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BMO-regularity for a degenerate transmission problem

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Abstract

We examine a transmission problem driven by a degenerate quasilinear operator with a natural interface condition. Two aspects of the problem entail genuine difficulties in the analysis: the absence of representation formulas for the operator and the degenerate nature of the diffusion process. Our arguments circumvent these difficulties and lead to new regularity estimates. For bounded interface data, we prove the local boundedness of weak solutions and establish an estimate for their gradient in \(\textrm{BMO}\)-spaces. The latter implies solutions are of class \(C^{0,\mathrm{Log-Lip}}\) across the interface. Relaxing the assumptions on the data, we establish local Hölder continuity for the solutions.

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Acknowledgements

The authors thank Paolo Baroni and Giuseppe Mingione for their insightful comments on the material in the paper.

Funding

EP is partially supported by FAPERJ (grants E26/200.002/2018 and E26/201.390/2021). JMU is partially supported by the King Abdullah University of Science and Technology (KAUST). All authors are partially supported by the Centre for Mathematics of the University of Coimbra (funded by the Portuguese Government through FCT/MCTES, https://doi.org/10.54499/UIDB/00324/2020).

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Authors and Affiliations

  1. Applied Mathematics and Computational Sciences Program (AMCS), Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), 23955-6900, Thuwal, Kingdom of Saudi Arabia

    José Miguel Urbano

  2. CMUC, Department of Mathematics, University of Coimbra, 3000-143, Coimbra, Portugal

    Vincenzo Bianca, Edgard A. Pimentel & José Miguel Urbano

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  1. Vincenzo Bianca
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  2. Edgard A. Pimentel
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  3. José Miguel Urbano
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Correspondence to José Miguel Urbano.

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Bianca, V., Pimentel, E.A. & Urbano, J.M. BMO-regularity for a degenerate transmission problem. Anal.Math.Phys. 14, 9 (2024). https://doi.org/10.1007/s13324-023-00867-x

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