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

Murat Akman; John Lewis; Olli Saari; Andrew Vogel

doi:10.1515/acv-2018-0064

Published by De Gruyter April 6, 2019

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

Murat Akman , John Lewis , Olli Saari and Andrew Vogel

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2018-0064

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Abstract

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 2≤n≤p<∞. For a convex compact set E in ℝn, 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:

[C𝒜⁢(λ⁢E1+(1-λ)⁢E2)]1p-n≥λ⁢[C𝒜⁢(E1)]1p-n+(1-λ)⁢[C𝒜⁢(E2)]1p-n

when n<p<∞, 0≤λ≤1, and E1,E2 are nonempty convex compact sets in ℝn. While p=n then

C𝒜⁢(λ⁢E1+(1-λ)⁢E2)≥λ⁢C𝒜⁢(E1)+(1-λ)⁢C𝒜⁢(E2),

where 0≤λ≤1 and E1,E2 are convex compact sets in ℝn 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 ℝn 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 n≤p<∞. 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

Communicated by Frank Duzaar

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

Funding statement: 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.

Acknowledgements

Research for this article was carried out while the first and fourth author were visiting the Department of Mathematics at the University of Kentucky, the authors would like to thank the department for its hospitality.

References

[1] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer, Berlin, 1996. 10.1007/978-3-662-03282-4Search in 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. 10.5186/aasfm.2014.3923Search in Google 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. 10.1090/memo/1348Search in Google Scholar

[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. 10.1016/j.aim.2017.01.013Search in Google 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. 10.1007/s13324-012-0023-8Search in Google Scholar

[6] C. Borell, Capacitary inequalities of the Brunn–Minkowski type, Math. Ann. 263 (1983), no. 2, 179–184. 10.1007/BF01456879Search in Google 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. 10.24033/asens.1480Search in Google 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. 10.1007/978-3-642-55925-9_40Search in Google Scholar

[9] A. Cianchi and P. Salani, Overdetermined anisotropic elliptic problems, Math. Ann. 345 (2009), no. 4, 859–881. 10.1007/s00208-009-0386-9Search in Google 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. 10.1007/s11118-004-1326-7Search in Google 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. 10.1016/j.aim.2015.06.022Search in Google 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. 10.1007/s00208-003-0460-7Search in Google 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. 10.5186/aasfm.1991.1609Search in Google 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. 10.1112/jlms/s1-30.4.388Search in Google Scholar

[15] R. J. Gardner, The Brunn–Minkowski inequality, Bull. Amer. Math. Soc. (N. S.) 39 (2002), no. 3, 355–405. 10.1090/S0273-0979-02-00941-2Search in Google Scholar

[16] J. B. Garnett and D. E. Marshall, Harmonic Measure, New Math. Monogr. 2, Cambridge University, Cambridge, 2008. Search in 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. 10.1002/cpa.3160400305Search in Google Scholar

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

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

[20] D. Jerison, A Minkowski problem for electrostatic capacity, Acta Math. 176 (1996), no. 1, 1–47. 10.1007/BF02547334Search in Google Scholar

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

[23] N. S. Landkof, Foundations of Modern Potential Theory, Grundlehren Math. Wiss. 180, Springer, New York, 1972. 10.1007/978-3-642-65183-0Search in 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. 10.4007/annals.2010.172.1907Search in Google 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. 10.1090/pspum/079/2500495Search in 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. 10.1016/j.ansens.2007.09.001Search in Google 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. Search in 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. 10.1090/S0894-0347-2011-00726-1Search in Google 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. 10.4171/JEMS/797Search in Google Scholar

[30] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203–1219. 10.1016/0362-546X(88)90053-3Search in Google 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. 10.1090/S0002-9947-02-03091-XSearch in Google Scholar

[32] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia Math. Appl. 44, Cambridge University, Cambridge, 1993. 10.1017/CBO9780511526282Search in Google Scholar

[33] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302. 10.1007/BF02391014Search in Google Scholar

[34] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150. 10.1016/0022-0396(84)90105-0Search in Google Scholar

Received: 2018-10-10

Revised: 2019-03-03

Accepted: 2019-03-07

Published Online: 2019-04-06

Published in Print: 2021-04-01

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

Abstract

Acknowledgements

References

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