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.

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