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

Ahmed Zabel ,Maryam A Alghamdi

doi:10.1155/2011/352590

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.

Highlights

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
We prove Garding’s inequality for this operator
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 10, 11

Summary

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 Garding 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. 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

Elliptic Differential Operator with Infinite Number of Variables

A Garding Inequality

Conclusions