#### Graduation Year

2020

#### Document Type

Dissertation

#### Degree

Ph.D.

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Degree Granting Department

Mathematics and Statistics

#### Major Professor

Thomas Bieske, Ph.D.

#### Committee Member

Bradley Kamp, Ph.D.

#### Committee Member

Yuncheng You, Ph.D.

#### Committee Member

Razvan Teodorescu, Ph.D.

#### Committee Member

Sherwin Kouchekian, Ph.D

#### Keywords

Non-linear potential theory, p(x)-Laplace equation, Removability, Sub-Riemannian Geometry, Viscosity solutions

#### Abstract

In this thesis, we examine the p(x)-Laplace equation in the context of Carnot groups. The p(x)-Laplace equation is the prototype equation for a class of nonlinear elliptic partial differential equations having so-called nonstandard growth conditions. An important and useful tool in studying these types of equations is viscosity theory. We prove a p()-Poincar´e-type inequality and use it to prove the equivalence of potential theoretic weak solutions and viscosity solutions to the p(x)-Laplace equation. We exploit this equivalence to prove a Rad´o-type removability result for solutions to the p-Laplace equation in the Heisenberg group. Then we extend this result to the p(x)-Laplace equation in the Heisenberg group.

#### Scholar Commons Citation

Freeman, Robert D., "On the p(x)-Laplace equation in Carnot groups" (2020). *Graduate Theses and Dissertations.*

https://scholarcommons.usf.edu/etd/8198