Integrable boundaries and fractals for Hölder classes; the Gauss–Green theorem

Abstract

We define a surface integral over the boundary of open bounded sets in \( R^d,\ d\ge {}2\), and provide a criterion that completely describes all the boundaries where this integral exists for the \( (d-1\))-differential forms from Holder classes. The surface integral and the Gauss–Green formula are used in many areas of mathematics and physics, and we give the proof for the Gauss–Green formula under the same condition as the existence of the newly defined integral. This integration process substantially extends the class of boundaries where surface integrals were previously defined for Holder differential forms and includes highly irregular boundaries like non-rectifiable Jordan curves, fractals, sets of finite perimeter boundaries and flat chains. In all known generalizations, to prove the existence of the surface or contour integral for a differential form defined on the boundary, the differential of the form or the differential of its Whitney extension has to be summable in the region. As we show, this significant restriction is superfluous in the new integral definition.