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

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

Jie Xiao 1,

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.
Keywords: Isoperimetric inequalities, $p$-capacitary potentials, convex plane rings..
Mathematics Subject Classification: Primary: 53A30; Secondary: 35J92, 31A1.
Citation: Jie Xiao. On the variational $p$-capacity problem in the plane. Communications on Pure & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992.  Google Scholar

[6]

M. Flucher, Variational Problems with Concentration, Birkhäuser, 1999. doi: 10.1007/978-3-0348-8687-1.  Google Scholar

[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.  Google Scholar

[8]

W. Hansen and N. Nadirashvili, Isoperimetric inequalities in potential theory, Potential Anal., 3 (1994), 1-14. doi: 10.1007/BF01047833.  Google Scholar

[9]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc., Mineola, New York, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal., 66 (1977), 201-224.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

G. A. Philippin and L. E. Payne, On the conformal capacity problem, Symposia Mathematica, Vol. XXX (Cortona, 1988), 119-136.  Google Scholar

[18]

G. Pólya, Estimating electrostatic capacity, Amer. Math. Monthly, 54 (1947), 201-206.  Google Scholar

[19]

T. Ransford, Potential Theory in the Complex Plane, London Math. Soc. Student Texts 28, Cambridge University Press, 1995. doi: 10.1017/CBO9780511623776.  Google Scholar

[20]

A. S. Romanov, Capacity relations in a flat quadrilateral, Sib. Math. J., 49 (2008), 709-717. doi: 10.1007/s11202-008-0068-y.  Google Scholar

[21]

J. Sarvas, Symmetrization of condensers in $n$-space, Ann. Acad. Sci. Fenn. Ser. AI, 522, 1972, 44 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Xiao, A maximum problem of S.-T. Yau for variational capacity,, , ().   Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992.  Google Scholar

[6]

M. Flucher, Variational Problems with Concentration, Birkhäuser, 1999. doi: 10.1007/978-3-0348-8687-1.  Google Scholar

[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.  Google Scholar

[8]

W. Hansen and N. Nadirashvili, Isoperimetric inequalities in potential theory, Potential Anal., 3 (1994), 1-14. doi: 10.1007/BF01047833.  Google Scholar

[9]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc., Mineola, New York, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal., 66 (1977), 201-224.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

G. A. Philippin and L. E. Payne, On the conformal capacity problem, Symposia Mathematica, Vol. XXX (Cortona, 1988), 119-136.  Google Scholar

[18]

G. Pólya, Estimating electrostatic capacity, Amer. Math. Monthly, 54 (1947), 201-206.  Google Scholar

[19]

T. Ransford, Potential Theory in the Complex Plane, London Math. Soc. Student Texts 28, Cambridge University Press, 1995. doi: 10.1017/CBO9780511623776.  Google Scholar

[20]

A. S. Romanov, Capacity relations in a flat quadrilateral, Sib. Math. J., 49 (2008), 709-717. doi: 10.1007/s11202-008-0068-y.  Google Scholar

[21]

J. Sarvas, Symmetrization of condensers in $n$-space, Ann. Acad. Sci. Fenn. Ser. AI, 522, 1972, 44 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Xiao, A maximum problem of S.-T. Yau for variational capacity,, , ().   Google Scholar

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