Quasilinear P.D.Es, Interpolation spaces and H\"olderian mappings

• Main Authors: Ahmed, Irshaad, Fiorenza, Alberto, Formica, Maria Rosaria, Gogatishvili, Amiran, Hamidi, Abdallah El, Rakotoson, Jean Michel
• Format: Journal Article
• Language: English
• Published: 02.11.2022
• Subjects: Mathematics - Analysis of PDEs; Mathematics - Functional Analysis
• DOI: 10.48550/arxiv.2211.01574

As in the work of Tartar ( Tartar L. Interpolation non lin\'eaire et r\'egularit\'e, 9, Journal of Functional Analysis, (1972), 469-489) we developed here some new results on non linear interpolation of $\alpha$-H\"olderian 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 $\alpha$-H\"olderian under suitable values of $\alpha$ and adequate hypothesis on $V$ and $\widehat a.$