- Research Article
- Open Access

- VladimirM Miklyukov
^{1}, - Antti Rasila
^{2}Email author and - Matti Vuorinen
^{3}

**2009**:853607

https://doi.org/10.1155/2009/853607

© Vladimir M. Miklyukov et al. 2009

**Received:**1 July 2009**Accepted:**15 November 2009**Published:**4 February 2010

## Abstract

## Keywords

- Generalize Solution
- Laplace Equation
- Lipschitz Function
- Smooth Boundary
- Type Theorem

## 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

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 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 -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 Phragmén-Lindelöf 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 -sphere in . In particular, the symbol denotes the -sphere with the radius , that is, the set

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

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 the*third 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.

Here is the unit normal vector of and is the volume element on .

Definition 1.2.

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. Saint-Venant's Principle

In this section, we will prove the Saint-Venant principle for solutions of the -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 . Fix a domain in with compact and smooth boundary, and write

Theorem 2.1.

Proof.

Case A.

to obtain an inequality similar to (2.12).

Case B.

## 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 -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

we have

and we denote

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.

### 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.

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 Phragmén-Lindelöf 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,

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

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

Similarly, for the solutions of the -Laplace equation with the boundary condition (1.24), we may prove that

It follows that

### 4.2. Phragmén-Lindelöf Type Theorems I

We prove Phragmén-Lindelöf 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 .

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.

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.

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.

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. Phragmén-Lindelöf Type Theorems II

We prove Phragmén-Lindelöf 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.

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 .

## Authors’ Affiliations

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