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Regularity in Orlicz spaces for nondivergence degenerate elliptic equations in carnot groups

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Abstract

In this paper, we consider nondivergence degenerate elliptic equations of the kind

$$\begin{aligned} \underset{i,j=1}{\overset{q}{\sum }}a_{ij}(\xi ) X_{i} X_{j}u=f,\quad \text{ in } \Omega \end{aligned}$$

where \(\{X_{1},\ldots ,X_{q}\} \) is the basis of the space of horizontal vector fields in a homogeneous Carnot group \(\mathbb G \,{=}\,(\mathbb R ^{n};\circ ) \), the coefficients \(a_{ij}(\xi )\) are real valued bounded measurable functions defined in \(\Omega \subset \mathbb G \), satisfying the uniform ellipticity. We establish the regularity in Orlicz spaces for the solutions if the coefficients \(a_{ij}(\xi ) \) belong to \(VMO_{loc}\).

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11271299, 11001221) and the Mathematical Tianyuan Foundation of China (Grant No. 11126027).

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Authors and Affiliations

  1. Department of Applied Mathematics, Key Laboratory of Space Applied Physics and Chemistry, Ministry of Education, Northwestern Polytechnical University, Xi’an, Shaanxi, 710129, China

    Pengcheng Niu & Maochun Zhu

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  1. Pengcheng Niu
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  2. Maochun Zhu
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Correspondence to Pengcheng Niu.

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Niu, P., Zhu, M. Regularity in Orlicz spaces for nondivergence degenerate elliptic equations in carnot groups. RACSAM 108, 577–601 (2014). https://doi.org/10.1007/s13398-013-0127-5

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