Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 1, pp. 1-17.

We obtain local and global W 1,q estimates on weighted Lebesgue spaces with certain Muckenhoupt weights for solutions to a nonhomogeneous p-Laplace type equation with VMO coefficients in a 𝒞 1 domain. These estimates can be viewed as weighted norm inequalities for certain nonlinear singular operators (without any explicit kernel) arising from the p-Laplacian, and are applicable to a quasilinear Riccati type equation.

Publié le :
Classification : 35R05, 35J92, 42B37, 35J15, 35J25, 42B25, 42B99
Nguyen, Cong Phuc 1

1 Department of Mathematics Louisiana State University 303 Lockett Hall Baton Rouge, LA 70803, USA
@article{ASNSP_2011_5_10_1_1_0,
     author = {Nguyen, Cong Phuc},
     title = {Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {1--17},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 10},
     number = {1},
     year = {2011},
     mrnumber = {2829320},
     zbl = {1228.35260},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2011_5_10_1_1_0/}
}
TY  - JOUR
AU  - Nguyen, Cong Phuc
TI  - Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2011
SP  - 1
EP  - 17
VL  - 10
IS  - 1
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2011_5_10_1_1_0/
LA  - en
ID  - ASNSP_2011_5_10_1_1_0
ER  - 
%0 Journal Article
%A Nguyen, Cong Phuc
%T Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2011
%P 1-17
%V 10
%N 1
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2011_5_10_1_1_0/
%G en
%F ASNSP_2011_5_10_1_1_0
Nguyen, Cong Phuc. Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 1, pp. 1-17. http://archive.numdam.org/item/ASNSP_2011_5_10_1_1_0/

[1] S. Byun and L. Wang, Quasilinear elliptic equations with BMO coefficients in Lipschitz domains, Trans. Amer. Math. Soc. 359 (2007), 5899–5913. | MR | Zbl

[2] S. Byun, L. Wang and S. Zhou, Nonlinear elliptic equations with BMO coefficients in Reifenberg domains, J. Funct. Anal. 250 (2007), 167–196. | MR | Zbl

[3] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. | EuDML | MR | Zbl

[4] E. DiBenedetto, C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827–850. | MR | Zbl

[5] E. DiBenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993), 1107–1134. | MR | Zbl

[6] L. D’Onofrio and T. Iwaniec, Notes on p-harmonic analysis, In: “The p-harmonic Equation and Recent Advances in Analysis”, Contemp. Math., Vol. 370, Amer. Math. Soc., Providence, RI, 2005, 25–49. | MR | Zbl

[7] L. Grafakos, “Classical and Modern Fourier Analysis”, Pearson Education, Inc., Upper Saddle River, NJ, 2004. | MR | Zbl

[8] T. Iwaniec, On L p -integrability in PDE’s and quasiregular mappings for large exponents, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), 301–322. | MR | Zbl

[9] T. Iwaniec, Projections onto gradient fields and L p -estimates for degenerated elliptic operators, Studia Math. 75 (1983), 293–312. | EuDML | MR | Zbl

[10] C. Fefferman and E. Stein, H p spaces of several variables, Acta Math. 129 (1972), 137–193. | MR | Zbl

[11] J. Kinnunen and S. Zhou, A local estimate for nonlinear equations with discontinuous coefficients, Comm. Partial Differential Equations 24 (1999), 2043–2068. | MR | Zbl

[12] J. Kinnunen and S. Zhou, A boundary estimate for nonlinear equations with discontinuous coefficients, Differential Integral Equations 14 (2001), 475–492. | MR | Zbl

[13] J. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J. 32 (1983), 849–858. | MR | Zbl

[14] V. G. Maz’ya and T. O. Shaposhnikova, “Theory of Multipliers in Spaces of Differentiable Functions”, Monographs and Studies in Mathematics, Vol. 23, Pitman Publishing Co., Brooklyn, Newyork, 1985. | MR | Zbl

[15] V. G. Maz’ya and E. I. Verbitsky, Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers, Ark. Mat. 33 (1995), 81–115. | MR | Zbl

[16] V. G. Maz’ya and E. I. Verbitsky, The Schrödinger operator on the energy space: boundedness and compactness criteria, Acta Math. 188 (2002), 263–302. | MR | Zbl

[17] V. G. Maz’ya and E. I. Verbitsky, Infinitesimal form boundedness and Trudinger’s subordination for the Schrödinger operator, Invent. Math. 162 (2005), 81–136 | MR | Zbl

[18] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. | MR | Zbl

[19] N. C. Phuc, Quasilinear Riccati type equations with super-critical exponents, Comm. Partial Differential Equations 35 (2010), 1958–1981. | MR | Zbl

[20] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405. | MR | Zbl

[21] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126–150. | MR | Zbl