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

2019/07/29 by Armstrong, Niles, Blank, Ivan
#35J15 #35R35 #58J05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1907.12523

Abstract

We give background which shows the connection between the mean value theorem and the obstacle problem, and then we prove that a set is a mean value set for an elliptic operator of the form Lu := ∂_i (a^ij(x) ∂_j u(x)) if and only if it arises as the noncontact set of an obstacle problem involving the Green's function of the operator.

Related

• The Mean Value Theorem and Basic Properties of the Obstacle Problem for Divergence Form Elliptic Operators
• Geometry of mean value sets for general divergence form uniformly elliptic operators
• Properties of Mean Value Sets: Angle Conditions, Blowup Solutions, and Nonconvexity
• The Caffarelli Alternative in Measure for the Nondivergence Form\n Elliptic Obstacle Problem with Principal Coefficients in VMO
• A brief survey on the obstacle problem