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arXiv · 2211.01574

Quasilinear P.D.Es, Interpolation spaces and Hölderian mappings

Irshaad Ahmed·Alberto Fiorenza·Maria Rosaria Formica·Amiran Gogatishvili·Abdallah El Hamidi·Jean Michel Rakotoson
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Abstract

As in the work of Tartar ( Tartar L. Interpolation non linéaire et régularité, 9, Journal of Functional Analysis, (1972), 469-489) we developed here some new results on non linear interpolation of $α$-Hölderian mappings between normed spaces, namely, by studying the action of the mappings on $K$-functionals and between interpolation spaces with logarithm functors. We apply those results to obtain regularity results on the gradient of the solution to quasilinear equations of the form $$-div(\widehat a(\nabla u ))+V(u)=f, $$ whenever $V$ is a nonlinear potential, $f$ belongs to non standard spaces as Lorentz-Zygmund spaces. We show among other that the mapping $T: \ Tf=\nabla u$ is locally or globally $α$-Hölderian under suitable values of $α$ and adequate hypothesis on $V$ and $\widehat a.$

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Irshaad Ahmed, Alberto Fiorenza, Maria Rosaria Formica, Amiran Gogatishvili, Abdallah El Hamidi, Jean Michel Rakotoson. 2023-10-29. Quasilinear P.D.Es, Interpolation spaces and Hölderian mappings. https://arxiv.org/abs/2211.01574

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