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 from [24] and [18], 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.
@article{ASNSP_2011_5_10_2_335_0, author = {Cianchi, Andrea}, title = {Nonlinear potentials, local solutions to elliptic equations and rearrangements}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {335--361}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 10}, number = {2}, year = {2011}, mrnumber = {2856151}, zbl = {1235.31009}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2011_5_10_2_335_0/} }
TY - JOUR AU - Cianchi, Andrea TI - Nonlinear potentials, local solutions to elliptic equations and rearrangements JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2011 SP - 335 EP - 361 VL - 10 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2011_5_10_2_335_0/ LA - en ID - ASNSP_2011_5_10_2_335_0 ER -
%0 Journal Article %A Cianchi, Andrea %T Nonlinear potentials, local solutions to elliptic equations and rearrangements %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2011 %P 335-361 %V 10 %N 2 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2011_5_10_2_335_0/ %G en %F ASNSP_2011_5_10_2_335_0
Cianchi, Andrea. Nonlinear potentials, local solutions to elliptic equations and rearrangements. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 2, pp. 335-361. http://www.numdam.org/item/ASNSP_2011_5_10_2_335_0/
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