## Arkiv för Matematik

• Ark. Mat.
• Volume 52, Number 2 (2014), 329-354.

### Global integral gradient bounds for quasilinear equations below or near the natural exponent

Nguyen Cong Phuc

#### Abstract

We obtain sharp integral potential bounds for gradients of solutions to a wide class of quasilinear elliptic equations with measure data. Our estimates are global over bounded domains that satisfy a mild exterior capacitary density condition. They are obtained in Lorentz spaces whose degrees of integrability lie below or near the natural exponent of the operator involved. As a consequence, nonlinear Calderón–Zygmund type estimates below the natural exponent are also obtained for $\mathcal{A}$-superharmonic functions in the whole space ℝn. This answers a question raised in our earlier work (On Calderón–Zygmund theory for p- and $\mathcal{A}$-superharmonic functions, to appear in Calc. Var. Partial Differential Equations, DOI 10.1007/s00526-011-0478-8) and thus greatly improves the result there.

#### Note

Supported in part by NSF grant DMS-0901083.

#### Article information

Source
Ark. Mat., Volume 52, Number 2 (2014), 329-354.

Dates
First available in Project Euclid: 30 January 2017

https://projecteuclid.org/euclid.afm/1485802674

Digital Object Identifier
doi:10.1007/s11512-012-0177-5

Mathematical Reviews number (MathSciNet)
MR3255143

Zentralblatt MATH identifier
1327.35384

Rights

#### Citation

Phuc, Nguyen Cong. Global integral gradient bounds for quasilinear equations below or near the natural exponent. Ark. Mat. 52 (2014), no. 2, 329--354. doi:10.1007/s11512-012-0177-5. https://projecteuclid.org/euclid.afm/1485802674

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