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). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/8198
