Nonlinear potentials, local solutions to elliptic equations, and rearrangements

A sharp rearrangement estimate for the nonlinear Havin-Maz’ya potentials is established. In particular, this estimate leads to a characterization of those rearrangement invariant spaces between which the nonlinear potentials are bounded. In combination with results by Kilpelainen and Maly, and by Duzaar and Mingione, it also enables us to derive local bounds for solutions to quasilinear elliptic PDE’s and for their gradient in rearrangement form. As a consequence, the local regularity of solutions to elliptic equations and for their gradient in arbitrary rearrangement invariant spaces is reduced to one-dimensional Hardy-type inequalities. Applications to the special cases of Lorentz and Orlicz spaces are presented. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM