- Research Article
- Open Access
© Vladimir M. Miklyukov et al. 2009
- Received: 1 July 2009
- Accepted: 15 November 2009
- Published: 4 February 2010
- Generalize Solution
- Laplace Equation
- Lipschitz Function
- Smooth Boundary
- Type Theorem
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
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 . 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.
(see ) 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
We call 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
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].
The second frequency is the following quantity:
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 .
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 .
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, .
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.
to obtain an inequality similar to (2.12).
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.
and we denote
By choosing the estimate as in (2.14), we also have
Thus we have the estimate
Similarly, from (2.4) we obtain
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
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.
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 .
As in (2.22), we may by (1.23) write
Thus by (1.16),
Now we note that
and by the Hölder inequality,
Next we will find that
Because by the Hölder inequality
From (4.14) we obtain
By using (4.11), we obtain the inequality
It follows that
4.2. Phragmén-Lindelöf Type Theorems I
By using (3.14), we obtain from this the inequality
We observe that in this case
It follows that
However here we do not have any identity similar to (4.28). We have the following.
As above, we find that
Thus we obtain the following.
4.3. Phragmén-Lindelöf Type Theorems II
By combining these inequalities, we obtain
We obtain the following.
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