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On The P(X)-Laplace Equation In Carnot Groups, Robert D. Freeman
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 …
Generalizations Of A Laplacian-Type Equation In The Heisenberg Group And A Class Of Grushin-Type Spaces, Kristen Snyder Childers
Generalizations Of A Laplacian-Type Equation In The Heisenberg Group And A Class Of Grushin-Type Spaces, Kristen Snyder Childers
USF Tampa Graduate Theses and Dissertations
In [2], Beals, Gaveau and Greiner find the fundamental solution to a 2-Laplace-type equation in a class of sub-Riemannian spaces. This fundamental solution is based on the well-known fundamental solution to the p-Laplace equation in Grushin-type spaces [4] and the Heisenberg group [6]. In this thesis, we look to generalize the work in [2] for a p-Laplace-type equation. After discovering that the "natural" generalization fails, we find two generalizations whose solutions are based on the fundamental solution to the p-Laplace equation.