As mentioned elsewhere on this site, Sauvigny's book Partial Differential Equations provides a proof of Green's theorem (or the more general Stokes's theorem) for oriented relatively compact open sets in manifolds, as long as the boundary has capacity zero. The precise definition is complicated, so you would need to read the book to get that, but it includes reasonable things like corners and cone points. He also indicates the problems that arise with capacity nonzero, and (if I remember correctly) there are always problems. Of course, there is a problem with making sense of the integration if you allow objects that are too wildly behaved.

As mentioned elsewhere on this site, Sauvigny's book Partial Differential Equations provides a proof of Green's theorem (or the more general Stokes's theorem) for oriented open sets in manifolds, as long as the boundary has capacity zero, and the differential form that you integrate has compact support. (Sauvigny just assumes that the open set is bounded in Euclidean space, but the same proof works without that hypothesis, as long as the differential form is compactly supported, $C^1$ in the interior and $C^0$ up to the boundary). The precise definition of capacity is complicated, so you would need to read the book to get that, but it includes reasonable things like corners and cone points. He also indicates the problems that arise with capacity nonzero, and (if I remember correctly) there are always problems. Of course, there is a problem with making sense of the integration if you allow objects that are too wildly behaved.

As mentioned elsewhere on this site, Sauvigny's book Partial Differential Equations provides a proof of Green's theorem (or the more general Stokes's theorem) for oriented relatively compact open sets in manifolds, as long as the boundary has capacity zero. The precise definition is complicated, so you would need to read the book to get that, but it includes reasonable things like corners and cone points. He also indicates the problems that arise with capacity nonzero, and (if I remember correctly) there are always problems. Of course, there is a problem with making sense of the integration if you allow objects that are too wildly behaved.

As mentioned elsewhere on this site, Sauvigny's book Partial Differential Equations provides a proof of Green's theorem (or the more general Stokes's theorem) for oriented relatively compact open sets in manifolds, as long as the boundary has capacity zero. The precise definition is complicated, so you would need to read the book to get that, but it includes reasonable things like corners and cone points. He also indicates the problems that arise with capacity nonzero, and (if I remember correctly) there are always problems. Of course, there is a problem with making sense of the integration if you allow objects that are too wildly behaved.