Universal potential estimates for $ 1 &lt; p\leq 2-\frac{1}{n} $

Quoc-Hung Nguyen; Nguyen Cong Phuc; Quoc-Hung Nguyen; Nguyen Cong Phuc

doi:10.3934/mine.2023057

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-24. doi: 10.3934/mine.2023057

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Universal potential estimates for $ 1 < p\leq 2-\frac{1}{n} $

Quoc-Hung Nguyen ^{1
,
,},
Nguyen Cong Phuc ²

1.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
2.
Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA 70803, USA

Received: 26 August 2022 Revised: 13 October 2022 Accepted: 14 October 2022 Published: 26 October 2022

We extend the so-called universal potential estimates of Kuusi-Mingione type (J. Funct. Anal. 262: 4205–4269, 2012) to the singular case $ 1 < p\leq 2-1/n $ for the quasilinear equation with measure data

$ \begin{equation*} -\operatorname{div}(A(x,\nabla u)) = \mu \end{equation*} $

in a bounded open subset $ \Omega $ of $ \mathbb{R}^n $, $ n\geq 2 $, with a finite signed measure $ \mu $ in $ \Omega $. The operator $ \operatorname{div}(A(x, \nabla u)) $ is modeled after the $ p $-Laplacian $ \Delta_p u: = {\rm div}\, (|\nabla u|^{p-2}\nabla u) $, where the nonlinearity $ A(x, \xi) $ ($ x, \xi \in \mathbb{R}^n $) is assumed to satisfy natural growth and monotonicity conditions of order $ p $, as well as certain additional regularity conditions in the $ x $-variable.
- pointwise estimate,
- potential estimate,
- Wolff's potential,
- Riesz's potential,
- fractional maximal function,
- Calderón space,
- $ p $-Laplacian,
- quasilinear equation,
- measure data
Citation: Quoc-Hung Nguyen, Nguyen Cong Phuc. Universal potential estimates for $ 1 < p\leq 2-\frac{1}{n} $[J]. Mathematics in Engineering, 2023, 5(3): 1-24. doi: 10.3934/mine.2023057

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Abstract

We extend the so-called universal potential estimates of Kuusi-Mingione type (J. Funct. Anal. 262: 4205–4269, 2012) to the singular case $ 1 < p\leq 2-1/n $ for the quasilinear equation with measure data

$ \begin{equation*} -\operatorname{div}(A(x,\nabla u)) = \mu \end{equation*} $

in a bounded open subset $ \Omega $ of $ \mathbb{R}^n $, $ n\geq 2 $, with a finite signed measure $ \mu $ in $ \Omega $. The operator $ \operatorname{div}(A(x, \nabla u)) $ is modeled after the $ p $-Laplacian $ \Delta_p u: = {\rm div}\, (|\nabla u|^{p-2}\nabla u) $, where the nonlinearity $ A(x, \xi) $ ($ x, \xi \in \mathbb{R}^n $) is assumed to satisfy natural growth and monotonicity conditions of order $ p $, as well as certain additional regularity conditions in the $ x $-variable.

References

[1]	R. A. DeVore, R. C. Sharpley, Maximal functions measuring smoothness, Mem. Amer. Math. Soc., 47 (1984), 293. http://doi.org/10.1090/memo/0293 doi: 10.1090/memo/0293
[2]	E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. Theor., 7 (1983), 827–850. https://doi.org/10.1016/0362-546X(83)90061-5 doi: 10.1016/0362-546X(83)90061-5
[3]	H. Dong, H. Zhu, Gradient estimates for singular $p$-Laplace type equations with measure data, Calc. Var., 61 (2021), 86. https://doi.org/10.1007/s00526-022-02189-5 doi: 10.1007/s00526-022-02189-5
[4]	F. Duzaar, G. Mingione, Gradient estimates via linear and nonlinear potentials, J. Funct. Anal., 259 (2010), 2961–2998. https://doi.org/10.1016/j.jfa.2010.08.006 doi: 10.1016/j.jfa.2010.08.006
[5]	F. Duzaar, G. Mingione, Gradient estimates via non-linear potentials, Amer. J. Math., 133 (2011), 1093–1149.
[6]	E. Giusti, Direct methods in the calculus of variations, River Edge, NJ: World Scientic Publishing Co., Inc., 2003. https://doi.org/10.1142/5002
[7]	T. Kilpeläinen, J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137–161. https://doi.org/10.1007/BF02392793 doi: 10.1007/BF02392793
[8]	T. Kuusi, G. Mingione, Universal potential estimates, J. Funct. Anal., 262 (2012), 4205–4269. https://doi.org/10.1016/j.jfa.2012.02.018 doi: 10.1016/j.jfa.2012.02.018
[9]	T. Kuusi, G. Mingione, Linear potentials in nonlinear potential theory, Arch. Rational Mech. Anal., 207 (2013), 215–246. https://doi.org/10.1007/s00205-012-0562-z doi: 10.1007/s00205-012-0562-z
[10]	P. Lindqvist, Notes on the p-Laplace equation, Univ. Jyväskylä, 2006.
[11]	J. J. Manfredi, Regularity of the gradient for a class of nonlinear possibly degenerate elliptic equations, Ph.D. Thesis of University of Washington, St. Louis, 1986.
[12]	G. Mingione, Gradient potential estimates, J. Eur. Math. Soc., 13 (2011), 459–486. http://doi.org/10.4171/JEMS/258 doi: 10.4171/JEMS/258
[13]	Q.-H. Nguyen, N. C. Phuc, Good-$\lambda$ and Muckenhoupt-Wheeden type bounds in quasilinear measure datum problems, with applications, Math. Ann., 374 (2019), 67–98. https://doi.org/10.1007/s00208-018-1744-2 doi: 10.1007/s00208-018-1744-2
[14]	Q.-H. Nguyen, N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equation with measure data, J. Funct. Anal., 278 (2020), 108391. https://doi.org/10.1016/j.jfa.2019.108391 doi: 10.1016/j.jfa.2019.108391
[15]	Q.-H. Nguyen, N. C. Phuc, A comparison estimate for singular p-Laplace equations and its consequences, submitted for publication.
[16]	N. S. Trudinger, X.-J. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math., 124 (2002), 369–410.

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