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Open AccessOpen Access JUSTC Mathematics 14 June 2023

Harnack inequality for polyharmonic equations

  • Department of Mathematics, Anhui University of Technology, Ma’anshan 243002, China

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https://doi.org/10.52396/JUSTC-2022-0114
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  • Author Bio:

    Jiamin Zeng is currently a postgraduate student at Anhui University of Technology. His research mainly focuses on elliptic partial differential equations

    Yi Fang is an Associate Professor at Anhui University of Technology. He received his Ph.D. degree from the University of Science and Technology of China in 2015. His research mainly focuses on elliptic partial differential equations

  • Corresponding author: E-mail: yif1915@ahut.edu.cn
  • Received Date: 08 August 2022
  • Accepted Date: 17 November 2022
    • Available Online: 14 June 2023
    Abstract

  • Abstract

    Some new types of mean value formulas for the polyharmonic functions were established. Based on the formulas, the Harnack inequality for the nonnegative solutions to the polyharmonic equations was proved.

    Graphical abstract

    Harnack inequality.

    Abstract

    Some new types of mean value formulas for the polyharmonic functions were established. Based on the formulas, the Harnack inequality for the nonnegative solutions to the polyharmonic equations was proved.

    • Public Summary

    • Some new type mean value formulas for polyharmonic functions were established.
    • The Harnack inequality for polyharmonic functions was proved.

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    • References(7)
  • [1]
    Han Q, Lin F. Elliptic Partial Differential Equations. 2nd edition. Providence, RI: American Mathematical Society, 2011.
    [2]
    Gilbarg D, Trudinger N. Elliptic Partial Differential Equations of Second Order. Berlin: Springer Verlag, 1983.
    [3]
    Caristi G, Mitidieri E. Harnack inequality and applications to solutions of biharmonic equations. In: Partial Differential Equations and Functional Analysis. Basel, Switzerland: Birkhäuser Verlag, 2006.
    [4]
    Karachik V V. On the mean value property for polyharmonic functions in the ball. Siberian Advances in Mathematics, 2014, 24 (3): 169–182. doi: 10.3103/S1055134414030031
    [5]
    Łysik G. On the mean value property for polyharmonic functions. Acta Math. Hung., 2011, 133: 133–139. doi: 10.1007/s10474-011-0138-7
    [6]
    Wei J, Xu X. Classification of solutions of higher order conformally invariant equations. Math. Ann., 1999, 313: 207–228. doi: 10.1007/s002080050258
    [7]
    Simader C G. Mean value formulas, Weyl’s lemma and Liouville theorems for δ2 and Stokes’ system. Results in Mathematics, 1992, 22: 761–780. doi: 10.1007/BF03323122
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    [1]
    Han Q, Lin F. Elliptic Partial Differential Equations. 2nd edition. Providence, RI: American Mathematical Society, 2011.
    [2]
    Gilbarg D, Trudinger N. Elliptic Partial Differential Equations of Second Order. Berlin: Springer Verlag, 1983.
    [3]
    Caristi G, Mitidieri E. Harnack inequality and applications to solutions of biharmonic equations. In: Partial Differential Equations and Functional Analysis. Basel, Switzerland: Birkhäuser Verlag, 2006.
    [4]
    Karachik V V. On the mean value property for polyharmonic functions in the ball. Siberian Advances in Mathematics, 2014, 24 (3): 169–182. doi: 10.3103/S1055134414030031
    [5]
    Łysik G. On the mean value property for polyharmonic functions. Acta Math. Hung., 2011, 133: 133–139. doi: 10.1007/s10474-011-0138-7
    [6]
    Wei J, Xu X. Classification of solutions of higher order conformally invariant equations. Math. Ann., 1999, 313: 207–228. doi: 10.1007/s002080050258
    [7]
    Simader C G. Mean value formulas, Weyl’s lemma and Liouville theorems for δ2 and Stokes’ system. Results in Mathematics, 1992, 22: 761–780. doi: 10.1007/BF03323122
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