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

TL;DR

This paper extends nonlinear interpolation theory for Hölderian mappings between normed spaces and applies it to establish regularity of solutions' gradients in quasilinear PDEs with nonlinear potentials.

Contribution

It introduces new results on nonlinear interpolation involving K-functionals and applies them to analyze regularity in quasilinear PDEs with nonstandard data.

Findings

Hölderian regularity of the gradient mapping in quasilinear PDEs.

Regularity results for solutions with Lorentz-Zygmund space data.

Conditions under which the gradient mapping is locally or globally Hölderian.

Abstract

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 $\widehat a.$