# American Institute of Mathematical Sciences

May  2015, 14(3): 959-968. doi: 10.3934/cpaa.2015.14.959

## On the variational $p$-capacity problem in the plane

 1 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7

Received  July 2014 Revised  December 2014 Published  March 2015

Under $1 < p \leq 2$, this note presents some new optimal isoperimetric type properties of the variational $p$-capacitary potentials on convex plane rings.
Citation: Jie Xiao. On the variational $p$-capacity problem in the plane. Communications on Pure and Applied Analysis, 2015, 14 (3) : 959-968. doi: 10.3934/cpaa.2015.14.959
##### References:
 [1] G. Alessandrini, Isoperimetric inequalities for the length of level lines of solutions of quasilinear capacity problems in the plane, Z. Angew. Math. Phys., 40 (1989), 920-924. doi: 10.1007/BF00945812. [2] V. Andrievskii, W. Hasen and N. Nadirashvilli, Isoperimetric inequalities for capacities in the plane, Math. Ann., 292 (1992), 191-195. doi: 10.1007/BF01444617. [3] R. W. Barnard, K. Pearce and A. Y. Solynin, An isoperimetric inequality for logarithmic capacity, Ann. Acad. Sci. Fenn. Math., 27 (2002), 419-436. [4] D. Betsakos, Geometric versions of Schwarz's lemma for quasiregular mappings, Proc. Amer. Math. Soc., 139 (2010), 1397-1407. doi: 10.1090/S0002-9939-2010-10604-4. [5] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992. [6] M. Flucher, Variational Problems with Concentration, Birkhäuser, 1999. doi: 10.1007/978-3-0348-8687-1. [7] L. E. Fraenkel, A lower bound for electrostatic capacity in the plane, Proc. Royal Soc. Edin., 88 (1981), 267-273. doi: 10.1017/S0308210500020114. [8] W. Hansen and N. Nadirashvili, Isoperimetric inequalities in potential theory, Potential Anal., 3 (1994), 1-14. doi: 10.1007/BF01047833. [9] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc., Mineola, New York, 2006. [10] A. Henrot and H. Shahgholian, Existence of classical solutions to a free boundary problem for the $p$-Laplace operator: I. the exterior convex case, J. Reine Angew. Math., 521 (2000), 85-97. doi: 10.1515/crll.2000.031. [11] P. Laurence, On the convexity of geometric functionals of level for solutions of certain elliptic partial differential equations, Z. Angew. Math. Phys., 40 (1989), 258-284. doi: 10.1007/BF00945002. [12] J. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal., 66 (1977), 201-224. [13] J. Lewis, Applications of Boundary Harnack Inequalities for $p$ Harmonic Functions and Related Topics, Regularity estimates for nonlinear elliptic and parabolic problems, 1-72, Lecture Notes in Math., 2045, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-27145-8_1. [14] M. Longinetti, Some isoperimetric inequalities for the level curves of capacity and Green's functions on convex plane domains, SIAM J. Math. Anal., 19 (1988), 377-389. doi: 10.1137/0519028. [15] V. Maz'ya, Conductor and capacitary inequalities for functions on topological spaces and their applications to Sobolev-type imbeddings, J. Funct. Anal., 224 (2005), 408-430. doi: 10.1016/j.jfa.2004.09.009. [16] V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, 2nd, revised and augmented edition, Springer, 2011. doi: 10.1007/978-3-642-15564-2. [17] G. A. Philippin and L. E. Payne, On the conformal capacity problem, Symposia Mathematica, Vol. XXX (Cortona, 1988), 119-136. [18] G. Pólya, Estimating electrostatic capacity, Amer. Math. Monthly, 54 (1947), 201-206. [19] T. Ransford, Potential Theory in the Complex Plane, London Math. Soc. Student Texts 28, Cambridge University Press, 1995. doi: 10.1017/CBO9780511623776. [20] A. S. Romanov, Capacity relations in a flat quadrilateral, Sib. Math. J., 49 (2008), 709-717. doi: 10.1007/s11202-008-0068-y. [21] J. Sarvas, Symmetrization of condensers in $n$-space, Ann. Acad. Sci. Fenn. Ser. AI, 522, 1972, 44 pp. [22] D. Smets and J. Schaftingen, Desingularization of vortices for the Euler equation, Arch. Rational Mech. Anal., 198 (2010), 869-925. doi: 10.1007/s00205-010-0293-y. [23] A. Y. Solynin and V. A. Zalgaller, An isoperimetric inequality for logarithmic capacity of polygons, Ann. Math., 159 (2004), 277-303. doi: 10.4007/annals.2004.159.277. [24] J. Xiao, A maximum problem of S.-T. Yau for variational capacity,, , ().

