Nonlinear potentials, local solutions to elliptic equations and rearrangements
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, p. 335-361

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.

Published online : 2018-08-07
Classification:  31C15,  35B45
@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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 10},
     number = {2},
     year = {2011},
     pages = {335-361},
     zbl = {1235.31009},
     mrnumber = {2856151},
     language = {en},
     url = {http://www.numdam.org/item/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, Serie 5, Volume 10 (2011) no. 2, pp. 335-361. http://www.numdam.org/item/ASNSP_2011_5_10_2_335_0/

[1] D. R.Adams and L. I. Hedberg, “Function Spaces and Potential Theory", Springer, Berlin, 1996. | MR 1411441 | Zbl 0834.46021

[2] D. R. Adams, Traces of potentials II, Indiana Univ. Math. J. 22 (1973), 907–918. | MR 313783 | Zbl 0265.46039

[3] D. R. Adams and N. G. Meyers, Thinnes and Wiener criteria for non-linear potentials, Indiana Univ. Math. J. 22 (1972), 132–158. | MR 316724 | Zbl 0244.31012

[4] A. Alberico and A. Cianchi, Optimal summability of solutions to nonlinear elliptic problems, Nonlinear Anal. 67 (2007), 1775–1790. | MR 2326030

[5] A. Alvino, V. Ferone and G. Trombetti, Estimates for the gradient of solutions of nonlinear elliptic equations with L 1 data, Ann. Mat. Pura. Appl. 178 (2000), 129–142. | MR 1849383 | Zbl 1220.35079

[6] A. Alvino, A. Cianchi, V. G. Maz’ya and A. Mercaldo, Well-posed elliptic Neumann problems involving irregular data and domains, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 1017–1054. | Numdam | MR 2659156 | Zbl 1200.35105

[7] C. Bennett and R. Sharpley, “Interpolation of Operators", Academic Press, Boston, 1988. | MR 928802 | Zbl 0647.46057

[8] M. F. Betta, V. Ferone and A. Mercaldo, Regularity for solutions of nonlinear elliptic equations, Bull. Sci. Math. 118 (1994), 539–567. | MR 1309088 | Zbl 0842.35014

[9] M. Carro, L. Pick, J. Soria and V. Stepanov, On embeddings between classical Lorentz spaces, Math. Inequal. Appl. 4 (2001), 397–428. | MR 1841071 | Zbl 0996.46013

[10] M. Carro and J. Soria, Boundedness of some integral operators, Canad. J. Math. 45 (1993), 1155–1166. | MR 1247539 | Zbl 0798.42010

[11] A. Cianchi, Maximizing the L norm of the gradient of solutions to the Poisson equation, J. Geom. Anal. 2 (1992), 499–515. | MR 1189042 | Zbl 0780.35009

[12] A. Cianchi, An optimal interpolation theorem of Marcinkiewicz type in Orlicz spaces, J. Funct. Anal. 153 (1998), 357–381. | MR 1614590 | Zbl 0913.46028

[13] A. Cianchi, Strong and weak type inequalities for some classical operators in Orlicz spaces, J. London Math. Soc. 60 (1999), 187–202. | MR 1721824 | Zbl 0940.46015

[14] A. Cianchi, Symmetrization and second order Sobolev inequalities, Ann. Mat. Pura Appl. 183 (2004), 45–77. | MR 2044332 | Zbl 1223.46033

[15] A. Cianchi, Sharp estimates for nonlinear potentials and applications, In: “Analyis, Partial Differential Equations, and Applications", A. Cialdea, F. Lanzara, P. E. Ricci (eds.), Birkhauser, Basel, 2009, 57–64. | MR 2760868 | Zbl 1182.31018

[16] A. Cianchi, R. Kerman and L. Pick, Boundary trace inequalities and rearrangements, J. Anal. Math. 105 (2008), 241–265. | MR 2438426 | Zbl 1169.46013

[17] F. Duzaar and G. Mingione, Gradient estimates in non-linear potential theory, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 20 (2009), 179–190. | MR 2506379 | Zbl 1173.35065

