We study stagnation zones of harmonic functions on canonical domains in the Euclidean dimensional space. PhragménLindelöf type theorems are proved.
1. Introduction
In this article we investigate solutions of the Laplace equation on canonical domains in the dimensional Euclidean space.
Suppose that is a domain in , and let be a function. For , a subset is called zone (stagnation zone with the deviation) of if there exists a constant such that the difference between and the function is smaller than on . We may, for example, consider difference in the sense of the sup norm
the norm
or the Sobolev norm
where is the dimensional Hausdorff measure in .
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 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 Laplace equation in subdomains of of a special form, canonical domains. In twodimensional 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 Laplace equation
(see [4]) with boundary conditions of types (see Definitions 1.1 and 1.2 below)
on canonical domains in the Euclidean dimensional space, where is a closed subset of . We will prove PhragménLindelöf type theorems for solutions of the Laplace equation with such boundary conditions.
1.1. Canonical Domains
Let . Fix an integer , and set
We call the set
a ball and
a sphere in . In particular, the symbol denotes the sphere with the radius , that is, the set
For every , we set
Let be fixed, and let (see Figure 1)
Figure 1. (a) and in .
For , we also assume that . Then for , the is the a layer between two parallel hyperplanes, and for the boundary of the domain consists of two coaxial cylindrical surfaces. The intersections are precompact for all . Thus, the functions are exhaustion functions for .
1.2. Structure Conditions
Let be a subdomain of and let
be a vector function such that for a.e. the function
is defined and is continuous with respect to . We assume that the function
is measurable in the Lebesgue sense for all and
Suppose that for a.e. and for all the following properties hold:
with and some constants . We consider the equation
An important special case of (1.17) is the Laplace equation
As in [4, Chapter 6], we call continuous weak solutions of (1.17) 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 and . Let be an open subset of (with respect to the relative topology of ), and let be a nonempty closed subset of . We set
where with . If , then we call the first frequency of the order of the set . If , then the quantity is thethird frequency.
The second frequency is the following quantity:
where the supremum is taken over all constants and . See also Pólya and Szegö [5] as well as Lax [6].
1.4. Generalized Boundary Conditions
Suppose that is a proper subdomain of . Let be a locally Lipschitz function. We denote by the set of all points at which does not have the differential. Let be a subset and let be its boundary with respect to . If is rectifiable, then it has locally finite perimeter in the sense of De Giorgi, and therefore a unit normal vector exists almost everywhere on [7, Sections 3.2.14, 3.2.15].
Let be a domain and let be a subset of the boundary of . Define the concept of a generalized solution of (1.17) with zero boundary conditions on . A subset is called admissible, if and have a rectifiable boundary with respect to .
Suppose that is unbounded. Let be a set closed in . We denote by the collection of all subdomains with and rectifiable boundaries .
Definition 1.1.
We say that a locally Lipschitz function is a generalized solution of (1.17) with the boundary condition
if for every subdomain ,
and for every locally Lipschitz function the following property holds:
Here is the unit normal vector of and is the volume element on .
Definition 1.2.
We say that a locally Lipschitz function is a generalized solution of (1.17) with the boundary condition
if for every subdomain with (1.22) and for every locally Lipschitz function the following property holds:
In the case of a smooth boundary , and , the relation (1.23) implies (1.17) with (1.21) everywhere on . This requirement (1.25) implies (1.17) with (1.24) on . See [8, Section 9.2.1].
The surface integrals exist by (1.22). Indeed, this assumption guarantees that exists a.e. on . The assumption that implies existence of a normal vector for a.e. points on [7, Chapter 2, Section 3.2]. Thus, the scalar product is defined and is finite a.e. on .
2. SaintVenant's Principle
In this section, we will prove the SaintVenant principle for solutions of the Laplace equation. The SaintVenant 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 SaintVenant 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 . Fix a domain in with compact and smooth boundary, and write
We write , , and . Let , and
For , we set
Theorem 2.1.
Let , and let . If is a generalized solution of (1.17) with the generalized boundary condition (1.21) on , then the inequality
holds for all .
If is a generalized solution of (1.17) with the generalized boundary condition (1.24), then
holds for all . Here
Proof.
Case A.
At first we consider the case in which is a generalized solution of (1.17) with the generalized boundary condition (1.24) on . It is easy to see that a.e. on ,
The domain belongs to . Let be a locally Lipschitz function. By (1.25) we have
But
For , we have by (1.16) and (1.25)
since for and for . We obtain
where
Note that we may also choose
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,
By using (1.19), we may write
By (2.12) and the Fubini theorem,
By integrating this differential inequality, we have
for arbitrary with . We have shown that
Case B.
Now we assume that is a generalized solution of (1.17) with the boundary condition (1.21) on . Fix . By choosing in (1.23), we see that
For an arbitrary constant , we get from this and (1.23)
Thus
where
or
As above, we obtain
By using (1.20), we get
where is the constant from (1.20). Then by (2.26) and (2.27),
and by (2.25) we have
or
By integrating this inequality, we have shown that
3. Stagnation Zones
Next we apply the SaintVenant 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 wellknown imbedding theorems.
