Stagnation Zones for -Harmonic Functions on Canonical Domains

Abstract

We study stagnation zones of -harmonic functions on canonical domains in the Euclidean -dimensional space. Phragmén-Lindelöf 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

(1.1)

the -norm

(1.2)

or the Sobolev norm

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

We call the set

(1.7)

a -ball and

(1.8)

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

(1.9)

For every , we set

(1.10)

Let be fixed, and let (see Figure 1)

Figure 1

(a) and in .

(1.11)

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

(1.12)

be a vector function such that for a.e. the function

(1.13)

is defined and is continuous with respect to . We assume that the function

(1.14)

is measurable in the Lebesgue sense for all and

(1.15)

Suppose that for a.e. and for all the following properties hold:

(1.16)

with and some constants . We consider the equation

(1.17)

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

(1.18)

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

(1.19)

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:

(1.20)

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

(1.21)

if for every subdomain ,

(1.22)

and for every locally Lipschitz function the following property holds:

(1.23)

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

(1.24)

if for every subdomain with (1.22) and for every locally Lipschitz function the following property holds:

(1.25)

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

(2.1)

We write , , and . Let , and

(2.2)

For , we set

(2.3)

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

(2.4)

holds for all .

If is a generalized solution of (1.17) with the generalized boundary condition (1.24), then

(2.5)

holds for all . Here

(2.6)

(2.7)

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 ,

(2.8)

The domain belongs to . Let be a locally Lipschitz function. By (1.25) we have

(2.9)

But

(2.10)

For , we have by (1.16) and (1.25)

(2.11)

since for and for . We obtain

(2.12)

where

(2.13)

Note that we may also choose

(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,

(2.15)

By using (1.19), we may write

(2.16)

(2.17)

By (2.12) and the Fubini theorem,

(2.18)

By integrating this differential inequality, we have

(2.19)

for arbitrary with . We have shown that

(2.20)

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

(2.21)

For an arbitrary constant , we get from this and (1.23)

(2.22)

Thus

(2.23)

where

(2.24)

or

(2.25)

As above, we obtain

(2.26)

By using (1.20), we get

(2.27)

where is the constant from (1.20). Then by (2.26) and (2.27),

(2.28)

and by (2.25) we have

(2.29)

or

(2.30)

By integrating this inequality, we have shown that

(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 -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

(3.1)

We write

(3.2)

For and

(3.3)

we have

(3.4)

and we denote

(3.5)

Let . We write

(3.6)

(3.7)

Let By (2.5) we have, for ,

(3.8)

where

(3.9)

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

(3.10)

where

(3.11)

By adding these inequalities and noting that , we obtain

(3.12)

Thus we have the estimate

(3.13)

Similarly, from (2.4) we obtain

(3.14)

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

(3.15)

or a solution of (1.17) on with the generalized boundary condition (1.24) on and

(3.16)

then the subdomain is an -zone with respect to the -norm, that is,

(3.17)

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

(3.18)

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

(3.19)

(3.20)

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

(3.21)

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

(3.22)

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

(3.23)

(3.24)

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

(4.1)

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

(4.2)

We choose

(4.3)

The function is admissible in Definition 1.1 for

(4.4)

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

(4.5)

By the construction of , (4.2), and (4.3), the surface integral is equal to zero, and we have

(4.6)

Thus by (1.16),

(4.7)

Now we note that

(4.8)

and by the Hölder inequality,

(4.9)

From this inequality and (4.7), we obtain

(4.10)

Because on , we have the following inequality:

(4.11)

Next we will find that

(4.12)

where the minimum is taken over all in (4.3). We have

(4.13)

(4.14)

Because by the Hölder inequality

(4.15)

we have

(4.16)

and hence,

(4.17)

It is easy to see that here the equality holds for a special choice of . Thus

(4.18)

Similarly,

(4.19)

From (4.14) we obtain

(4.20)

By using (4.11), we obtain the inequality

(4.21)

where is an arbitrary constant. From this we obtain

(4.22)

where .

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

(4.23)

It follows that

(4.24)

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

(4.25)

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)

(4.26)

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

(4.27)

We observe that in this case

(4.28)

and hence,

(4.29)

It follows that

(4.30)

By letting , we obtain the following statement.

Theorem 4.1.

Fix a domain in with compact and smooth boundary. Let

(4.31)

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

(4.32)

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

(4.33)

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:

(4.34)

As above, we find that

(4.35)

Thus we obtain the following.

Corollary 4.3.

Fix a domain in with compact and smooth boundary. Let

(4.36)

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

(4.37)

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

(4.38)

where . As in (3.14), we obtain from (2.4) the estimate

(4.39)

By combining these inequalities, we obtain

(4.40)

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

(4.41)

We obtain the following.

Theorem 4.5.

Fix a domain in , where , with compact and smooth boundary. Let

(4.42)

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 .