Quasilinear PDEs, Interpolation Spaces and Hölderian mappings

in the work of Tartar [ 59 ], we develop here some new results on nonlinear interpolation of α -Hölderian 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(∇ 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:Tf=∇ u is locally or globally α -Hölderian under suitable values of α and appropriate hypotheses on V and â .