Abstract
This paper builds a bridge between partial regularity theory and nonlinear potential theory for the following generalized stationary Stokes system with super-quadratic growth and continuous coefficients
where Du is the symmetric part of the gradient ∇u. We first establish an ε-regularity criterion involving both the excess functional of the symmetric gradient Du and Wolff potentials of the nonhomogeneous term f to guarantee the local vanishing mean oscillation (VMO)-regularity of Du in an open subset Ωu of Ω with full measure. Such an ε-regularity criterion leads to a pointwise Wolff potential estimate of Du, which immediately infers that Du is partially C0-regular under appropriate assumptions. Finally, we give a local continuous modulus estimate of Du.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 12101452 and 12071229). The authors are very grateful to Professor Giuseppe Mingione for suggesting this interesting problem to them and sincerely thank all the reviewers for their insightful and constructive comments.
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Ma, L., Zhang, Z. Partial regularity and nonlinear potential estimates for Stokes systems with super-quadratic growth. Sci. China Math. 67, 1525–1554 (2024). https://doi.org/10.1007/s11425-022-2137-x
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DOI: https://doi.org/10.1007/s11425-022-2137-x