Date of Graduation
1-2021
Document Type
Dissertation
Degree Name
Doctor of Philosophy in Mathematics (PhD)
Degree Level
Graduate
Department
Mathematical Sciences
Advisor/Mentor
Ariel Barton
Committee Member
Andrew Raich
Second Committee Member
Zachary Bradshaw
Keywords
Differential Equations
Abstract
Here we generalize the higher-order divergence-form elliptic differential equations studied by Barton in [4] by the inclusion of certain lower-order terms. The methods used here compare to those used in [4], with the addition of further Sobolev-type estimates to handle included lower-order terms. In section 3 we derive a Caccioppoli inequality in which we bound the L2 norm of the mth order gradient, in terms of the L2 norm of the solution. In section 5 we adapt some of the ideas from [9] to derive Lp bounds on gradients of solutions as a substitute for a reverse Holder inequality. Finally in section 4 we study the fundamental solution of the operator L. We prove existence and bounds first in the case that L is of sufficiently high order (2m > d), then in section 6.2 we extend these results to operators of lower order where 2m ≤ d.
Citation
Duffy, M. J. (2021). Gradient Estimates And The Fundamental Solution For Higher-Order Elliptic Systems With Lower-Order Terms. Graduate Theses and Dissertations Retrieved from https://scholarworks.uark.edu/etd/4173