Date of Graduation


Document Type


Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level



Mathematical Sciences


Ariel Barton

Committee Member

Andrew Raich

Second Committee Member

Zachary Bradshaw


Differential Equations


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.