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Advances in Calculus of Variations

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The Brunn--Minkowski inequality and a Minkowski problem for 𝒜-harmonic Green's function

Murat Akman / John Lewis / Olli Saari / Andrew Vogel
Published Online: 2019-04-06 | DOI: https://doi.org/10.1515/acv-2018-0064


In this article we study two classical problems in convex geometry associated to 𝒜-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modelled on the p-Laplace equation. Let p be fixed with 2np<. For a convex compact set E in n, we define and then prove the existence and uniqueness of the so-called 𝒜-harmonic Green’s function for the complement of E with pole at infinity. We then define a quantity C𝒜(E) which can be seen as the behavior of this function near infinity. In the first part of this article, we prove that C𝒜() satisfies the following Brunn–Minkowski-type inequality:


when n<p<, 0λ1, and E1,E2 are nonempty convex compact sets in n. While p=n then


where 0λ1 and E1,E2 are convex compact sets in n containing at least two points. Moreover, if equality holds in the either of the above inequalities for some E1 and E2, then under certain regularity and structural assumptions on 𝒜 we show that these two sets are homothetic. The classical Minkowski problem asks for necessary and sufficient conditions on a non-negative Borel measure on the unit sphere 𝕊n-1 to be the surface area measure of a convex compact set in n under the Gauss mapping for the boundary of this convex set. In the second part of this article we study a Minkowski-type problem for a measure associated to the 𝒜-harmonic Green’s function for the complement of a convex compact set E when np<. If μE denotes this measure, then we show that necessary and sufficient conditions for existence under this setting are exactly the same conditions as in the classical Minkowski problem. Using the Brunn–Minkowski inequality result from the first part, we also show that this problem has a unique solution up to translation.

Keywords: Brunn–Minkowski inequality; Minkowski problem; inequalities and extremum problems; potentials and capacities; variational formula, Hadamard variational formula

MSC 2010: 35J60; 31B15; 39B62; 52A40; 35J20; 52A20; 35J92


  • [1]

    D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer, Berlin, 1996. Google Scholar

  • [2]

    M. Akman, On the dimension of a certain measure in the plane, Ann. Acad. Sci. Fenn. Math. 39 (2014), no. 1, 187–209. CrossrefGoogle Scholar

  • [3]

    M. Akman, J. Gong, J. Hineman, J. Lewis and A. Vogel, The Brunn–Minkowski inequality and a Minkowski problem for nonlinear capacity, preprint (2017), https://arxiv.org/abs/1709.00447; to appear in Mem. Amer. Math. Soc.

  • [4]

    M. Akman, J. Lewis and A. Vogel, σ-finiteness of elliptic measures for quasilinear elliptic PDE in space, Adv. Math. 309 (2017), 512–557. CrossrefWeb of ScienceGoogle Scholar

  • [5]

    M. Akman, J. L. Lewis and A. Vogel, On the logarithm of the minimizing integrand for certain variational problems in two dimensions, Anal. Math. Phys. 2 (2012), no. 1, 79–88. CrossrefWeb of ScienceGoogle Scholar

  • [6]

    C. Borell, Capacitary inequalities of the Brunn–Minkowski type, Math. Ann. 263 (1983), no. 2, 179–184. CrossrefGoogle Scholar

  • [7]

    C. Borell, Hitting probabilities of killed Brownian motion: A study on geometric regularity, Ann. Sci. Éc. Norm. Supér. (4) 17 (1984), no. 3, 451–467. CrossrefGoogle Scholar

  • [8]

    L. A. Caffarelli, D. Jerison and E. H. Lieb, On the case of equality in the Brunn–Minkowski inequality for capacity, Adv. Math. 117 (1996), no. 2, 193–207. CrossrefGoogle Scholar

  • [9]

    A. Cianchi and P. Salani, Overdetermined anisotropic elliptic problems, Math. Ann. 345 (2009), no. 4, 859–881. CrossrefWeb of ScienceGoogle Scholar

  • [10]

    A. Colesanti and P. Cuoghi, The Brunn–Minkowski inequality for the n-dimensional logarithmic capacity of convex bodies, Potential Anal. 22 (2005), no. 3, 289–304. CrossrefGoogle Scholar

  • [11]

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

  • [12]

    A. Colesanti and P. Salani, The Brunn–Minkowski inequality for p-capacity of convex bodies, Math. Ann. 327 (2003), no. 3, 459–479. CrossrefGoogle Scholar

  • [13]

    A. Eremenko and J. L. Lewis, Uniform limits of certain A-harmonic functions with applications to quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), no. 2, 361–375. CrossrefGoogle Scholar

