Abstract
This paper is concerned with partial regularity of weak solutions to nonlinear sub-elliptic systems under sub-quadratic natural growth conditions. We begin with establishing a Sobolev-Poincaré type inequality associated with Hörmander’s vector fields for \(u\in HW^{1,m}(\Omega , \mathbb {R}^N)\) with \(1<m<2\). Then \(\mathcal {A}\)-harmonic approximation method is applied, and partial Hölder continuity with optimal local Hölder exponent for gradients of weak solutions to the systems is established.
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We would like to express our gratitude to referees for their valuable comments and suggestions.
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The Project was supported by National Natural Science Foundation of China (No. 11126294 and No. 11201081), and supported by Science and Technology Planning Project of Jiangxi Province, China (No. GJJ13657).
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Wang, J., Liao, D., Guo, Z. et al. Hölder continuity for nonlinear sub-elliptic systems with sub-quadratic growth. RACSAM 109, 27–42 (2015). https://doi.org/10.1007/s13398-014-0162-x
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DOI: https://doi.org/10.1007/s13398-014-0162-x
Keywords
- Partial regularity
- Nonlinear sub-elliptic systems
- Hörmander’s vector fields
- Sub-quadratic natural growth condition
- \(\mathcal {A}\)-harmonic approximation technique