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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 |
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,, , ().
|
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. |
[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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