3.1. Stagnation Zones with Respect to the Norm
We rewrite (2.4) and (2.5) in another form. Let and let . Fix a domain in with compact and smooth boundary, and write
We write
For and
we have
and we denote
Let . We write
Let By (2.5) we have, for ,
where
By choosing the estimate as in (2.14), we also have
where
By adding these inequalities and noting that , we obtain
Thus we have the estimate
Similarly, from (2.4) we obtain
From this we obtain the following theorem on stagnation zones.
Theorem 3.1.
Let , , and let where is as in (3.3). If is a solution of (1.17) on with the generalized boundary condition (1.21) on , where and
or a solution of (1.17) on with the generalized boundary condition (1.24) on and
then the subdomain is an zone with respect to the norm, that is,
where is as in (3.6).
3.2. Stagnation Zones with Respect to the Norm
Let , and let where is as in (3.3).
Denote by the best constant of the imbedding theorem from to that is in the inequality
if such constant exists (see Maz'ya [11] or [12]). Then we obtain from (3.13), (3.14)
These relations can be used to obtain information about stagnation zones with respect to the norm. Namely, we have the following.
Theorem 3.2.
Let , and let
where is a domain in with compact and smooth boundary. If is a solution of (1.17) on , with the generalized boundary condition, (1.21) or (1.24), on , where , and the right side of, (3.19) or (3.20), is smaller than , then the domain is a stagnation zone with the deviation in the sense of the norm on .
3.3. Stagnation Zones for Bounded, Uniformly Continuous Functions
Let , and let where is as in (3.3).
As before, denote by the best constant of the imbedding theorem from to , that is in the inequality
if such constant exists. For example, if the domain is convex, then (3.22) holds for (see Maz'ya [11] or [12, page 85]).
In this case from (3.13), (3.14), we obtain
These relations can be used to obtain theorems about stagnation zones for bounded, uniformly continuous functions.
Theorem 3.3.
Let . If is a solution of (1.17), , on where is as before with the generalized boundary condition, (1.21) or (1.24), on where and the right side of, (3.23) or (3.24), is smaller than , then the domain is a stagnation zone with the deviation in the sense of the norm .
4. Other Applications
Next we prove PhragménLindelöf type theorems for the solutions of the Laplace equation with boundary conditions (1.21) and (1.24).
4.1. Estimates for Norms
Let , and let be a domain in with compact and smooth boundary. Write
Suppose that is as in (3.3). First we will prove some estimates of the norm of a solution. Let be a solution of (1.17) on with the generalized boundary condition (1.21) on . Fix and estimate .
Let be a Lipschitz function such that
We choose
The function is admissible in Definition 1.1 for
As in (2.22), we may by (1.23) write
By the construction of , (4.2), and (4.3), the surface integral is equal to zero, and we have
Thus by (1.16),
Now we note that
and by the Hölder inequality,
From this inequality and (4.7), we obtain
Because on , we have the following inequality:
Next we will find that
where the minimum is taken over all in (4.3). We have
Because by the Hölder inequality
we have
and hence,
It is easy to see that here the equality holds for a special choice of . Thus
Similarly,
From (4.14) we obtain
By using (4.11), we obtain the inequality
where is an arbitrary constant. From this we obtain
where .
Similarly, for the solutions of the Laplace equation with the boundary condition (1.24), we may prove that
It follows that
4.2. PhragménLindelöf Type Theorems I
We prove PhragménLindelöf type theorems for cylindrical domains. Let . Fix a domain in with compact and smooth boundary. Consider the domain
Let be a generalized solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition (1.21) on .
Fix , and let be as in (3.3). Let , where is the th unit coordinate vector, and let By (4.22)
By using (3.14), we obtain from this the inequality
We observe that in this case
and hence,
It follows that
By letting , we obtain the following statement.
Theorem 4.1.
Fix a domain in with compact and smooth boundary. Let
and let be a generalized solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition (1.21) on . If for a constant the right side of (4.30) goes to as , then on .
Similarly for a solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition (1.24), we may write
However here we do not have any identity similar to (4.28). We have the following.
Theorem 4.2.
Fix a domain in with compact and smooth boundary. Let
and let be a generalized solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition (1.24) on . If the right side of (4.32) tends to as , then on .
If everywhere on , then an identity similar to (4.28) holds in the following form:
As above, we find that
Thus we obtain the following.
Corollary 4.3.
Fix a domain in with compact and smooth boundary. Let
and let be a generalized solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition on . If the right side of (4.35) tends to as , then on .
4.3. PhragménLindelöf Type Theorems II
We prove PhragménLindelöf type theorems for canonical domains of an arbitrary form. Let . We consider a domain
where is a domain in with compact and smooth boundary. Let be a generalized solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition (1.21) on .
Fix . Let By (4.22) we may write
where . As in (3.14), we obtain from (2.4) the estimate
By combining these inequalities, we obtain
The inequality (4.40) holds for arbitrary constant and every . Thus the following statement holds.
Theorem 4.4.
Let be a generalized solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition (1.21) on , . If for a constant the right side of (4.40) tends to as , then on .
If satisfies (1.17) with (1.15), (1.16) and the boundary condition (1.24) on , then we have
We obtain the following.
Theorem 4.5.
Fix a domain in , where , with compact and smooth boundary. Let
and let be a generalized solution of (1.17) with (1.15) and (1.16) satisfying the boundary condition (1.24) on . If for a constant the right side of (4.41) tends to as , then on .
References