[18] F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials, Amer. J. Math., 39 (2010), 379–418. | MR 2729305 | Zbl 1204.35100

[19] F. Duzaar and G. Mingione, Gradient contunuity estimates, Cal. Var. Partial Differential Equations 13 (2011), 459–486. | MR 2729305 | Zbl 1217.35077

[20] V. P. Havin and V. G. Maz’ya, Nonlinear potential theory, Usp. Mat. Nauk 27 (1972), 67–138 (Russian); English translation: Russian Math. Surveys 27 (1972), 71–148. | Zbl 0269.31004

[21] L. I. Hedberg and Th. H. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (1983), 161–187. | Numdam | MR 727526 | Zbl 0508.31008

[22] T. Holmstedt, Interpolation of quasi-normed spaces, Math. Scand. 26 (1970), 177–199. | MR 415352 | Zbl 0193.08801

[23] R. Kerman and L. Pick, Optimal Sobolev imbeddings, Forum Math. 18 (2006), 535–570. | MR 2254384 | Zbl 1120.46018

[24] T. Kilpelainen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137–161. | MR 1264000 | Zbl 0820.35063

[25] D. Labutin, Potential estimates for a class of fully nonlinear equations, Duke Math. J. 111 (2002), 1–49. | MR 1876440 | Zbl 1100.35036

[26] J. Malý and W. P. Ziemer, “Fine Regularity of Solutions to Elliptic Partial Differential Equations", American Mathematical Society, Providence, 1997. | MR 835882 | MR 1461542 | Zbl 0882.35001

[27] V. G. Maz’ya, Some estimates of solutions of second-order elliptic equations, Dokl. Akad. Nauk. SSSR 137 (1961), 1057–1059 (Russian); English translation: Soviet Math. Dokl. 2 (1961), 413–415. | MR 140940 | Zbl 0115.08701

[28] V. G. Maz’ya, On weak solutions of the Dirichlet and Neumann problems, Trusdy Moskov. Mat. Obšč. 20 (1969), 137–172 (Russian); English translation: Trans. Moscow Math. Soc. 20 (1969), 135-172. | MR 259329 | Zbl 0226.35027

[29] V. G. Maz’ya, “Sobolev Spaces", Springer-Verlag, Berlin, 1985. | MR 817985 | Zbl 0692.46023

[30] G. Mingione, The Calderon-Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 6 (2007), 195–261. | Numdam | MR 2352517 | Zbl 1178.35168

[31] G. Mingione, Gradient estimates below the duality exponent, Math. Ann. 346 (2010), 571–627. | MR 2578563 | Zbl 1193.35077

[32] G. Mingione, Gradient potential estimates, J. Eur. Math. Soc. 13 (2011), 459–486. | MR 2746772 | Zbl 1217.35077

[33] R. O’Neil, Convolution operators in L(p,q) spaces, Duke Math. J. 30 (1963), 129–142. | MR 146673 | Zbl 0178.47701

[34] B. Opic and L. Pick, On generalized Lorentz-Zygmund spaces, Math. Inequal. Appl. 2 (1999), 391–467. | MR 1698383 | Zbl 0956.46020

[35] N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math. 168 (2008), 859–914. | MR 2456885 | Zbl 1175.31010

[36] N. C. Phuc and I. E. Verbitsky, Singular quasilinear and Hessian equations and inequalities, J. Funct. Anal. 256 (2009), 1875–1906. | MR 2498563 | Zbl 1169.35026

[37] R. S. Strichartz, A note on Trudinger’ s extension of Sobolev’ s inequality, Indiana Univ. Math. J. 21 (1972), 841–842. | MR 293389 | Zbl 0241.46028

[38] G. Talenti, Elliptic equations and rearrangements, Ann. Sc. Norm. Super. Pisa (4) 3 (1976), 697–718. | Numdam | MR 601601 | Zbl 0341.35031

[39] G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl. 120 (1979), 159–184. | MR 551065 | Zbl 0419.35041

[40] N. S. Trudinger and X. J. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math. 124 (2002), 369–410. | MR 1890997 | Zbl 1067.35023