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##### References:
 [1] G. Alessandrini, Isoperimetric inequalities for the length of level lines of solutions of quasilinear capacity problems in the plane, Z. Angew. Math. Phys., 40 (1989), 920-924. doi: 10.1007/BF00945812. [2] V. Andrievskii, W. Hasen and N. Nadirashvilli, Isoperimetric inequalities for capacities in the plane, Math. Ann., 292 (1992), 191-195. doi: 10.1007/BF01444617. [3] R. W. Barnard, K. Pearce and A. Y. Solynin, An isoperimetric inequality for logarithmic capacity, Ann. Acad. Sci. Fenn. Math., 27 (2002), 419-436. [4] D. Betsakos, Geometric versions of Schwarz's lemma for quasiregular mappings, Proc. Amer. Math. Soc., 139 (2010), 1397-1407. doi: 10.1090/S0002-9939-2010-10604-4. [5] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992. [6] M. Flucher, Variational Problems with Concentration, Birkhäuser, 1999. doi: 10.1007/978-3-0348-8687-1. [7] L. E. Fraenkel, A lower bound for electrostatic capacity in the plane, Proc. Royal Soc. Edin., 88 (1981), 267-273. doi: 10.1017/S0308210500020114. [8] W. Hansen and N. Nadirashvili, Isoperimetric inequalities in potential theory, Potential Anal., 3 (1994), 1-14. doi: 10.1007/BF01047833. [9] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc., Mineola, New York, 2006. [10] A. Henrot and H. Shahgholian, Existence of classical solutions to a free boundary problem for the $p$-Laplace operator: I. the exterior convex case, J. Reine Angew. Math., 521 (2000), 85-97. doi: 10.1515/crll.2000.031. [11] P. Laurence, On the convexity of geometric functionals of level for solutions of certain elliptic partial differential equations, Z. Angew. Math. Phys., 40 (1989), 258-284. doi: 10.1007/BF00945002. [12] J. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal., 66 (1977), 201-224. [13] J. Lewis, Applications of Boundary Harnack Inequalities for $p$ Harmonic Functions and Related Topics, Regularity estimates for nonlinear elliptic and parabolic problems, 1-72, Lecture Notes in Math., 2045, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-27145-8_1. [14] M. Longinetti, Some isoperimetric inequalities for the level curves of capacity and Green's functions on convex plane domains, SIAM J. Math. Anal., 19 (1988), 377-389. doi: 10.1137/0519028. [15] V. Maz'ya, Conductor and capacitary inequalities for functions on topological spaces and their applications to Sobolev-type imbeddings, J. Funct. Anal., 224 (2005), 408-430. doi: 10.1016/j.jfa.2004.09.009. [16] V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, 2nd, revised and augmented edition, Springer, 2011. doi: 10.1007/978-3-642-15564-2. [17] G. A. Philippin and L. E. Payne, On the conformal capacity problem, Symposia Mathematica, Vol. XXX (Cortona, 1988), 119-136. [18] G. Pólya, Estimating electrostatic capacity, Amer. Math. Monthly, 54 (1947), 201-206. [19] T. Ransford, Potential Theory in the Complex Plane, London Math. Soc. Student Texts 28, Cambridge University Press, 1995. doi: 10.1017/CBO9780511623776. [20] A. S. Romanov, Capacity relations in a flat quadrilateral, Sib. Math. J., 49 (2008), 709-717. doi: 10.1007/s11202-008-0068-y. [21] J. Sarvas, Symmetrization of condensers in $n$-space, Ann. Acad. Sci. Fenn. Ser. AI, 522, 1972, 44 pp. [22] D. Smets and J. Schaftingen, Desingularization of vortices for the Euler equation, Arch. Rational Mech. Anal., 198 (2010), 869-925. doi: 10.1007/s00205-010-0293-y. [23] A. Y. Solynin and V. A. Zalgaller, An isoperimetric inequality for logarithmic capacity of polygons, Ann. Math., 159 (2004), 277-303. doi: 10.4007/annals.2004.159.277. [24] J. Xiao, A maximum problem of S.-T. Yau for variational capacity,, , ().
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