Executive Summary : | Partial differential equations (PDEs) are crucial in modeling the spatial and temporal evolution of physical systems, including advection, diffusion, thermodynamics, and electrodynamics. They also play a significant role in research areas such as artificial intelligence, stochastic processes, finance, machine learning, representation theory, and geometric measure theory. The study of elliptic and parabolic equations is significant due to their applications in real-world processes like electrorheological fluid flows, image processing, optimal investment strategies, artificial neural network architectures, and heterogeneous biological interactions. This project proposes studying the existence, multiplicity, and regularity properties of solutions of mixed local-nonlocal double-phase elliptic and parabolic equations. The study aims to develop a new framework of energy/function spaces and study their continuous and compact embeddings. The lack of compact embeddings of energy spaces corresponding to the critical growth of source nonlinearities makes PDEs interesting and challenging.
The project also focuses on the regularity properties of the weak solution of mixed local-nonlocal double-phase problems, examining how the vanishing and nonvanishing regions of modulating coefficients affect the Holder regularity of the solution and its gradient. These results have numerous mathematical applications, such as the proof of the Hopf-type maximum principle and the strong comparison principle. |