Miklyukov VM (ed.): Proceedings of the Seminar 'Superslow Processes', Issue 1, Volgograd State University, Volgograd, Russia (2006).

Miklyukov VM (ed.): Proceedings of the Seminar 'Superslow Processes', Issue 2, Volgograd State University, Volgograd, Russia (2007).

Miklyukov, VM: Stagnation zones of solutions, Memory I.N. Vekua. Georgian Mathematical Journal. 14(3), 519–531 (2007)

Heinonen, J, Kilpeläinen, T, Martio, O: Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs,p. vi+363. The Clarendon Press, Oxford, UK (1993)

Pólya, G, Szegö, G: Isoperimetric Inequalities in Mathematical Physics,p. xvi+279. Princeton University Press, Princeton, NJ, USA (1951)

Lax, PD: A PhragménLindelöf theorem in harmonic analysis and its application to some questions in the theory of elliptic equations. Communications on Pure and Applied Mathematics. 10, 361–389 (1957). Publisher Full Text

Federer, H: Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften,p. xiv+676. Springer, New York, NY, USA (1969)

Miklyukov, VM: Introduction to Nonsmooth Analysis, Izdvo VolGU, Volgograd, Russia (2008)

Barbarosie, C, Toader, AM: SaintVenant's principle and its connections to shape and topology optimization. Zeitschrift für Angewandte Mathematik und Mechanik. 88(1), 23–32 (2008). PubMed Abstract  Publisher Full Text

Oleĭnik, OA, Yosifian, GA: Boundary value problems for second order elliptic equations in unbounded domains and SaintVenant's principle. Annali della Scuola Normale Superiore, Classe di Scienze. 4(4), 269–290 (1977)

Maz'ya, VG: Sobolev Spaces, Springer Series in Soviet Mathematics,p. xix+486. Springer, Berlin, Germany (1985)

Adams, R, Fournier, J: Sobolev Spaces, Pure and Applied Mathematics,p. xiv+305. Academic Press, New York, NY, USA (2003)