Nonlinear Subelliptic Analysis

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ABSTRACT A major part of the success in the linear and quasi-linear theory comes from interpreting derivatives in the generalized sense of distributions, allowing for a more powerful calculus. Distributions do not seem, in general well suited to non-linear problems because they cannot be multiplied. Jets are generalized (local) point-wise derivatives that allow for the interpretation and calculation of non-linear functions of derivatives. In this proposal, the PI presents a project touse jets to develop some basic Analysis tools in 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 sub-elliptic jets. Basic analysis topics like Taylor developments and maximum principles have to be adapted to conform to the new sub-elliptic geometry. Topics studied include: sub-elliptic extensions of the uniqueness theorem of R. Jensen for viscosity solutions, regularity for the sub-elliptic p-Laplacian, sub-elliptic convex functions, and Cordes sub-elliptic 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. In this proposal the PI proposes to develop tools to study partial differential equations written in terms of vector fields. These equations have applications to problems in Robotics, Control Theory and Mathematical Finance.