Mean Value Theorems for Riemannian Manifolds Via the Obstacle Problem

Abstract

We develop some of the basic theory for the obstacle problem on Riemannian Manifolds, and we use it to establish a mean value theorem. Our mean value theorem works for a very wide class of Riemannian manifolds and has no weights at all within the integral.

Nondegenerate motion of singular points in obstacle problems with varying data

“…The upshot is that all of the information contained in the operator must be contained within the collection of mean value sets and vice versa. Theorem 1.1 was proven by Blank and Hao in detail in [7], and an analogous result was shown in [5] for the Laplace-Beltrami operator on Riemannian manifolds. Since then, the authors have been studying properties of these mean value sets.…”

“…Under the assumptions of Theorem 1.1 along with the assumption that the a ij belong to C 1,1 the family {D r (x 0 )} is always strictly increasing in the sense that r < s implies D r (x 0 ) ⊂ D s (x 0 ). Now within[5] part of the main result states the following:Theorem 1.4 (Mean Value Theorem on Riemannian manifolds). Given a point x 0 in a complete Riemannian manifold M (possibly with boundary), there exists a maximal number r 0 > 0 (which is finite if M is compact) and a family of open sets {D r (x 0 )} for…”

Geometry of Mean Value Sets for General Divergence Form Uniformly Elliptic Operators

“…Very recently, in joint work with Benson and LeCrone, the second author has extended many of the results within this work to Riemannian manifolds [2] and there exist 0 < λ ≤ µ < ∞ such that…”

A Uniqueness Theorem for Mean Value Sets for Elliptic Divergence Form Operators

“…Remark 2.5 (Green's functions are preferable to a fundamental solution). Although Caffarelli and Blank-Hao use the fundamental solution and work on IR n in [6] and [5], especially within [4] it becomes clear that the best way to find the D r (x 0 )'s is by solving the problem given in Equation (2.8) above where a Green's function is used. One might worry about the effect that changing Ω for an operator defined on all of IR n might change the D r (x 0 )'s, but a key point observed within [4] is that it has no effect as long as both domains are "big enough."…”

“…In fact, the theorem cannot be found there (or in other similar texts), so in [5] the second author of this paper and Z. Hao proved the theorem in detail. Of course, even after stating what "comparable to B r (x 0 )" means precisely, it is still clear that the more that is known about the collection of these D r (x 0 )'s, the more useful this theorem becomes, and since making this observation, both authors of the current paper have been studying the properties of these sets [1,2,3,4]. Of course, there is a natural definition to make: Definition 1.1 (Mean value sets).…”