Home Extremals in nonlinear potential theory
Article
Licensed
Unlicensed Requires Authentication

Extremals in nonlinear potential theory

  • Ryan Hynd and Francis Seuffert ORCID logo EMAIL logo
Published/Copyright: March 9, 2021

Abstract

We consider the PDE - Δ p u = ρ , where 𝜌 is a signed Borel measure on R n . For each p > n , we characterize solutions as extremals of a generalized Morrey inequality determined by 𝜌.

MSC 2010: 35J62; 49J40; 31C45

Award Identifier / Grant number: DMS-1554130

Funding statement: Partially supported by NSF grant DMS-1554130.

  1. Communicated by: Juha Kinnunen

References

[1] A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts Math. 17, European Mathematical Society, Zürich, 2011. 10.4171/099Search in Google Scholar

[2] V. Bögelein and J. Habermann, Gradient estimates via non standard potentials and continuity, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 2, 641–678. 10.5186/aasfm.2010.3541Search in Google Scholar

[3] A. Cianchi, Sharp Morrey–Sobolev inequalities and the distance from extremals, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4335–4347. 10.1090/S0002-9947-08-04491-7Search in Google Scholar

[4] A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, The sharp Sobolev inequality in quantitative form, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 5, 1105–1139. 10.4171/JEMS/176Search in Google Scholar

[5] A. Cianchi and V. G. Maz’ya, Optimal second-order regularity for the 𝑝-Laplace system, J. Math. Pures Appl. (9) 132 (2019), 41–78. 10.1016/j.matpur.2019.02.015Search in Google Scholar

[6] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), no. 3, 396–414. 10.1090/S0002-9947-1936-1501880-4Search in Google Scholar

[7] F. Duzaar and G. Mingione, Gradient continuity estimates, Calc. Var. Partial Differential Equations 39 (2010), no. 3–4, 379–418. 10.1007/s00526-010-0314-6Search in Google Scholar

[8] F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials, Amer. J. Math. 133 (2011), no. 4, 1093–1149. 10.1353/ajm.2011.0023Search in Google Scholar

[9] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Textb. Math., CRC Press, Boca Raton, 2015. 10.1201/b18333Search in Google Scholar

[10] A. Figalli and R. Neumayer, Gradient stability for the Sobolev inequality: The case p 2 , J. Eur. Math. Soc. (JEMS) 21 (2019), no. 2, 319–354. 10.4171/JEMS/837Search in Google Scholar

[11] G. B. Folland, Real Analysis, 2nd ed., Pure Appl. Math. (New York), John Wiley & Sons, New York, 1999. Search in Google Scholar

[12] N. Fusco, The quantitative isoperimetric inequality and related topics, Bull. Math. Sci. 5 (2015), no. 3, 517–607. 10.1007/s13373-015-0074-xSearch in Google Scholar

[13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Class. Math., Springer, Berlin, 2001. 10.1007/978-3-642-61798-0Search in Google Scholar

[14] R. Hynd and F. Seuffert, Extremals of Morrey’s inequality, preprint (2018), https://arxiv.org/abs/1810.04393. Search in Google Scholar

[15] R. Hynd and F. Seuffert, Asymptotic flatness of Morrey extremals, Calc. Var. Partial Differential Equations 59 (2020), no. 5, Paper No. 159. 10.1007/s00526-020-01827-0Search in Google Scholar

[16] M. K.-H. Kiessling, On the quasi-linear elliptic PDE - ( u / 1 - | u | 2 ) = 4 π k a k δ s k in physics and geometry, Comm. Math. Phys. 314 (2012), no. 2, 509–523. 10.1007/s00220-018-3261-2Search in Google Scholar

[17] M. K.-H. Kiessling, Correction to: On the quasi-linear elliptic PDE - ( u / 1 - | u | 2 ) = 4 π k a k δ s k in physics and geometry [2958962], Comm. Math. Phys. 364 (2018), no. 2, 825–833. 10.1007/s00220-018-3261-2Search in Google Scholar

[18] J. Kinnunen and O. Martio, Nonlinear potential theory on metric spaces, Illinois J. Math. 46 (2002), no. 3, 857–883. 10.1215/ijm/1258130989Search in Google Scholar

[19] T. Kuusi and G. Mingione, Linear potentials in nonlinear potential theory, Arch. Ration. Mech. Anal. 207 (2013), no. 1, 215–246. 10.1007/s00205-012-0562-zSearch in Google Scholar

[20] C. B. Morrey, Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166. 10.1090/S0002-9947-1938-1501936-8Search in Google Scholar

[21] C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Class. Math., Springer, Berlin, 2008. Search in Google Scholar

[22] R. Neumayer, A note on strong-form stability for the Sobolev inequality, Calc. Var. Partial Differential Equations 59 (2020), no. 1, Paper No. 25. 10.1007/s00526-019-1686-xSearch in Google Scholar

[23] Q.-H. Nguyen and N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equations with measure data, J. Funct. Anal. 278 (2020), no. 5, Article ID 108391. 10.1016/j.jfa.2019.108391Search in Google Scholar

[24] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3) 13 (1959), 115–162. 10.1007/978-3-642-10926-3_1Search in Google Scholar

[25] F. Seuffert, A stability result for a family of sharp Gagliardo–Nirenberg inequalities, preprint (2016), https://arxiv.org/abs/1610.06869. Search in Google Scholar

[26] F. Seuffert, An extension of the Bianchi–Egnell stability estimate to Bakry, Gentil, and Ledoux’s generalization of the Sobolev inequality to continuous dimensions, J. Funct. Anal. 273 (2017), no. 10, 3094–3149. 10.1016/j.jfa.2017.07.001Search in Google Scholar

Received: 2020-06-10
Revised: 2021-02-04
Accepted: 2021-02-06
Published Online: 2021-03-09
Published in Print: 2022-10-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 9.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/acv-2020-0063/html
Scroll to top button