Subelliptic Jets and Applications

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

A major part of the success in the linear and quasilinear theory of partial differential equations is based upon interpreting derivatives in the generalized sense of distributions, allowing for a more powerful calculus. However, Distributions do not seem, in general, well suited to non-linear problems because they cannot be multiplied. "Jets" are generalized (local) pointwise derivatives that allow for the interpretation and calculation of non-linear functions of derivatives. In this proposal, the PI presents a project to extend the notion of jets from the Euclidean or Riemannian case to more general state spaces. Typically, in these spaces higher derivatives with respect different parameters do not commute, as in the Euclidean case, but rather satisfy more complicated algebraic relations. Jets adapted to the geometry of a state space endowed with a family of vector fields satisfying a non-degeneracy condition are called "subelliptic jets". Basic analysis topics like Taylor developments and maximum principles have to be adapted to conform to the new subelliptic geometry. The PI proposes to study Hamilton-Jacobi equations, to provide with a natural approximation procedure for semi-continuous functions, and to explore notions of "subelliptic convexity" to help formulate appropriate second derivative subelliptic estimates. The derivative is a basic tool in mathematical analysis, used to measure the growth and decay of functions. Knowledge of the derivative of a function allows for its recovery by means of integration. When trying to model complex scientific phenomena it is often necessary to write down equations satisfied by derivatives, and derivatives of derivatives, of functions with respect to several parameters. These equations are called partial differential equations. If these equations are linear, a well developed theory exists to study their solutions. Much less in known in the more interesting case of nonlinear equations, although when the state space is Euclidean, there is vast fully-nonlinear theory developed in the last twenty years. However, in applications coming from Control Theory, Robotics and Finance, very often we find derivatives which do not commute. The analysis of nonlinear operator of these derivatives is the main object of this project. The significance of this proposal relies in the interconnection between areas of classical mathematical analysis and applied mathematics as well as the use of computational tools not available until now in mathematical analysis.