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

Jie Xiao

doi:10.3934/cpaa.2015.14.959

Previous Article
Time-dependent singularities in the heat equation
CPAA Home
This Issue
Next Article
KAM Tori for generalized Benjamin-Ono equation

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

Abstract
Related Papers

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. 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. 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. Google Scholar
[4]	D. Betsakos, Geometric versions of Schwarz's lemma for quasiregular mappings,, Proc. Amer. Math. Soc., 139 (2010), 1397. 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. doi: 10.1017/S0308210500020114. Google Scholar
[8]	W. Hansen and N. Nadirashvili, Isoperimetric inequalities in potential theory,, Potential Anal., 3 (1994), 1. doi: 10.1007/BF01047833. Google Scholar
[9]	J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Dover Publications, (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. 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. doi: 10.1007/BF00945002. Google Scholar
[12]	J. Lewis, Capacitary functions in convex rings,, Arch. Rational Mech. Anal., 66 (1977), 201. 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, 2045 (2012), 1. 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. 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. doi: 10.1016/j.jfa.2004.09.009. Google Scholar
[16]	V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations,, 2nd, (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, (1988), 119. Google Scholar
[18]	G. Pólya, Estimating electrostatic capacity,, Amer. Math. Monthly, 54 (1947), 201. Google Scholar
[19]	T. Ransford, Potential Theory in the Complex Plane,, London Math. Soc. Student Texts 28, 28 (1995). doi: 10.1017/CBO9780511623776. Google Scholar
[20]	A. S. Romanov, Capacity relations in a flat quadrilateral,, Sib. Math. J., 49 (2008), 709. 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). Google Scholar
[22]	D. Smets and J. Schaftingen, Desingularization of vortices for the Euler equation,, Arch. Rational Mech. Anal., 198 (2010), 869. 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. 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. 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. 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. Google Scholar
[4]	D. Betsakos, Geometric versions of Schwarz's lemma for quasiregular mappings,, Proc. Amer. Math. Soc., 139 (2010), 1397. 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. doi: 10.1017/S0308210500020114. Google Scholar
[8]	W. Hansen and N. Nadirashvili, Isoperimetric inequalities in potential theory,, Potential Anal., 3 (1994), 1. doi: 10.1007/BF01047833. Google Scholar
[9]	J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Dover Publications, (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. 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. doi: 10.1007/BF00945002. Google Scholar
[12]	J. Lewis, Capacitary functions in convex rings,, Arch. Rational Mech. Anal., 66 (1977), 201. 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, 2045 (2012), 1. 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. 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. doi: 10.1016/j.jfa.2004.09.009. Google Scholar
[16]	V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations,, 2nd, (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, (1988), 119. Google Scholar
[18]	G. Pólya, Estimating electrostatic capacity,, Amer. Math. Monthly, 54 (1947), 201. Google Scholar
[19]	T. Ransford, Potential Theory in the Complex Plane,, London Math. Soc. Student Texts 28, 28 (1995). doi: 10.1017/CBO9780511623776. Google Scholar
[20]	A. S. Romanov, Capacity relations in a flat quadrilateral,, Sib. Math. J., 49 (2008), 709. 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). Google Scholar
[22]	D. Smets and J. Schaftingen, Desingularization of vortices for the Euler equation,, Arch. Rational Mech. Anal., 198 (2010), 869. 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. doi: 10.4007/annals.2004.159.277. Google Scholar
[24]	J. Xiao, A maximum problem of S.-T. Yau for variational capacity,, , (). Google Scholar

[1]	Matteo Focardi, Maria Stella Gelli, Giovanni Pisante. On a 1-capacitary type problem in the plane. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1319-1333. doi: 10.3934/cpaa.2010.9.1319
[2]	Marita Thomas. Uniform Poincaré-Sobolev and isoperimetric inequalities for classes of domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2741-2761. doi: 10.3934/dcds.2015.35.2741
[3]	Jorge Tejero. Reconstruction of rough potentials in the plane. Inverse Problems & Imaging, 2019, 13 (4) : 863-878. doi: 10.3934/ipi.2019039
[4]	Eric Baer, Alessio Figalli. Characterization of isoperimetric sets inside almost-convex cones. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 1-14. doi: 10.3934/dcds.2017001
[5]	Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129
[6]	Lorenzo Brasco, Eleonora Cinti. On fractional Hardy inequalities in convex sets. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4019-4040. doi: 10.3934/dcds.2018175
[7]	Alessandro Ferriero, Nicola Fusco. A note on the convex hull of sets of finite perimeter in the plane. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 103-108. doi: 10.3934/dcdsb.2009.11.103
[8]	Lauri Harhanen, Nuutti Hyvönen. Convex source support in half-plane. Inverse Problems & Imaging, 2010, 4 (3) : 429-448. doi: 10.3934/ipi.2010.4.429
[9]	Misha Bialy. Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3903-3913. doi: 10.3934/dcds.2013.33.3903
[10]	Kohei Nakamura. An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020385
[11]	Anderson Silva, C. Polcino Milies. Cyclic codes of length $ 2p^n $ over finite chain rings. Advances in Mathematics of Communications, 2020, 14 (2) : 233-245. doi: 10.3934/amc.2020017
[12]	Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5049-5058. doi: 10.3934/dcds.2013.33.5049
[13]	Annalisa Cesaroni, Matteo Novaga. The isoperimetric problem for nonlocal perimeters. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 425-440. doi: 10.3934/dcdss.2018023
[14]	Jorge A. Becerril, Javier F. Rosenblueth. Necessity for isoperimetric inequality constraints. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1129-1158. doi: 10.3934/dcds.2017047
[15]	Duanzhi Zhang. $P$-cyclic symmetric closed characteristics on compact convex $P$-cyclic symmetric hypersurface in R²ⁿ. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 947-964. doi: 10.3934/dcds.2013.33.947
[16]	Vitali Liskevich, Igor I. Skrypnik, Zeev Sobol. Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1731-1744. doi: 10.3934/cpaa.2013.12.1731
[17]	Simona Fornaro, Federica Gregorio, Abdelaziz Rhandi. Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2357-2372. doi: 10.3934/cpaa.2016040
[18]	Julian Braun, Bernd Schmidt. On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth. Networks & Heterogeneous Media, 2013, 8 (4) : 879-912. doi: 10.3934/nhm.2013.8.879
[19]	Woocheol Choi, Yong-Cheol Kim. $L^p$ mapping properties for nonlocal Schrödinger operators with certain potentials. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5811-5834. doi: 10.3934/dcds.2018253
[20]	Gerhard Knieper, Norbert Peyerimhoff. Ergodic properties of isoperimetric domains in spheres. Journal of Modern Dynamics, 2008, 2 (2) : 339-358. doi: 10.3934/jmd.2008.2.339

2019 Impact Factor: 1.105

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors