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

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 $\alpha$-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 $$-div(\hat{a}(\nabla u))+V(u)=f,$$ 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 $\alpha$-Hölderian under suitable values of $\alpha$ and adequate hypothesis on $V$ and $\hat{a}.$