Regularity for $p$-Laplace Type Equations with Applications in Composite MaterialsdissertationsZhu, HanyecreatorDong, HongjieAdvisorGuo, YanReaderPausader, BenoitReaderBrown University. Department of Applied Mathematicssponsor2024ix, 266 p.born digitalThesis (Ph. D.)--Brown University, 2024thesesIn this dissertation, we study regularity theory for equations of $p$-Laplace type. In the first part of the dissertation, we address elliptic and parabolic $p$-Laplace type equations with measure data. We establish pointwise gradient estimates via linear Riesz potentials and optimal gradient continuity criteria using the Riesz potentials. In the second part of the dissertation, we investigate the insulated and perfect conductivity problems with closely spaced insulators (or perfect conductors) embedded in a homogeneous matrix where the current-electric field relation is the power law $J=\sigma |E|^{p-2} E$. We quantitatively characterize the blow-up phenomena of the electric field as the distance between the inclusions approaches $0$.Applied mathematicsEnglishRPB20240505