Niles Armstrong, Ph.D.
Assistant Professor
Education, Licensure and Certification
Ph.D.
Kansas State University
Mathematics
2019
B.S.
Black Hills State University
Mathematics
2010
Biography
Areas of Expertise
Research Interests
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
Selected Publications
Kinodynamic Control Systems and Discontinuities in Clearance
SIAM Journal on Control & OptimizationNiles Armstrong, Jory Denny and Jeremy LeCrone
2024
We investigate the structure of discontinuities in clearance (or minimum time) functions for nonlinear control systems with general, closed obstacles (or targets). We establish general results regarding interactions between admissible trajectories and clearance discontinuities, e.g., instantaneous increases in clearance when passing through a discontinuity, and propagation of discontinuity along optimal trajectories. Then, investigating sufficient conditions for discontinuities, we explore a common directionality condition for velocities at a point, characterized by strict positivity of the minimal Hamiltonian. Elementary consequences of this common directionality assumption are explored before demonstrating how, in concert with corresponding obstacle configurations, it gives rise to clearance discontinuities both on the surface of the obstacle and propagating out into free space. Minimal assumptions are made on the topological structure of obstacle sets.
Nondegenerate motion of singular points in obstacle problems with varying data
Journal of Differential EquationsArmstrong, 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.
A Uniqueness Theorem for Mean Value Sets for Elliptic Divergence Form Operators
arXiv preprint arXiv:1907.12523Armstrong, 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.
Nonconvexity and compact containment of mean value sets for second order uniformly elliptic operators in divergence form
K-State Electronic Theses, Dissertations, and ReportsArmstrong, 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.
Angle Conditions, Blowup Solutions, and Nonconvexity.
Potential AnalysisArmstrong, N.
2018
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
