2025

Zhou, Xilin

This dissertation explores two interconnected topics, both centered around the Sobolev regularity of solutions to certain mathematical equations. The first topic focuses on establishing criteria for the existence of a homeomorphic Sobolev extension for the boundary parametrization of a Jordan curve. This problem plays a central role in geometric function theory and nonlinear elasticity. In the context of nonlinear elasticity, the existence of a homeomorphic Sobolev extension is essential for modeling elastic deformations, ensuring the physical feasibility of such deformations within the Sobolev framework. We establish sharp criteria in terms of the integrability properties of the hyperbolic metric of the Jordan domain. The second topic examines the regularity properties of solutions to forward-backward stochastic differential equations (FBSDEs), with a particular emphasis on Malliavin Sobolev differentiability. FBSDEs play a crucial role in optimal control theory and mathematical finance, where understanding their regularity properties is key to both theoretical developments and practical applications. In this part, we employ the coupling method introduced by S. Geiss and J. Ylinen, which provides a powerful approach to the study of FBSDEs with random coefficients. By means of this method, we establish a new characterization of the Malliavin Sobolev space D_1,2, and derive regularity results for SDEs, decoupled FBSDEs, and fully coupled FBSDEs.