Niles Armstrong, Ph.D.

Mean Value Sets for Elliptic Divergence Form Operators

A mean value theorem for elliptic divergence form operators is a generalization of the famous mean value theorem for harmonic functions. While this generalization applies to a much larger class of functions, the integral that appears in the generalized theorem is taken over a set that come from the noncontact set of an obstacle problem. We call these set the mean value sets. In order to make the most use of the generalized mean value theorem properties of these mean value sets are needed. So far little is known about these set in general. In fact, most of which is currently known about the mean value sets can be found in my recent publications and those reference within by Ivan Blank et al.

A mean value theorem for elliptic divergence form operators

The classical obstacle problem is to find the equilibrium position of an elastic membrane with fixed boundary, and which is constrained to lie above a given obstacle. Such a problem can be rephrased as a minimization problem of the Dirichlet energy functional over a set of functions with given boundary condition and obstacle condition. The Dirichlet energy functional can be generalized to the associated energy functionals of other divergence form uniformly elliptic operators, extending the theory of the obstacle problem

Armstrong, N., Blank, I.

2019

Recent work by Serfaty and Serra give a formula for the velocity of the free boundary of the obstacle problem at regular points ([19]), and much older work by King, Lacey, and Vázquez gives an example of a singular free boundary point (in the Hele-Shaw flow) that remains stationary for a positive amount of time ([13]). The authors show how singular free boundaries in the obstacle problem in some settings move immediately in response to varying data. Three applications of this result are given, and in particular, the authors show a uniqueness result: For sufficiently smooth elliptic divergence form operators on domains in and for the Laplace-Beltrami operator on a smooth manifold, the boundaries of distinct mean value sets (of the type found in [7] and [5]) which are centered at the same point do not intersect.

Armstrong, N., Blank, I.

2019

We study the mean value sets of a particular second order divergence form elliptic operator whose principal coefficients are discontinuous. Such an operator is of special interest in the study of composite materials. We show that the mean value sets associated to such an operator need not be convex. This example then leads to the construction of an operator with smooth coefficients whose mean value sets remain nonconvex.

Armstrong, N.

2019

The mean value theorem for harmonic functions has historically been an important and powerful result. As such, a generalization of this theorem that was stated by Caffarelli in 1998 and later proved by Blank-Hao in 2015 is of immediate interest. However, in order to make more use of this new general mean value theorem, more information about the mean value sets that appear in the theorem is needed. We present here a few new results regarding properties of such mean value sets.

2018-10-01

Armstrong, N.

We study the mean values sets of the second order divergence form elliptic operator with principal coefficients defined as
aijk(x):={αkδij(x)βkδij(x)xn>0xn