Consumer Optimization and a First-Order PDE with a Non-Smooth System

We study a first-order nonlinear partial differential equation and present a necessary and sufficient condition for the global existence of its solution in a non-smooth environment. Using this result, we prove a local existence theorem for a solution to this differential equation. Moreover, we present two applications of this result. The first concerns an inverse problem called the integrability problem in microeconomic theory and the second concerns an extension of Frobenius’ theorem.

Integrable solution for light shaping based on a Fourier-pair mapping

In far-field light shaping, one of the design methods is based on a one-to-one map between the irradiance of the source and target. However, an integrability issue may occur in this kind of algorithms, either in the ray mapping method for designing a freeform surface or in those geometric-optics-based methods for achieving a required output phase. We introduce another mapping-type algorithm to tackle the integrability problem, which instead of establishing a mapping between both the source and target irradiance in the space domain, the mapping is assumed on electric fields of a Fourier pair between the space domain and the spatial-frequency domain. By solving the mapping from the Fourier pair, the gradient of the output phase is achieved, that the gradient is equivalent to the obtained mapping function. Moreover, the existence and the characterization of the mapping guarantees the integrability of the gradient so that a smooth output phase can be directly integrated. Based on the obtained smooth output phase, a freeform surface can then be designed for the light-shaping task. Numerical examples are demonstrated for the comparison of the approaches with different mapping assumptions.

On the integrability problem for systems of partial differential equations in one unknown function, I

We discuss the integration problem for systems of partial differential equations in one unknown function and special attention is given to the first order systems. The Grassmannian contact structures are the basic setting for our discussion and the major part of our considerations inquires on the nature of the Cauchy characteristics in view of obtaining the necessary criteria that assure the existence of solutions. In all the practical applications of partial differential equations, what is mostly needed and what in fact is hardest to obtains are the solutions of the system or, occasionally, some specific solutions. This work is based on four most enlightening Mémoires written by Élie Cartan in the beginning of the last century.

On the integrability problem for systems of partial differential equations in one unknown function, II

We continue here our discussion of Part I, [18], by examining the local equivalence problem for partial differential equations and illustrating it with some examples, since almost any integration process or method is actually a local equivalence problem involving a suitable model. We terminate the discussion by inquiring on non-integrable Pfaffian systems and on their integral manifolds of maximal dimension.

Integrability of exponential process and its application to backward stochastic differential equations

We consider the integrability problem of an exponential process with unbounded coefficients. The integrability is established under weaker conditions of Kazamaki type, which complements the results of Yong obtained under a Novikov type condition. As applications, we consider the solvability of linear backward stochastic differential equations (BSDEs) and market completeness, the solvability of a Riccati BSDE and optimal investment, all in the setting of unbounded coefficients.

On the Formal Integrability Problem for Planar Differential Systems

We study the analytic integrability problem through the formal integrability problem and we show its connection, in some cases, with the existence of invariant analytic (sometimes algebraic) curves. From the results obtained, we consider some families of analytic differential systems inℂ2, and imposing the formal integrability we find resonant centers obviating the computation of some necessary conditions.