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Gradient estimates for the p-laplace equation with a singular source

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Abstract

In this paper we will employ the method of “DeGiorgi–Nash–Moser” to establish a \(L^\infty \) local estimate for the gradient of local weak solutions to p-Laplacian equation

$$\begin{aligned} -\text{ div }\left\{ |\nabla u|^{p-2}\nabla u\right\} =f(x,u,\nabla u)\,. \end{aligned}$$
(1.1)

In this work we extend to \(1<p<2\) the local estimates of the \(\Vert \nabla u\Vert _\infty \) of the local weak solutions of (1.1), obtained by Bhattacharya, DiBenedetto and Manfredi in Limits as \(p\rightarrow \infty \) of \(\Delta _p u_p=f\) and related extremal problems (1989), for \(p>2\).

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Notes

  1. In [3] the symmetric gradient \(D(\vec {u})=\frac{1}{2}\bigl (\nabla \vec {u}+(\nabla \vec {u})^t\bigr )\) appears instead of the gradient \(\nabla u\).

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Funding

This work was supported by public call no 003/2017 - CAPES/FAPEAP.

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All authors contributed to the study conception and design. All authors read and approved the final manuscript.

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Correspondence to Gilberlandio J. Dias.

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Dias, G.J., Duarte, Í.B.M. Gradient estimates for the p-laplace equation with a singular source. J Elliptic Parabol Equ (2024). https://doi.org/10.1007/s41808-024-00284-6

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