### Abstract

A Hadamard variational formula for p-capacity of convex bodies in Rn is established when 1 < p< n. The formula is applied to solve the Minkowski problem for p-capacity which involves a degenerate Monge-Ampère type equation. Uniqueness for the Minkowski problem for p-capacity is established when 1 < p< n and existence and regularity when 1 < p< 2. These results are (non-linear) extensions of the now classical solution of Jerison of the Minkowski problem for electrostatic capacity (p = 2).

Original language | English (US) |
---|---|

Article number | 5138 |

Pages (from-to) | 1511-1588 |

Number of pages | 78 |

Journal | Advances in Mathematics |

Volume | 285 |

DOIs | |

State | Published - Nov 5 2015 |

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### Keywords

- Convex domain
- Existence
- Minkowski inequality
- Minkowski problem
- Monge-Ampére equation
- P-capacitary measure
- P-capacity
- P-equilibrium potential
- P-Laplacian
- Regularity
- Uniqueness
- Variational formula

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*285*, 1511-1588. [5138]. https://doi.org/10.1016/j.aim.2015.06.022

**The Hadamard variational formula and the Minkowski problem for p-capacity.** / Colesanti, A.; Nyström, K.; Salani, P.; Xiao, J.; Yang, D.; Zhang, G.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 285, 5138, pp. 1511-1588. https://doi.org/10.1016/j.aim.2015.06.022

}

TY - JOUR

T1 - The Hadamard variational formula and the Minkowski problem for p-capacity

AU - Colesanti, A.

AU - Nyström, K.

AU - Salani, P.

AU - Xiao, J.

AU - Yang, D.

AU - Zhang, G.

PY - 2015/11/5

Y1 - 2015/11/5

N2 - A Hadamard variational formula for p-capacity of convex bodies in Rn is established when 1 < p< n. The formula is applied to solve the Minkowski problem for p-capacity which involves a degenerate Monge-Ampère type equation. Uniqueness for the Minkowski problem for p-capacity is established when 1 < p< n and existence and regularity when 1 < p< 2. These results are (non-linear) extensions of the now classical solution of Jerison of the Minkowski problem for electrostatic capacity (p = 2).

AB - A Hadamard variational formula for p-capacity of convex bodies in Rn is established when 1 < p< n. The formula is applied to solve the Minkowski problem for p-capacity which involves a degenerate Monge-Ampère type equation. Uniqueness for the Minkowski problem for p-capacity is established when 1 < p< n and existence and regularity when 1 < p< 2. These results are (non-linear) extensions of the now classical solution of Jerison of the Minkowski problem for electrostatic capacity (p = 2).

KW - Convex domain

KW - Existence

KW - Minkowski inequality

KW - Minkowski problem

KW - Monge-Ampére equation

KW - P-capacitary measure

KW - P-capacity

KW - P-equilibrium potential

KW - P-Laplacian

KW - Regularity

KW - Uniqueness

KW - Variational formula

UR - http://www.scopus.com/inward/record.url?scp=84941662511&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84941662511&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2015.06.022

DO - 10.1016/j.aim.2015.06.022

M3 - Article

AN - SCOPUS:84941662511

VL - 285

SP - 1511

EP - 1588

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 5138

ER -