The Equivalence of Weak and Viscosity Solutions to the p(x)-Laplace Equation in Carnot Groups

Carnot groups are a generalization of Euclidean R^n in that we maintain an algebraic group law but restrict the tangent space to a dimension strictly less than n. Carnot groups are used to model activities such as how the human brain processes visual images and the procedure to parallel park a car. In this environment, we discuss the key properties of weak and viscosity solutions to the so-called p(x)-Laplace equation, a key equation in nonlinear potential theory. The p(x)-Laplace equation is also used to model such activities as fluid flow with eddies. We then show that under a natural assumption, viscosity and weak solutions coincide.