on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq129_HTML.gif . The assumption that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq130_HTML.gif implies existence of a normal vector http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq131_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq132_HTML.gif a.e. points on http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq137_HTML.gif . Fix a domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq138_HTML.gif in http://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

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

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

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

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

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

        Theorem 2.1.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq146_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq147_HTML.gif . If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq149_HTML.gif , then the inequality
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ29_HTML.gif
        (2.4)

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

        If http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ30_HTML.gif
        (2.5)
        holds for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq152_HTML.gif . Here
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ31_HTML.gif
        (2.6)
        http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq155_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ33_HTML.gif
        (2.8)
        The domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq156_HTML.gif belongs to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq157_HTML.gif . Let http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ34_HTML.gif
        (2.9)
        But
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ35_HTML.gif
        (2.10)
        For http://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)
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ36_HTML.gif
        (2.11)
        since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq160_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq161_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq162_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq163_HTML.gif . We obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ37_HTML.gif
        (2.12)
        where
        http://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
        http://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,
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ41_HTML.gif
        (2.16)
        http://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,
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ44_HTML.gif
        (2.19)
        for arbitrary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq164_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq165_HTML.gif . We have shown that
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq167_HTML.gif . Fix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq168_HTML.gif . By choosing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq169_HTML.gif in (1.23), we see that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ46_HTML.gif
        (2.21)
        For an arbitrary constant http://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)
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ47_HTML.gif
        (2.22)
        Thus
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ48_HTML.gif
        (2.23)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ49_HTML.gif
        (2.24)
        or
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ50_HTML.gif
        (2.25)
        As above, we obtain
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ52_HTML.gif
        (2.27)
        where http://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),
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ54_HTML.gif
        (2.29)
        or
        http://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
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq173_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq174_HTML.gif . Fix a domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq175_HTML.gif in http://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

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

        We write

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

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

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

        we have

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

        and we denote

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

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

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

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

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

        where

        http://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

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

        where

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq181_HTML.gif , we obtain

        http://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

        http://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

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq182_HTML.gif -zones.

        Theorem 3.1.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq183_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq184_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq185_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq186_HTML.gif is as in (3.3). If http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq189_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq190_HTML.gif and
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq192_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ72_HTML.gif
        (3.16)
        then the subdomain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq193_HTML.gif is an http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq194_HTML.gif -zone with respect to the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq195_HTML.gif -norm, that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ73_HTML.gif
        (3.17)

        where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq197_HTML.gif -Norm

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq198_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq199_HTML.gif where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq202_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq203_HTML.gif that is in the inequality

        http://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)

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ75_HTML.gif
        (3.19)
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq205_HTML.gif , and let
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ77_HTML.gif
        (3.21)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq206_HTML.gif is a domain in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq207_HTML.gif with compact and smooth boundary. If http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq210_HTML.gif , where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq212_HTML.gif , then the domain http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq214_HTML.gif in the sense of the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq215_HTML.gif -norm on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq217_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq218_HTML.gif where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq221_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq222_HTML.gif , that is in the inequality

        http://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 http://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 http://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

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ79_HTML.gif
        (3.23)
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq225_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq226_HTML.gif is a solution of (1.17), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq227_HTML.gif , on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq228_HTML.gif where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq230_HTML.gif where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq232_HTML.gif , then the domain http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq234_HTML.gif in the sense of the norm http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq237_HTML.gif -Norms

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

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

        Suppose that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq242_HTML.gif -norm of a solution. Let http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq245_HTML.gif . Fix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq246_HTML.gif and estimate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq247_HTML.gif .

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

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

        We choose

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

        The function http://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

        http://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

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

        By the construction of http://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

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

        Thus by (1.16),

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

        Now we note that

        http://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,

        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ90_HTML.gif
        (4.10)

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

        http://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

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq253_HTML.gif in (4.3). We have

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

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

        we have

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_Equ96_HTML.gif
        (4.16)
        and hence,
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq254_HTML.gif . Thus

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

        Similarly,

        http://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

        http://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

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

        where http://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

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

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

        Similarly, for the solutions of the http://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

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

        It follows that

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq258_HTML.gif . Fix a domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq259_HTML.gif in http://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

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

        Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq262_HTML.gif .

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

        http://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

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

        and hence,

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

        It follows that

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

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

        and let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq273_HTML.gif . If for a constant http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq275_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq276_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq277_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq278_HTML.gif .

        Similarly for a solution http://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

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

        and let http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq284_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq285_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq286_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq287_HTML.gif .

        If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq288_HTML.gif everywhere on http://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:

        http://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

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

        and let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq293_HTML.gif on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq295_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq296_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq297_HTML.gif on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq299_HTML.gif . We consider a domain

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

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq300_HTML.gif is a domain in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq301_HTML.gif with compact and smooth boundary. Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq303_HTML.gif .

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

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

        where http://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

        http://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

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq307_HTML.gif and every http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq310_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq311_HTML.gif . If for a constant http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq313_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq314_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq315_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq316_HTML.gif .

        If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq318_HTML.gif , then we have

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

        and let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq323_HTML.gif . If for a constant http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq325_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq326_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F853607/MediaObjects/13661_2009_Article_884_IEq327_HTML.gif on http://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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        Copyright

        © Vladimir M. Miklyukov et al. 2009

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