×

Prescribing capacitary curvature measures on planar convex domains. (English) Zbl 1445.52001

Let \(\mu\) be a finite non-negative Borel measure on the one-dimensional unit sphere. The author considers the question if there exists a bounded non-empty open convex set \(\Omega \subset \mathbb{R}^2\) such that \(d\mu_p(\overline{\Omega}, \cdot) = d\mu(\cdot)\). He shows that the answer is positive if and only if \(\mu\) has the centroid at the origin and its support \(\mathrm{supp}(\mu)\) does not comprise any pair of antipodal points.

MSC:

52A15 Convex sets in \(3\) dimensions (including convex surfaces)
52A10 Convex sets in \(2\) dimensions (including convex curves)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adamowicz, T., On \(p\)-harmonic mappings in the plane, Nonlinear Anal., 71, 502-511 (2009) · Zbl 1170.35381
[2] Adamowicz, T., The geometry of planar \(p\)-harmonic mappings: convexity, level curves and the isoperimetric inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 14, 263-292 (2015) · Zbl 1323.31007
[3] Aronsson, G.: Aspects of p-harmonic functions in the plane. Summer School in Potential Theory (Joensuu, 1990), 9-34, Joensuun Yliop. Luonnont. Julk., 26, Univ. Joensuu, Joensuu (1992) · Zbl 0756.31007
[4] Barnard, RW; Pearce, K.; Solynin, AY, An isoperimetric inequality for logarithmic capacity, Ann. Acad. Sci. Fenn. Math., 27, 419-436 (2002) · Zbl 1017.30033
[5] Borell, C., Hitting probability of killed Brownian motion: a study on geometric regularity, Ann. Sci. Ecole Norm. Supér., 17, 451-467 (1984) · Zbl 0573.60067
[6] Böröczky, KJ; Trinh, HT, The planar \(L_p\)-Minkowski problem for \(0<p<1\), Adv. Appl. Math., 87, 58-81 (2017) · Zbl 1376.52013
[7] Caffarelli, L., Interior a priori estimates for solutons of fully non-linear equations, Ann. Math., 131, 189-213 (1989) · Zbl 0692.35017
[8] Caffarelli, L., A localization property of viscosity solutions to the Monge-Ampére equation and their strict convexity, Ann. Math., 131, 129-134 (1990) · Zbl 0704.35045
[9] Caffarelli, L., Interior \(W^{2, p}\) estimates for solutions of the Monge-Ampére equation, Ann. Math., 131, 135-150 (1990) · Zbl 0704.35044
[10] Caffarelli, L., Some regularity properties of solutions to the Monge-Ampére equation, Commun. Pure Appl. Math., 44, 965-969 (1991) · Zbl 0761.35028
[11] Caffarelli, L.; Jerison, D.; Lieb, EH, On the case of equality in the Brunn-Minkowski inequality for capacity, Adv. Math., 117, 193-207 (1996) · Zbl 0847.31005
[12] Cheng, S-Y; Yau, S-T, On the regularity of the solution of the \(n\)-dimensional Minkiwski problem, Commun. Pure Appl. Math., 29, 495-561 (1976) · Zbl 0363.53030
[13] Colesanti, A.; Cuoghi, P., The Brunn-Minkowski inequality for the \(n\)-dimensional loarithmic capacity, Potential Anal., 22, 289-304 (2005) · Zbl 1074.31007
[14] 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) · Zbl 1327.31024
[15] Colesanti, A.; Salani, P., The Brunn-Minkowski inequality for \(p\)-capacity of convex bodies, Math. Ann., 327, 459-479 (2003) · Zbl 1052.31005
[16] Dahlberg, BEJ, Estimates of harmonic measure, Arch. Ration. Mech. Anal., 65, 275-288 (1977) · Zbl 0406.28009
[17] Evans, LC; Gariepy, RF, Measure Theory and Fine Properties of Functions (1992), Boca Raton: CRC Press, Boca Raton · Zbl 0804.28001
[18] Gardner, RJ; Hartenstine, D., Capacities, surface area, and radial sums, Adv. Math., 221, 601-626 (2009) · Zbl 1163.52001
[19] Gutiérrez, C.E.: The Monge-Ampère Equation. Progress in Nonlinear Differential Equations and Their Applications, Vol. 44, Birkhäuser (2001) · Zbl 0989.35052
[20] Gutiérrez, CE; Hartenstine, D., Regularity of weak solutions to the Monge-Ampère equation, Trans. Am. Math. Soc., 355, 2477-2500 (2003) · Zbl 1088.35012
[21] Jerison, D., Prescribing harmonic measure on convex domains, Invent. Math., 105, 375-400 (1991) · Zbl 0754.31007
[22] Jerison, D., A Minkowski problem for electrostatic capacity, Acta Math., 176, 1-47 (1996) · Zbl 0880.35041
[23] Jerison, D., The direct method in the calculus of variations for convex bodies, Adv. Math., 122, 262-279 (1996) · Zbl 0920.35056
[24] Klain, DA, The Minkowski problem for polytopes, Adv. Math., 185, 270-288 (2004) · Zbl 1053.52015
[25] Lewis, JL, Capacitary functions in convex rings, Arch. Ration. Mech. Anal., 66, 201-224 (1977) · Zbl 0393.46028
[26] Lewis, JL; Nyström, K., Boundary behaviour for \(p\)-harmonic functions in Lipschitz and starlike Lipschitz ring domains, Ann. Sci. École Norm. Sup., 40, 765-813 (2007) · Zbl 1134.31008
[27] Lewis, JL; Nyström, K., Regularity and free boundary regularity for the \(p\)-Laplacian in Lipschitz and \(C^1\)-domains, Ann. Acad. Sci. Fenn. Math., 33, 523-548 (2008) · Zbl 1202.35110
[28] Lieberman, GM, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12, 1203-1219 (1988) · Zbl 0675.35042
[29] Ludwig, M.; Xiao, J.; Zhang, G., Sharp convex Lorentz-Sobolev inequalities, Math. Ann., 350, 169-197 (2011) · Zbl 1220.26020
[30] Lutwak, E., The Brunn-Minkowski-Firey theory. I. mixed volumes and the Minkowksi problem, J. Differ. Geom., 38, 131-150 (1993) · Zbl 0788.52007
[31] Schneider, R., Convex Bodies: The Brunn-Minkowski Theory (1993), Cambridge: Cambridge Univ. Press, Cambridge · Zbl 0798.52001
[32] Solynin, AY; Zalgaller, VA, An isoperimetric inequality for logarithmic capacity of polygons, Ann. Math., 159, 277-303 (2004) · Zbl 1060.31001
[33] Umanskiy, V., On solvability of two-dimensional \(L_p\)-Minkwoski problem, Adv. Math., 180, 176-186 (2003) · Zbl 1048.52001
[34] Xiao, J., On the variational \(p\)-capacity problem in the plane, Commun. Pure Appl. Anal., 14, 959-968 (2015) · Zbl 1315.31001
[35] Xiao, J., Exploiting log-capacity in convex geometry, Asian J. Math., 22, 955-980 (2018) · Zbl 1402.31006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.