A new approach to weighted Sobolev spaces

We present in this paper a new way to define weighted Sobolev spaces when the weight functions are arbitrary small. This new approach can replace the old one consisting in modifying the domain by removing the set of points where at least one of the weight functions is very small. The basic idea is to replace the distributional derivative with a new notion of weak derivative. In this way, non-locally integrable functions can be considered in these spaces. Indeed, assumptions under which a degenerate elliptic partial differential equation has a unique non-locally integrable solution are given. Tools like a Poincaré inequality and a trace operator are developed, and density results of smooth functions are established.