Publication Cover
Applicable Analysis
An International Journal
Volume 98, 2019 - Issue 10
285
Views
14
CrossRef citations to date
0
Altmetric
Articles

Radial symmetry for p-harmonic functions in exterior and punctured domains

Pages 1785-1798 | Received 22 Jan 2018, Accepted 30 Mar 2018, Published online: 15 Apr 2018

References

  • Garofalo N, Sartori E. Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a-priori estimates. Adv Differ Equ. 1999;4:137–161.
  • Payne LE, Philippin GA. On some maximum principles involving harmonic functions and their derivatives. SIAM J Math Anal. 1979;10:96–104.
  • Philippin GA. On a free boundary problem in electrostatics. Math Methods Appl Sci. 1990;12:387–392.
  • Weinberger HF. Remark on the preceding paper of Serrin. Arch Ration Mech Anal. 1971;43:319–320.
  • Magnanini R, Poggesi G. On the stability for Alexandrov’s Soap Bubble theorem, to appear in J Anal Math. Forthcoming 2016, arxiv:1610.07036.
  • Magnanini R, Poggesi G. Serrin’s problem and Alexandrov’s Soap Bubble Theorem: stability via integral identities. Forthcoming 2017, arxiv:1708.07392.
  • Reichel W. Radial symmetry for elliptic boundary-value problems on exterior domains. Arch Ration Mech Anal. 1997;137:381–394.
  • Reichel W. Radial symmetry for an electrostatic, a capillarity and some fully nonlinear overdetermined problems on exterior domains. Z Anal Anwend. 1996;15:619–635.
  • Serrin J. A symmetry problem in potential theory. Arch Ration Mech Anal. 1971;43:304–318.
  • Payne LE, Schaefer PW. Duality theorems in some overdetermined boundary value problems. Math Methods Appl Sci. 1989;11:805–819.
  • Garofalo N, Lewis JL. A symmetry result related to some overdetermined boundary value problems. Amer J Math. 1989;111:9–33.
  • Damascelli L, Pacella F. Monotonicity and symmetry results for p-Laplace equations and applications. Adv Differ Equ. 2000;5:1179–1200.
  • Brock F, Henrot A. A symmetry result for an overdetermined elliptic problem using continuous rearrangements and domain derivative. Rend Circ Mat Palermo. 2002;51:375–390.
  • Brandolini B, Nitsch C, Salani P, et al. Serrin-type overdetermined problems: an alternative proof. Arch Rat Mech Anal. 2008;190:267–280.
  • Farina A, Kawohl B. Remarks on an overdetermined boundary value problem. Calc Var Partial Differ Equ. 2008;89:351–357.
  • Cianchi A, Salani P. Overdetermined anisotropic elliptic problems. Math Ann. 2009;345:859–881.
  • Wang G, Xia C. A characterization of the Wulff shape by an overdetermined anisotropic PDE. Arch Ration Mech Anal. 2011;199:99–115.
  • Bianchini C, Ciraolo G. Wulff shape characterizations in overdetermined anisotropic elliptic problems. Comm Par Differ Equ. Forthcoming 2017, arxiv:1703.07111.
  • Magnanini R. Alexandrov, Serrin, Weinberger, Reilly: symmetry and stability by integral identities. Bruno Pini Math Anal Semin. Forthcoming 2017, arxiv:1709.073939.
  • Nitsch C, Trombetti C. The classical overdetermined Serrin problem. Complex Var Elliptic Equ. Forthcoming 2017, arxiv:1711.10787.
  • Kawohl B. Overdetermined problems and the p-Laplacian. Proc Equadiff. 2005;11:1–6.
  • Mendez O, Reichel W. Electrostatic characterization of spheres. Forum Math. 2000;12:223–245.
  • Bianchini C, Ciraolo G, Salani P. An overdetermined problem for the anisotropic capacity. Calc Var Partial Differ Equ. 2016;55(84):24.
  • Colesanti A, Cuoghi P. The Brunn-Minkowski inequality for n-dimensional logarithmic capacity of convex bodies. Potential Anal. 2005;22:289–304.
  • Martensen E. Eine Integralgleichung für die logarithmische Gleichgewichtsbelegung und die Krümmung der Randkurve eines ebenen Gebiets. ZAMM Z Angew Math Mech. 1992;72:T596–T599.
  • Payne LE, Philippin GA. On the conformal capacity problem. In: Talenti G, editor.Geometry of solutions to partial differential equations. Vol. 30, Symp. Mat. London: Academic Press; 1989. p. 119–136.
  • Alessandrini G, Rosset E. Symmetry of singular solutions of degenerate quasilinear elliptic equations. Rend Istit Mat Univ Trieste. 2007;39:1–8.
  • Enciso A, Peralta-Salas D. Symmetry for an overdetermined boundary problem in a punctured domain. Nonlinear Anal. 2009;70:1080–1086.
  • Pólya G, Szegö G. Isoperimetric inequalities in mathematical physics. Vol. 27, Annals of mathematics studies. Princeton: Princeton University Press; 1951.
  • Jauregui JL. The capacity-volume inequality of Poincaré-Faber-Szegö. Notes; 2012. Available from: http://www.math.union.edu/jaureguj/capacity-volume.pdf.
  • Henrot A, Pierre M. Variation et Optimization de Formes: Une Analyse Géométrique. Berlin Heidelberg: Springer-Verlag; 2005.
  • Crasta G, Fragalà I, Gazzola F. On a long standing conjecture by Pölya-Szegö and related topics. Z Angew Math Phys. 2005;56:763–782.
  • Douglas JF, Zhou H-X, Hubbard JB. Hydrodynamic friction and the capacitance of arbitrarily shaped objects. Phys Rev E. 1994;49:5319–5331.
  • Di Benedetto E. C1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 1983;7:827–850.
  • Lewis JL. Regularity of the derivatives of solutions to certain degenerate elliptic equations. Indiana Univ Math J. 1983;32:849–858.
  • Tolksdorf P. Regularity for a more general class of quasilinear elliptic equations. J Differ Equ. 1984;51:126–150.
  • Gilbarg D, Trudinger NS. Elliptic partial differential equations of second order. Berlin: Springer-Verlag; 1998.
  • Vogel AL. Symmetry and regularity for general regions having a solution to certain overdetermined boundary value problems. Atti Sem Mat Fis Univ Modena. 1992;50:443–484.
  • Lieberman GM. Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 1988;12:1203–1219.
  • Alt HW, Caffarelli LA. Existence and regularity for a minimum problem with free boundary. J Reine Angew Math. 1981;325:105–144.
  • Alt HW, Caffarelli LA, Friedman A. A free boundary problem for quasilinear elliptic equations. Ann Sc Norm Sup Pisa. 1984;11:1–44.
  • Kichenassamy S, Véron L. Singular solutions of the p-Laplace equation. Math Ann. 1986;275:599–615.
  • Colesanti A, Salani P. The Brunn-Minkowski inequality for p-capacity of convex bodies. Math Ann. 2003;327:459–479.
  • Heinonen J, Kilpeläinen T, Martio O. Nonlinear potential theory of degenerate elliptic equations. Oxford: Clarendon Press; 1993.
  • Burago YuD, Zalgaller VA. Geometric inequalities. Berlin: Springer-Verlag; 1988.
  • Gehring F. Inequalities for condensers, hyperbolic capacity, and extremal length. Michigan Math J. 1971;18:1–20.
  • Alexandrov AD. A characteristic property of spheres. Ann Math Pura Appl. 1962;58:303–315.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.