Degenerate Sobolev spaces and regularity of subelliptic equations

Sawyer, Eric; Wheeden, Richard

doi:10.1090/S0002-9947-09-04756-4

Degenerate Sobolev spaces and regularity of subelliptic equations
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by Eric T. Sawyer and Richard L. Wheeden PDF

Trans. Amer. Math. Soc. 362 (2010), 1869-1906 Request permission

Abstract:

We develop a notion of degenerate Sobolev spaces naturally associated with nonnegative quadratic forms that arise from a large class of linear subelliptic equations with rough coefficients. These Sobolev spaces allow us to make the widest possible definition of a weak solution that leads to local Hölder continuity of solutions, extending our results in an earlier work, where we studied regularity of classical weak solutions. In cases when the quadratic forms arise from collections of rough vector fields, we study containment relations between the degenerate Sobolev spaces and the corresponding spaces defined in terms of weak derivatives relative to the vector fields.

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