ON SOME UNBOUNDED DOMAINS FOR A MAXIMUM PRINCIPLE

Abstract

In this paper, we study some characterizations of unbounded domains. Among these, so-called G-domain is introduced by Cabre for the Aleksandrov-Bakelman-Pucci maximum principle of second order linear elliptic operator in a non-divergence form. This domain is generalized to wG-domain by Vitolo for the maximum principle of an unbounded domain, which contains G-domain. We study the properties of these domains and compare some other characterizations. We prove that sA-domain is wG-domain, but using the Cantor set, we are able to construct a example which is wG-domain but not sA-domain.

1. INTRODUCTION

We consider the second order elliptic operator in the following non-divergence form

in a given domain Ω in ℝn, where , Dij = DiDj . The operator is called a uniformly elliptic if, for some positive constants λ,Λ,

For the elliptic operator, there is a well known property called maximum principles. For the bounded domain, it can be written as follows:

Theorem 1.1. Let Lu ≥ 0 for some bounded domain Ω, then

For an unbounded domain, one can consider the following simple example:

Here, ∆ is the Laplace operator, is an upper half plane. For the Dirichlet value problem, we have infinitely many solutions of the form u(x) = u(x1, x2, ..., xn) := kxn for any k ∈ ℝ.

Thus, unlike bounded domains, the maximum principle is not easy to obtain for the unbounded domains, hence there are recent publications regarding the subject. For example, one may refer to [1, 2, 3, 5] and references therein.

Definition 1.1 ([1]). We say that the maximum principle holds for the operator L in Ω if

imply w ≤ 0 in Ω.

Using an improved classical Alexandrov-Bakelman-Pucci maximum principle, Cabre [2] obtained the maximum principle above for the following type of domains, which will be denoted by G-domain hereafter.

Definition 1.2 ([2]). We say that Ω satisfies a condition G if there exist positive constants σ < 1, τ < 1 and R0 such that

where BRx is a ball containing x of radius Rx ≤ R0 and Ωx,τ is the component of Ω ∩ BRx/τ to which x belongs.

As noted by Vitolo [5], the G-domain contains connected open sets with finite measure, infinite cylinders, and strips. The explanations are presented in the next section.

By Cafagna and Vitolo [3], G-domain was generalized to wG-domain, and they obtained the maximum principle.

Definition 1.3 ([3]). We say that Ω satisfies a condition wG if there exist positive constants σ < 1 and τ < 1 such that

where BRx is a ball containing x of radius Rx and Ωx,τ is the component of Ω∩BRx/τ to which x belongs.

Observe that in the definition, we do not impose any restriction on the boundedness of radius R unlike G-domain. It is immediate to see that G-domain is wG-domain. In the next section, we present examples of wG-domain, which is not G-domain.

The following A-domain appear in the book by O. A. Ladyzhenskaya and N. N. Uraltseva [4].

Definition 1.4. A domain Ω is called A-domain if there exists a constant σ > 0 and R > 0, such that for each y ∈ ∂Ω and r ∈ (0, R), the Lebesgue measure

where Br(y) is the ball of radius r > 0, centered at y.

Similar to wG-domain, we may also consider the following sA-domain. But in this case, the condition is stronger than A condition unlike G condition.

Definition 1.5. A domain Ω is called sA-domain if there exists a constant σ > 0, such that for each y ∈ ∂Ω and r > 0, the Lebesgue measure

where Br(y) is the ball of radius r > 0, centered at y.

There are examples which are sA-domain, but not A-domain, which is also presented in the next section.

So far, we introduce 4 types of condition, G, wG, A, sA conditions. By its definition, it is rather easy to tell the inclusion of between G and wG, A and sA. In the paper, we show that the sA condition imply the wG condition, but the converse does not hold.

Theorem 1.2. Any sA domain is wG domain, but the converse does not hold.

It is proved in Theorem 2.2 and Theorem 2.3. The main idea for a counter example is to use Cantor set for its construction, such that, we are able to construct locally A-domain, but not sA-domain for big r for any σ.

 

2. MAIN RESULTS

In this section, we prove main results of the paper, and some known and unknown but simple related facts. Firstly, we enlist some known examples of G-domain.

