Global Regularity of Weak Solutions to Quasilinear Elliptic and Parabolic Equations with Controlled Growth

Abstract

We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to x.

Keywords:

• BMO coefficients
• Boundary value problems
• Quasilinear elliptic and parabolic equations
• Sobolev spaces

Mathematics Subject Classification:

• 35K59
• 35J62
• 35B65

Acknowledgement

We would like to thank Dian K. Palagachev for informing us his recent papers [32, 33], and the anonymous referees for their helpful comments. H. Dong was partially supported by NSF grant number DMS-0800129.