Graduation Year


Document Type




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


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


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.

Included in

Mathematics Commons