Example 2.1. Any connected open set Ω with finite measure is a G-domain. Namely Ω does satisfy Definition 1.2. Let the Lebesgue measure of Ω, |Ω| = m, choose sufficiently large R such that |BR| ≥ 2m. Then Ω satisfies Definition with , for any τ > 1, R0 = R. Note that for any x ∈ Ω, there exists BRx = BR(x),

Example 2.2. Any infinite cylinder and strips are G-domain. Let Ω = {x ∈ ℝn| |x'| ≤ r, r > 0, x = (x', xn)}. Then for each x ∈ Ω, there exists BRx = B2r(x) such that, for any τ > 1,

for some y ∈ ℝn \ Ω. Domains of strips case is similar.

Example 2.3. A checked domain is also G-domain. Let

Ω1 := {x ∈ ℝ | x ∈ (2i, 2i + 1) for some integer i}, Ω := Ω1 × Ω1.

Note that for any x ∈ Ω, B10(x) contains a unit square in ℝn \ Ω. Similarly, one can prove n-dimensional case.

The G-domain (Definition 1.2) is generalized to wG-domain (Definition 1.3). The following examples show that the converse does not hold.

Example 2.4. Any open connected cone whose closure is different from the whole space is wG-domain, but not G-domain. For example we consider 2-dimensional case. Let Ω := {x ∈ ℝ2| x2 > x1 ∧ x2 > −x1, x = (x1, x2)}. For any x ∈ Ω, choose B2|x|(0). With this ball, it is easy to that it satisfies the definition 1.3. But, note that for any positive y, the point (0, ) in Ω has a distance of y to its boundary. Thus at least By/2 is needed to touch outside of Ω containing the point. This means that one can not impose the boundedness of R in the definition 1.2. Thus in all Ω is not G-domain.

Example 2.5. Let

Ω1 := {x ∈ ℝ | x ∈ (22i, 22i+1) for some natural number i}, Ω := Ω1 × Ω1.

Similar to the previous example, Ω is wG-domain, but not G-domain.

As discussed in the introduction, there are examples which are sA-domain, but not A-domain.

Theorem 2.1. There exist a domain which does satisfies A condition, but not sA condition.

Proof. Consider the following domain in ℝ2

It is immediate to see that Ω is A-domain considering the unit ball centering on its boundary. Considering the unit ball centered at the origin,

Thus,

□

The next theorem implies that the sA condition imply the wG condition.

Theorem 2.2. Any domain Ω which satisfies Definition 1.5 does satisfy Definition 1.3.

Proof. Let Ω be a sA-domain, x be an arbitrary point in Ω, d(x) be a distance of x to ∂Ω, and |x − y| = d(x) for some y ∈ ∂Ω. Choose R = 2d(x), then x ∈ B2d(x)(y) and

by Definition 1.5. Thus in all, Ω is wG domain with the same σ and for any τ < 1. □

For the next, we will present an example which is wG-domain, but not sA-domain. Thus, the converse of the previous theorem does not hold.

First consider the domain in ℝ2 using the Cantor set. The Cantor set C is defined by

For each positive integer m, let

Note that |C| = 0, |D| = 1. Now we define a set operations as follows: for any set S, we set −S = {−x | x ∈ S}, m + S = S + m := {m + x | x ∈ S}.

We define an open set in ℝ2.

Note that Ω1 satisfies Definition 1.3, but is not connected. To connect these components, we add the following sets: for any m ∈ ℕ,

Now we take to be . Then is wG-domain. For , it is not easy to check that it satisfies Definition 1.5 due to the irregularity of the domain. It is difficult to estimate .

We will modify the above idea to obtain the following theorem.

Theorem 2.3. There is wG-domain, which is not sA-domain.

Proof. First recall that the Cantor set C is defined by

For each positive integer m, let

Note that |C| = 0, |D| = 1.

Now we define a set operations as follows: for any set S, we set m + S = S + m := {m + x | x ∈ S}. Also we define

Let

Ω := {x ∈ ℝn| |x| ∈ E}∪{x ∈ ℝn| x = (x1, x'), |x1| ∈ [m−1, m), for some m ∈ ℕ}.

It is easy to see that Ω is connected since x' = 0 is contained in Ω. Observe that

For any x ∈ Ω and |x| ≤ m, then for some y ∈ ∂Ω, and , . This is due to the fact that if |x| ≤ m, then x belongs to a locally connected component of width . Thus in all, Ω is wG-domain.

But 0 ∈ Ω and Br(0) is a disjoint union of Br \ Ω and Br ∩ Ω, we have that

due to (2.2). Thus Ω is not sA-domain. □

Remark 2.6. In the definition of sA-domain, one may replace Ω by Ωx,τ as in the definition of G or wG-domain. But the above example in the proof still works as a counterexample.