On the local Hölder continuity of the inverse of the 𝑝-Laplace operator

Lê, An

doi:10.1090/S0002-9939-07-08913-7

On the local Hölder continuity of the inverse of the $p$-Laplace operator
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Proc. Amer. Math. Soc. 135 (2007), 3553-3560 Request permission

Abstract:

We prove an interpolation type inequality between $C^\alpha$, $L^\infty$ and $L^p$ spaces and use it to establish the local Hölder continuity of the inverse of the $p$-Laplace operator: $\|(-\Delta _p)^{-1}(f) - (-\Delta _p)^{-1}(g)\|_{C^{1}(\bar {\Omega })} \leq C \| f - g \|^r_{L^\infty (\Omega )}$, for any $f$ and $g$ in a bounded set in $L^\infty (\Omega )$.

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