A characterization of balls through optimal concavity for potential functions
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- by Paolo Salani
- Proc. Amer. Math. Soc. 143 (2015), 173-183
- DOI: https://doi.org/10.1090/S0002-9939-2014-12196-4
- Published electronically: August 28, 2014
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Abstract:
In this short note two unconventional overdetermined problems are considered. Let $p\in (1,n)$; first, the following is proved: if $\Omega$ is a bounded domain in $\mathbb {R}^n$ whose $p$-capacitary potential function $u$ has two homotetic convex level sets, then $\Omega$ is a ball. Then, as an application, we obtain the following: if $\Omega$ is a convex domain in $\mathbb {R}^n$ whose $p$-capacitary potential function $u$ is $(1-p)/(n-p)$-concave (i.e. $u^{(1-p)/(n-p)}$ is convex), then $\Omega$ is a ball.References
- Virginia Agostiniani and Rolando Magnanini, Symmetries in an overdetermined problem for the Green’s function, Discrete Contin. Dyn. Syst. Ser. S 4 (2011), no. 4, 791–800. MR 2746441, DOI 10.3934/dcdss.2011.4.791
- Chiara Bianchini, A Bernoulli problem with non-constant gradient boundary constraint, Appl. Anal. 91 (2012), no. 3, 517–527. MR 2876741, DOI 10.1080/00036811.2010.549479
- Christer Borell, Capacitary inequalities of the Brunn-Minkowski type, Math. Ann. 263 (1983), no. 2, 179–184. MR 698001, DOI 10.1007/BF01456879
- Luis A. Caffarelli, David Jerison, and Elliott H. Lieb, On the case of equality in the Brunn-Minkowski inequality for capacity, Adv. Math. 117 (1996), no. 2, 193–207. MR 1371649, DOI 10.1006/aima.1996.0008
- Andrea Cianchi and Paolo Salani, Overdetermined anisotropic elliptic problems, Math. Ann. 345 (2009), no. 4, 859–881. MR 2545870, DOI 10.1007/s00208-009-0386-9
- G. Ciraolo, R. Magnanini, S. Sakaguchi, Symmetry of minimizers with a level surface parallel to the boundary, preprint, 2012. arXiv:1203.5295v2
- Andrea Colesanti, Brunn-Minkowski inequalities for variational functionals and related problems, Adv. Math. 194 (2005), no. 1, 105–140. MR 2141856, DOI 10.1016/j.aim.2004.06.002
- Andrea Colesanti and Paolo Salani, The Brunn-Minkowski inequality for $p$-capacity of convex bodies, Math. Ann. 327 (2003), no. 3, 459–479. MR 2021025, DOI 10.1007/s00208-003-0460-7
- Cristian Enache and Shigeru Sakaguchi, Some fully nonlinear elliptic boundary value problems with ellipsoidal free boundaries, Math. Nachr. 284 (2011), no. 14-15, 1872–1879. MR 2838287, DOI 10.1002/mana.200810170
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- R. M. Gabriel, A result concerning convex level surfaces of $3$-dimensional harmonic functions, J. London Math. Soc. 32 (1957), 286–294. MR 90662, DOI 10.1112/jlms/s1-32.3.286
- R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 355–405. MR 1898210, DOI 10.1090/S0273-0979-02-00941-2
- Antoine Henrot and Gérard A. Philippin, Some overdetermined boundary value problems with elliptical free boundaries, SIAM J. Math. Anal. 29 (1998), no. 2, 309–320. MR 1616562, DOI 10.1137/S0036141096307217
- Bernhard Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, vol. 1150, Springer-Verlag, Berlin, 1985. MR 810619, DOI 10.1007/BFb0075060
- Oliver Dimon Kellogg, Foundations of potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. Reprint from the first edition of 1929. MR 0222317
- Alan U. Kennington, Power concavity and boundary value problems, Indiana Univ. Math. J. 34 (1985), no. 3, 687–704. MR 794582, DOI 10.1512/iumj.1985.34.34036
- John L. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal. 66 (1977), no. 3, 201–224. MR 477094, DOI 10.1007/BF00250671
- Osvaldo Mendez and Wolfgang Reichel, Electrostatic characterization of spheres, Forum Math. 12 (2000), no. 2, 223–245. MR 1740890, DOI 10.1515/form.2000.005
- Paolo Salani, Convexity of solutions and Brunn-Minkowski inequalities for Hessian equations in $\Bbb R^3$, Adv. Math. 229 (2012), no. 3, 1924–1948. MR 2871162, DOI 10.1016/j.aim.2011.12.009
- Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
- James Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318. MR 333220, DOI 10.1007/BF00250468
- Philip W. Schaefer, On nonstandard overdetermined boundary value problems, Proceedings of the Third World Congress of Nonlinear Analysts, Part 4 (Catania, 2000), 2001, pp. 2203–2212. MR 1971630, DOI 10.1016/S0362-546X(01)00345-5
- Henrik Shahgholian, Diversifications of Serrin’s and related symmetry problems, Complex Var. Elliptic Equ. 57 (2012), no. 6, 653–665. MR 2916825, DOI 10.1080/17476933.2010.504848
Bibliographic Information
- Paolo Salani
- Affiliation: DiMaI - Departimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy
- Email: paolo.salani@unifi.it
- Received by editor(s): October 28, 2012
- Received by editor(s) in revised form: March 6, 2013
- Published electronically: August 28, 2014
- Communicated by: Joachim Krieger
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 143 (2015), 173-183
- MSC (2010): Primary 35N25, 35R25, 35R30, 35B06, 52A40
- DOI: https://doi.org/10.1090/S0002-9939-2014-12196-4
- MathSciNet review: 3272742