Physical Sciences and Mathematics Commons^™

On The P(X)-Laplace Equation In Carnot Groups, Robert D. Freeman

USF Tampa Graduate Theses and Dissertations

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 …