2017/04/25 by Aryal, Ashok, Blank, Ivan · 1 citation
#35J15 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1704.07929
In the Fermi Lectures on the obstacle problem in 1998, Caffarelli gave a proof of the mean value theorem which extends to general divergence form uniformly elliptic operators. In the general setting, the result shows that for any such operator L and at any point x0 in the domain, there exists a nested family of sets \ Dr(x0) \ where the average over any of those sets is related to the value of the function at x0. Although it is known that the \ Dr(x0) \ are nested and are comparable to balls in the sense that there exists c, C depending only on L such that Bcr(x0) ⊂ Dr(x0) ⊂ BCr(x0) for all r > 0 and x0 in the domain, otherwise their geometric and topological properties are largely unknown. In this paper we begin the study of these topics and we prove a few results about the geometry of these sets and give a couple of applications of the theorems.