International Journal of Mathematics and Mathematical Sciences
Volume 2011, Issue 1 352590
Research Article
Open Access

Garding′s Inequality for Elliptic Differential Operator with Infinite Number of Variables

Ahmed Zabel, 

Ahmed Zabel

Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt azhar.edu.eg

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Maryam Alghamdi, 

Corresponding Author

Maryam Alghamdi

Department of Mathematics, King Abdulaziz University, P.O. Box 4087, Jeddah 21491, Saudi Arabia kau.edu.sa

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First published: 29 December 2010
https://doi.org/10.1155/2011/352590
Academic Editor: Zayid Abdulhadi

Abstract

We formulate the elliptic differential operator with infinite number of variables and investigate that it is well defined on infinite tensor product of spaces of square integrable functions. Under suitable conditions, we prove Garding′s inequality for this operator.

1. Introduction

In order to solve the Dirichlet problem for a differential operator by using Hilbert space methods (sometimes called the direct methods in the calculus of variations), Garding′s inequality represents an essential tool [1, 2]. For strongly elliptic differential operators, Garding′s inequality was proved by Gärding [3] and its converse by Agmon [4]. One can find a proof for Garding′s inequality and its converse in the work of Stummel [5] for strongly semielliptic operators. Two examples for strongly elliptic and semielliptic operators are studied in [6]. More recent results on this subject can be found in [7, 8] for a class of differential operators containing some non-hypoelliptic operators which were first introduced by Dynkin [9] and for differential operators in generalized divergence form (see also [10, 11]).

The aim of this work is to study the existence of the weak solution of the Dirichlet problem for a second-order elliptic differential operator with infinite number of variables.

2. Some Function Spaces

In this paper, we will consider spaces of functions of infinitely many variables, see [12, 13]. For this purpose we introduce the product measure
(2.1)
defined on the space R∞ = R1 × R1 × ⋯ of points , where is a fixed sequence of weights, such that
(2.2)
For k = 1,2, …, we put
(2.3)
We can write x ∈ R∞, by , where
(2.4)
and .
With respect to dρ we construct on R∞ the Hilbert space of functions of infinitely many variables
(2.5)
which can be understood as the infinite tensor product
(2.6)
with the identity stabilization , , e(k) = 1. To say that the function f ∈ L2(R∞, dρ(x)) is cylindrical, it means that there exist an m = 1,2, …, and an , such that f(x) = fc(x(m)), x ∈ R∞.
On the collection of functions which are l = 1,2, … times continuously differentiable up to the boundary Γ of Rm for sufficiently large m, we introduce the scalar product
(2.7)
where
(2.8)
The differentiation is taken in the sense of generalized functions, and after the completion we obtain the Sobolev spaces , l = 1,2, …   .
Sobolev space of order l on R∞ is defined by
(2.9)
endowed with the scalar product (2.7) forming a dense subspace of L2(R∞, dρ(x)), with
(2.10)
for .
We use the technique of [13] to construct chains of spaces
(2.11)
where are the duals of .

3. Elliptic Differential Operator with Infinite Number of Variables

Consider to be a sequence of nonnegative locally bounded functions in R∞ (i.e., they are bounded on each compact subset) with derivatives (∂/∂xk)ak ∈ Lp,loc  for any p ≥ 1 and k = 1,2, …,    and for a suitable x0 ∈ R∞ it satisfies the following conditions:
  • (1)

    there exists a constant c1 > 0 such that

    (3.1)

  • (2)

    let c1 be the constant in condition (1), and there is n0 belonging to ℕ such that

    (3.2)

Now, we define on L2(R∞, dρ(x)) an elliptic differential operator with infinitely many variables
(3.3)
where
(3.4)

Theorem 3.1. Assume that satisfy the condition that

(3.5)
converges in L2(R∞, dρ(x)). Then the operator L in (3.3) is well defined and admits a closure in L2(R∞, dρ(x)).

Proof. The mapping

(3.6)
is an isometry between the two spaces of square integrable functions. It carries (∂U/∂xk)(xk) into the sandwiched (by means of pk) derivative
(3.7)
and it carries
(3.8)
into the corresponding Dk derivative:
(3.9)
Denote by the linear span of the set of all cylindrical infinitely differentiable finite functions dense in , that is, all the functions of the form
(3.10)
where n depends on u and . Condition (3.5) implies that , (see [13, Lemma (3.2)]). We note that the action of L on the function u(x) = uc(x(n)) has the form
(3.11)
then in view of condition (3.5), the operator is well defined in L2(R∞, dρ(x)) and admits a closure which is again denoted by L.

4. A Garding Inequality

In our consideration, we have an operator of the form
(4.1)
with .

Lemma 4.1. The operator L is Hermitian.

Proof. It is sufficient to verify the Hermitianness on functions of the form u(x) = uc(x(n)), v(x) = vc(x(m)), where; for example, we take it that m ≤ n.

Using (3.11), we obtain

(4.2)
where
(4.3)
Hence, we have
(4.4)

Now, we can define on the bilinear form
(4.5)
where
(4.6)
then
(4.7)

Lemma 4.2. The bilinear form (4.7) is continuous on .

Proof. For ,

(4.8)
Thus B has a continuous extension onto which is again denoted by B.

Theorem 4.3. Suppose that L is given as in (4.1). In particular assume that (3.5) holds. Then there exist positive constants c0 > 0 and c1 ≥ 0 such that

(4.9)
holds for all .

Proof. For ,

(4.10)
and using conditions (1) and (2),
(4.11)
and with c0 = c1(1 − 1/2n0), we finally obtain (4.9).

5. Conclusions

In view of our recent achievement, we recommend to extend this approach to include the linear partial differential operators in generalized divergence form , where Γ is finite, and nonempty collection of α = (α1, …, αn), αi = 1,2, …, and aαβ  (α, β ∈ Γ × Γ) are real locally bounded functions on R∞.

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