Quasilinear PDEs, Interpolation Spaces and Holderian mappings

As in the work of Tartar [59], we develop here some new results on nonlinear interpolation of alpha-Holderian mappings between normed spaces, by studying the action of the mappings on K-functionals and between interpolation spaces with logarithm functions. We apply these results to obtain some regularity results on the gradient of the solutions to quasilinear equations of the form - div((a) over cap(del u)) + V (u) = f, where V is a nonlinear potential and f belongs to non-standard spaces like Lorentz-Zygmund spaces. We show several results; for instance, that the mapping T : T f = del u is locally or globally alpha-Holderian under suitable values of a and appropriate hypotheses on V and (a) over cap.