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

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.


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.

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 1 is a fixed sequence of weights, such that For k 1, 2, . . ., we put and dρ x dρ x k × dρ x . With respect to dρ we construct on R ∞ the Hilbert space of functions of infinitely many variables which can be understood as the infinite tensor product with the identity stabilization e e k ∞ 1 , e k ∈ L 2 R 1 , dρ k x k , e k 1. To say that the function f ∈ L 2 R ∞ , dρ x is cylindrical, it means that there exist an m 1, 2, . . ., and an On the collection of functions which are l 1, 2, . . . times continuously differentiable up to the boundary Γ of R m for sufficiently large m, we introduce the scalar product International Journal of Mathematics and Mathematical Sciences The differentiation is taken in the sense of generalized functions, and after the completion we obtain the Sobolev spaces W l 2 R ∞ , l 1, 2, . . . . Sobolev space of order l on R ∞ is defined by We use the technique of 13 to construct chains of spaces

Elliptic Differential Operator with Infinite Number of Variables
Consider a k ∞ k 1 to be a sequence of nonnegative locally bounded functions in R ∞ i.e., they are bounded on each compact subset with derivatives ∂/∂x k a k ∈ L p,loc for any p ≥ 1 and k 1, 2, . . . , and for a suitable x 0 ∈ R ∞ it satisfies the following conditions: 2 let c 1 be the constant in condition 1 , and there is n 0 belonging to N such that International Journal of Mathematics and Mathematical Sciences Now, we define on L 2 R ∞ , dρ x an elliptic differential operator with infinitely many variables Then the operator L 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 ∂U/∂x k x k into the sandwiched by means of p k derivative Denote by C ∞ c,0 R ∞ the linear span of the set of all cylindrical infinitely differentiable finite functions dense in W l 2 R ∞ , that is, all the functions u ∈ W l 2 R ∞ of the form where n depends on u and u c ∈ C ∞ 0 R n , n 1, 2, . . . . Condition 3.5 implies that D k 1, D 2 k 1 ∈ L 2 R 1 , dρ k x k , see 13, Lemma 3.2 . We note that the action of L on the function u x u c x n has the form then in view of condition 3.5 , the operator C ∞ c,0 R ∞ u x −→ Lu x − ∞ k 1 D k a k D k u x ∈ L 2 R ∞ , dρ x is well defined in L 2 R ∞ , dρ x and admits a closure which is again denoted by L.

A Garding Inequality
In our consideration, we have an operator of the form Proof. It is sufficient to verify the Hermitianness on functions of the form u x for example, we take it that m ≤ n.
Using 3.11 , we obtain × v c x p k x k dx k p 1 x 1 · · · p k−1 x k−1 × p k 1 x k 1 · · · p n x n dx 1 · · · dx k−1 dx k 1 · · · dx n International Journal of Mathematics and Mathematical Sciences

4.3
Hence, we have

4.4
Now, we can define on W 1 2 R ∞ the bilinear form

4.8
Thus B has a continuous extension onto W 1 2 R ∞ 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 c 0 > 0 and c 1 ≥ 0 such that holds for all u ∈ W 1 2 R ∞ .

International Journal of Mathematics and Mathematical Sciences 9
Proof. For u ∈ C ∞ c,0 R ∞ , 4.10 and using conditions 1 and 2 , and with c 0 c 1 1 − 1/2n 0 , we finally obtain 4.9 .