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On Brunn–Minkowski Type Inequalities and Overdetermined Problem for p-Capacity

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Abstract

In this paper, we mainly study some properties of p-capacity. Firstly, the p-capacitary difference Brunn–Minkowski and Minkowski inequalities are established. Secondly, we propose the p-capacitary overdetermined problem and prove the Serrin type symmetry result for the problem. Finally, we also make some considerations for the polar set of p-difference body for capacity.

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References

  1. Beckenbach, E.F., Bellman, R.: Inequalities, 2nd edn. Springer, Berlin (1965)

    Book  Google Scholar 

  2. Borell, C.: Capacitary inequalities of the Brunn–Minkowski type. Math. Ann. 263, 179–184 (1983)

    Article  MathSciNet  Google Scholar 

  3. Bianchini, C., Colesanti, A.: A sharp Rogers and Shephard inequality for the \(p\)-difference body of planar convex bodies. Proc. Am. Math. Soc. 136, 2575–2582 (2008)

    Article  MathSciNet  Google Scholar 

  4. Brandolini, B., Nitsch, C., Salani, P., Trombetti, C.: Serrin-type overdetermined problems: analternative proof. Arch. Ration. Mech. Anal. 190, 267–280 (2008)

    Article  MathSciNet  Google Scholar 

  5. Choulli, M., Henrot, A.: Use of the domain derivative to prove symmetry results in partial differential equations. Math. Nachr. 192, 91–103 (1998)

    Article  MathSciNet  Google Scholar 

  6. Caffarelli, L.A., Jerison, D., Lieb, E.H.: On the case of equality in the Brunn–Minkowski inequality for capacity. Adv. Math. 117, 193–207 (1996)

    Article  MathSciNet  Google Scholar 

  7. Colesanti, A., Cuoghi, P.: Brunn–Minkowski inequalities for two functionals involving the P-Laplace operator. Appl. Anal. 327, 459–479 (2003)

    MATH  Google Scholar 

  8. Colesanti, A., Salani, P.: The Brunn–Minkowski inequality for \(p\)-capacity of convex bodies. Math. Ann. 327, 459–479 (2003)

    Article  MathSciNet  Google Scholar 

  9. Colesanti, A.: Brunn–Minkowski inequalities for variational functionals and related problems. Adv. Math. 194, 105–140 (2005)

    Article  MathSciNet  Google Scholar 

  10. Colesanti, A., Cuoghi, P.: The Brunn–Minkowski inequality for the \(n\)-dimensional logarithmic capacity of convex bodies. Potential Anal. 22, 289–304 (2005)

    Article  MathSciNet  Google Scholar 

  11. Colesanti, A., Nyström, K., Salani, P., Xiao, J., Yang, D., Zhang, G.: The Hadamard variational formula and the Minkowski problem for \(p\)-capacity. Adv. Math. 285, 1511–1588 (2015)

    Article  MathSciNet  Google Scholar 

  12. Dahlberg, B.E.J.: Estimates of harmonic measure. Arch. Ration. Mech. Anal. 65, 275–288 (1977)

    Article  MathSciNet  Google Scholar 

  13. Firey, W.J.: Mean cross-section measures of harmonic means of convex bodies. Pac. J. Math. 11, 1263–1266 (1961)

    Article  MathSciNet  Google Scholar 

  14. Firey, W.J.: \(p\)-means of convex bodies. Math. Scand 10, 17–24 (1962)

    Article  MathSciNet  Google Scholar 

  15. Fragalà, I.: Symmetry results for overdetermined problems on convex domains via Brunn–Minkowski inequalities. J. Math. Pures Appl. 97, 55–65 (2012)

    Article  MathSciNet  Google Scholar 

  16. Gardner, R.: The Brunn–Minkowski inequality. Bull. Am. Math. Soc. 39, 355–405 (2002)

    Article  MathSciNet  Google Scholar 

  17. Hardy, G.H., Littlewood, J.E.: Pólya, Inequalities. Cambridge University Press, Cambridge (1934)

    Google Scholar 

  18. Hernández, M., Yepes Nicolás, J.: On Brunn–Minkowski type Inequalities for polar bodies. J. Geom. Anal. 26, 1–13 (2016)

    MathSciNet  MATH  Google Scholar 

  19. Han, H., Ye, D., Zhang, N.: The \(p\)-capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems. Calc. Var. Partial Differ. 57, 1–31 (2018)

    Article  MathSciNet  Google Scholar 

  20. Hong, H., Ye, D.: Sharp Geometric Inequalities for the General \(p\)-Affine Capacity. J. Geom. Anal 28, 2254–2287 (2018)

    Article  MathSciNet  Google Scholar 

  21. Jerison, D.: A Minkowski problem for electrostatic capacity. Acta. Math. 176, 1–47 (1996)

    Article  MathSciNet  Google Scholar 

  22. Jerison, D.: The direct method in the calculus of variations for convex bodies. Adv. Math. 122, 262–279 (1996)

    Article  MathSciNet  Google Scholar 

  23. Lewis, J.L.: Capacitary functions in convex rings. Arch. Ration. Mech. Anal. 66, 201–224 (1977)

    Article  MathSciNet  Google Scholar 

  24. Ludwig, M., Xiao, J., Zhang, G.: Sharp convex Lorentz–Sobolev inequalities. Math. Ann. 350, 169–197 (2011)

    Article  MathSciNet  Google Scholar 

  25. Lutwak, E.: Inequalities for Hadwiger’s harmonic quermassintegrals. Math. Ann. 280, 165–175 (1988)

    Article  MathSciNet  Google Scholar 

  26. Lutwak, E.: Centroid bodies and dual mixed volumes. Proc. Lond. Math. Soc. 60, 365–391 (1990)

    Article  MathSciNet  Google Scholar 

  27. Lutwak, E.: The Brunn–Minkowski firey theory I: mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993)

    Article  MathSciNet  Google Scholar 

  28. Payne, L.E., Schaefer, P.W.: Duality theorems in some overdetermined boundary value problems. Math. Methods Appl. Sci. 11, 805–819 (1989)

    Article  MathSciNet  Google Scholar 

  29. Rogers, C., Shephard, G.: The difference body of a convex body. Arch. Math. 8, 220–223 (1957)

    Article  MathSciNet  Google Scholar 

  30. Serrin, J.: A symmetry problem in potenial theory. Arch. Ration. Mech. Anal. 43, 304–318 (1971)

    Article  Google Scholar 

  31. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, 2nd edn. Cambridge University Press, Cambridge (2014)

    MATH  Google Scholar 

  32. Weinberger, H.: Remark on the preceding paper of Serrin. Arch. Ration. Mech. Anal. 43, 319–320 (1971)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We would like to thank the referees for valuable suggestions and comments that lead to improvement of the original manuscript.

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Authors and Affiliations

  1. Department of Mathematics, East China University of Technology, Nanchang, 330013, China

    Lewen Ji

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Lewen Ji

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  1. Lewen Ji
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Correspondence to Lewen Ji.

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Research is supported by the Natural Science Foundation of China (Grant No. 71762001), the Science and Technology Project of Education Department of Jiangxi Province (Grant No. GJJ180414) and PhD research startup foundation of East China University of Technology (Grant No. DHBK2018050).

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Ji, L. On Brunn–Minkowski Type Inequalities and Overdetermined Problem for p-Capacity. Results Math 75, 51 (2020). https://doi.org/10.1007/s00025-020-1177-6

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