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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 352590, 10 pages
http://dx.doi.org/10.1155/2011/352590
Research Article

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

1Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt
2Department of Mathematics, King Abdulaziz University, P.O. Box 4087, Jeddah 21491, Saudi Arabia

Received 21 July 2010; Accepted 17 November 2010

Academic Editor: Zayid Abdulhadi

Copyright © 2011 Ahmed Zabel and Maryam Alghamdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 defined on the space of points , where is a fixed sequence of weights, such that For k , we put We can write , by , where and .

With respect to we construct on the Hilbert space of functions of infinitely many variables which can be understood as the infinite tensor product with the identity stabilization , , . To say that the function is cylindrical, it means that there exist an , and an , such that , .

On the collection of functions which are times continuously differentiable up to the boundary Γ of for sufficiently large , we introduce the scalar product where The differentiation is taken in the sense of generalized functions, and after the completion we obtain the Sobolev spaces , .

Sobolev space of order on is defined by

endowed with the scalar product (2.7) forming a dense subspace of , with for .

We use the technique of [13] to construct chains of spaces 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 (i.e., they are bounded on each compact subset) with derivatives for any and and for a suitable it satisfies the following conditions:(1)there exists a constant such that (2)let be the constant in condition (1), and there is belonging to such that

Now, we define on an elliptic differential operator with infinitely many variables where

Theorem 3.1. Assume that satisfy the condition that converges in . Then the operator in (3.3) is well defined and admits a closure in .

Proof. The mapping is an isometry between the two spaces of square integrable functions. It carries into the sandwiched (by means of ) derivative and it carries into the corresponding derivative: 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 where depends on and . Condition (3.5) implies that , (see [13, Lemma (3.2)]). We note that the action of on the function has the form then in view of condition (3.5), the operator is well defined in and admits a closure which is again denoted by .

4. A Garding Inequality

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

Lemma 4.1. The operator is Hermitian.

Proof. It is sufficient to verify the Hermitianness on functions of the form , where; for example, we take it that .
Using (3.11), we obtain where Hence, we have

Now, we can define on the bilinear form where then

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

Proof. For , Thus has a continuous extension onto which is again denoted by .

Theorem 4.3. Suppose that is given as in (4.1). In particular assume that (3.5) holds. Then there exist positive constants and such that holds for all .

Proof. For , and using conditions (1) and (2), and with , 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 is finite, and nonempty collection of , , and are real locally bounded functions on .

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