Remarks on regularity for $p$-Laplacian type equations in non-divergence form

Amal Attouchi; Eero Ruosteenoja

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arXiv · 1710.07506

Remarks on regularity for $p$-Laplacian type equations in non-divergence form

Amal Attouchi·Eero Ruosteenoja

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Abstract

We study a singular or degenerate equation in non-divergence form modeled by the $p$-Laplacian, $$-|Du|^\gamma\left(\Delta u+(p-2)\Delta_\infty^N u\right)=f\ \ \ \ \text{in}\ \ \ \Omega.$$ We investigate local $C^{1,\alpha}$ regularity of viscosity solutions in the full range $\gamma>-1$ and $p>1$, and provide local $W^{2,2}$ estimates in the restricted cases where $p$ is close to 2 and $\gamma$ is close to 0.

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Amal Attouchi, Eero Ruosteenoja. 2017-10-20. Remarks on regularity for $p$-Laplacian type equations in non-divergence form. https://arxiv.org/abs/1710.07506

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