  • [14]

    R. M. Gabriel, An extended principle of the maximum for harmonic functions in 3-dimensions, J. London Math. Soc. 30 (1955), 388–401. Google Scholar

  • [15]

    R. J. Gardner, The Brunn–Minkowski inequality, Bull. Amer. Math. Soc. (N. S.) 39 (2002), no. 3, 355–405. CrossrefGoogle Scholar

  • [16]

    J. B. Garnett and D. E. Marshall, Harmonic Measure, New Math. Monogr. 2, Cambridge University, Cambridge, 2008. Google Scholar

  • [17]

    N. Garofalo and F.-H. Lin, Unique continuation for elliptic operators: A geometric-variational approach, Comm. Pure Appl. Math. 40 (1987), no. 3, 347–366. CrossrefGoogle Scholar

  • [18]

    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer, Berlin, 2001. Google Scholar

  • [19]

    J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Mineola, 2006. Google Scholar

  • [20]

    D. Jerison, A Minkowski problem for electrostatic capacity, Acta Math. 176 (1996), no. 1, 1–47. CrossrefGoogle Scholar

  • [21]

    T. Kilpeläinen and X. Zhong, Growth of entire 𝒜-subharmonic functions, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 1, 181–192. Google Scholar

  • [22]

    I. N. Krol, The behavior of the solutions of a certain quasilinear equation near zero cusps of the boundary, Trudy Mat. Inst. Steklov. 125 (1973), 140–146, 233. Google Scholar

  • [23]

    N. S. Landkof, Foundations of Modern Potential Theory, Grundlehren Math. Wiss. 180, Springer, New York, 1972. Google Scholar

  • [24]

    J. Lewis and K. Nyström, Boundary behavior and the Martin boundary problem for p harmonic functions in Lipschitz domains, Ann. of Math. (2) 172 (2010), no. 3, 1907–1948. Web of ScienceCrossrefGoogle Scholar

  • [25]

    J. L. Lewis, N. Lundström and K. Nyström, Boundary Harnack inequalities for operators of p-Laplace type in Reifenberg flat domains, Perspectives in Partial Differential Equations, Harmonic Analysis and Applications, Proc. Sympos. Pure Math. 79, American Mathematical Society, Providence (2008), 229–266. Google Scholar

  • [26]

    J. L. Lewis and K. Nyström, Boundary behaviour for p harmonic functions in Lipschitz and starlike Lipschitz ring domains, Ann. Sci. Éc. Norm. Supér. (4) 40 (2007), no. 5, 765–813. CrossrefGoogle Scholar

  • [27]

    J. L. Lewis and K. Nyström, Regularity and free boundary regularity for the p Laplacian in Lipschitz and C1 domains, Ann. Acad. Sci. Fenn. Math. 33 (2008), no. 2, 523–548. Google Scholar

  • [28]

    J. L. Lewis and K. Nyström, Regularity and free boundary regularity for the p-Laplace operator in Reifenberg flat and Ahlfors regular domains, J. Amer. Math. Soc. 25 (2012), no. 3, 827–862. CrossrefGoogle Scholar

  • [29]

    J. L. Lewis and K. Nyström, Quasi-linear PDEs and low-dimensional sets, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 7, 1689–1746. CrossrefGoogle Scholar

  • [30]

    G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203–1219. CrossrefGoogle Scholar

  • [31]

    J. Malý, D. Swanson and W. P. Ziemer, The co-area formula for Sobolev mappings, Trans. Amer. Math. Soc. 355 (2003), no. 2, 477–492. CrossrefGoogle Scholar

  • [32]

    R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia Math. Appl. 44, Cambridge University, Cambridge, 1993. Google Scholar

  • [33]

    J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302. CrossrefGoogle Scholar

  • [34]

    P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150. CrossrefGoogle Scholar

About the article

Received: 2018-10-10

Revised: 2019-03-03

Accepted: 2019-03-07

Published Online: 2019-04-06

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: DFG-SFB 1060

Funding Source: Division of Mathematical Sciences

Award identifier / Grant number: DMS-1265996

Award identifier / Grant number: DMS-1440140

This material is based upon work supported by National Science Foundation under Grant No. DMS-1440140 while the first and the third authors were in residence at the MSRI in Berkeley, California, during the Spring 2017 semester. The second author was partially supported by NSF DMS-1265996. The third author was partially supported by the Hausdorff Center for Mathematics as well as DFG-SFB 1060.

Citation Information: Advances in Calculus of Variations, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2018-0064.

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