Boundary Riesz potential estimates for parabolic equations with measurable nonlinearities

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

We obtain a boundary pointwise gradient estimate on a parabolic half cube $Q_{2 R} \cap\left\{\left(x^{1}, x^{\prime}, t\right) \in\right.\left.\mathbb{R}^{n+1}: x^{1}>0\right\}$ for nonlinear parabolic equations with measurable nonlinearities, which are only assumed to be measurable in x¹-variable. The estimates are obtained in terms of Riesz potential of the right-hand side measure and the oscillation of the boundary data, where the boundary data is given on $Q_{2 R} \cap\left\{\left(x^{1}, x^{\prime}, t\right) \in \mathbb{R}^{n+1}: x^{1}=0\right\} .$

Keywords

nonlinear parabolic equations / Riesz potentials / measurable nonlinearities

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] F. Duzaar, G. Mingione, Gradient estimates via non-linear potentials, Amer. J. Math., 133 (2011), 1093-1149. https://doi.org/10.1353/ajm.2011.0023

[2] T. Kuusi, G. Mingione, Potential estimates and gradient boundedness for nonlinear parabolic systems, Rev. Mat. Iberoam., 28 (2012), 535-576. https://doi.org/10.4171/RMI/684

[3] T. Kuusi, G. Mingione, The Wolff gradient bound for degenerate parabolic equations, J. Eur. Math. Soc., 16 (2014), 835-892. https://doi.org/10.4171/JEMS/449

[4] T. Kuusi, G. Mingione, Riesz potentials and nonlinear parabolic equations, Arch. Ration. Mech. Anal., 212 (2014), 727-780. https://doi.org/10.1007/s00205-013-0695-8

[5] Y. Kim, Y. Youn, Boundary Riesz potential estimates for elliptic equations with measurable nonlinearities, Nonlinear Anal., 194 (2020), 111445. https://doi.org/10.1016/j.na.2019.02.001

[6] Y. Kim, Riesz potential type estimates for nonlinear elliptic equations, J. Differential Equations, 263 (2017), 6844-6884. https://doi.org/10.1016/j.jde.2017.07.031

[7] S. S. Byun, L. G. Softova, Parabolic obstacle problem with measurable data in generalized Morrey spaces, Z. Anal. Anwend., 35 (2016), 153-171. https://doi.org/10.4171/zaa/1559

[8] S. S. Byun, D. Palagachev, L. G. Softova, Global gradient estimates in weighted Lebesgue spaces for parabolic operators, Ann. Acad. Sci. Fenn. Math., 41 (2016), 67-83. https://doi.org/10.5186/aasfm.2016.4102

[9] H. Xu, Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials, Commun. Anal. Mech., 15 (2023), 132-161. https://doi.org/10.3934/cam.2023008

[10] G. M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. https://doi.org/10.1142/3302

[11] Q. Han, F. Lin, Elliptic partial differential equations, Second edition, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.

[12] S. S. Byun, Y. Kim, Elliptic equations with measurable nonlinearities in nonsmooth domains, Adv. Math., 288 (2016), 152-200. https://doi.org/10.1016/j.aim.2015.10.015

[13] S. S. Byun, Y. Kim, Riesz potential estimates for parabolic equations with measurable nonlinearities, Int. Math. Res. Not., 2018 (2018), 6737-6779. https://doi.org/10.1093/imrn/rnx080

[14] E. DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. https://doi.org/10.1007/978-1-4612-0895-2

[15] E. Giusti, Direct methods in the calculus of variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. https://doi.org/10.1142/9789812795557

[16] G. Mingione, The Calderón-Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 195-261. https://doi.org/10.2422/2036-2145.2007.2.01

[17] T. Kuusi, G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl., 98 (2012), 390-427. https://doi.org/10.1016/j.matpur.2012.02.004