Abstract
In this paper we study second order partial differential operators in the uniform norm. We solve the problem of constructing a closed extension of the operator and of characterizing its essential spectrum. This program has been extensively developed in the £p, case.But the existing methods are not strong nough to dealwith th £∞ case. The uniform norm presents the interesting new feature that the boundary condition that the functions in the definition space go to zero at infinity (in £ sense) is not assumed, allowing for example to have functions in the domain of the operator. Since most of the techniques used in £p, are not applicable in £∞we create a new theory.
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