Borderline gradient regularity estimates for quasilinear parabolic systems with data independent of time

Karthik Adimurthi; Wontae Kim

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arXiv · 2307.02420

Borderline gradient regularity estimates for quasilinear parabolic systems with data independent of time

Karthik Adimurthi·Wontae Kim

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Abstract

In this paper, we study some regularity issues concerning the gradient of weak solutions of $u_t - {\rm div} \mathcal{A}(x,t,\nabla u) = g$, where $\mathcal{A}(x,t,\nabla u)$ is modeled after the $p$-Laplace operator. The main results we are interested in is to obtain optimal conditions on the datum $g$ (independent of time) such that borderline higher integrability of the gradient and Lipschitz estimates for the weak solution holds. Moreover, we develop a theory where we can obtain elliptic type estimates using parabolic theory, which gives improved potential estimates for the elliptic systems.

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Karthik Adimurthi, Wontae Kim. 2023-07-05. Borderline gradient regularity estimates for quasilinear parabolic systems with data independent of time. https://arxiv.org/abs/2307.02420

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