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Good-\(\lambda \) and Muckenhoupt–Wheeden type bounds in quasilinear measure datum problems, with applications

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Abstract

Weighted good-\(\lambda \) type inequalities and Muckenhoupt–Wheeden type bounds are obtained for gradients of solutions to a class of quasilinear elliptic equations with measure data. Such results are obtained globally over sufficiently flat domains in \(\mathbb {R}^n\) in the sense of Reifenberg. The principal operator here is modeled after the p-Laplacian, where for the first time singular case \(\frac{3n-2}{2n-1}<p\le 2-\frac{1}{n}\) is considered. Those bounds lead to useful compactness criteria for solution sets of quasilinear elliptic equations with measure data. As an application, sharp existence results and sharp bounds on the size of removable singular sets are deduced for a quasilinear Riccati type equation having a gradient source term with linear or super-linear power growth.

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Acknowledgements

The authors kindly thank the anonymous referee for his/her comments that help improve the quality of the paper. Q.-H. Nguyen is supported by the Centro De Giorgi, Scuola Normale Superiore, Pisa, Italy. N. C. Phuc is supported in part by Simons Foundation, award number 426071.

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  1. Scuola Normale Superiore, Centro Ennio de Giorgi, Piazza dei Cavalieri 3, 56100, Pisa, Italy

    Quoc-Hung Nguyen

  2. Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA, 70803, USA

    Nguyen Cong Phuc

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  1. Quoc-Hung Nguyen
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Correspondence to Quoc-Hung Nguyen.

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Communicated by Loukas Grafakos.

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Nguyen, QH., Phuc, N.C. Good-\(\lambda \) and Muckenhoupt–Wheeden type bounds in quasilinear measure datum problems, with applications. Math. Ann. 374, 67–98 (2019). https://doi.org/10.1007/s00208-018-1744-2

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  • DOI: https://doi.org/10.1007/s00208-018-1744-2

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