Sobolev Spaces on the Geometric Measure Theoretic Boundary of Sets of Locally Finite Perimeter

Abstract

AbstractWhile there is an ample literature on Sobolev spaces on open sets (see, e.g., [4], [3], [49], [125], [130], [166], [183] and the references cited therein), here the goal is to introduce a scale of Sobolev spaces on the geometric measure theoretic boundaries of sets of locally finite perimeter in the Euclidean setting and on Riemannian manifolds. This builds and expands on the work in [97], [139], and [141]. Our brand of “boundary” Sobolev spaces are analytic and geometric in nature, in the sense that they are defined using “weak derivatives” (and integration by parts along the boundary) which, in turn, are manufactured using the scalar components of the geometric measure theoretic outward unit normal to the set of locally finite perimeter in question. Corresponding to a domain with flat boundary (i.e., a half-space) our definition reduces precisely to the classical definition of Sobolev spaces (of order one) in the Euclidean setting, based on ordinary weak derivatives. Such a compatibility reinforces the idea that this is indeed a natural generalization of the standard scale of Sobolev spaces from the (entire) Euclidean ambient to sets exhibiting a much more intricate geometry (both in the Euclidean setting and that of manifolds). In contrast to other types of Sobolev spaces on generic measure metric spaces which have been introduced and studied elsewhere in the literature (see, e.g., [79], [81] and the references there) our brand of Sobolev spaces allows for integration by parts (on the boundary). Such a feature is critical in light of the applications to singular integral operators and boundary value problems we have in mind.