Geodesic
Nonlinear potentials, local solutions to elliptic equations and rearrangements
Cianchi, Andrea1
1Dipartimento di Matematica “U. Dini” Università di Firenze Piazza Ghiberti, 27 50122 Firenze, Italia
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 2, pp. 335-361

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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.

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Classification : 31C15, 35B45

Cianchi, Andrea  1

1 Dipartimento di Matematica “U. Dini” Università di Firenze Piazza Ghiberti, 27 50122 Firenze, Italia 
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://geodesic.mathdoc.fr/item/ASNSP_2011_5_10_2_335_0/
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     title = {Nonlinear potentials, local solutions to elliptic equations and rearrangements},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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     url = {http://geodesic.mathdoc.fr/item/ASNSP_2011_5_10_2_335_0/}
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