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

A. Colesanti; K. Nyström; P. Salani; J. Xiao; D. Yang; G. Zhang

doi:10.1016/j.aim.2015.06.022

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

A. Colesanti, K. Nyström, P. Salani, J. Xiao, D. Yang, G. Zhang

Mathematics

Research output: Contribution to journal › Article › peer-review

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)
Pages (from-to)	1511-1588
Number of pages	78
Journal	Advances in Mathematics
Volume	285
DOIs	https://doi.org/10.1016/j.aim.2015.06.022
State	Published - Nov 5 2015

Keywords

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

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2015.06.022

Cite this

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title = "The Hadamard variational formula and the Minkowski problem for p-capacity",

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-Amp{\`e}re 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).",

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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-Laplacian

KW - P-capacitary measure

KW - P-capacity

KW - P-equilibrium potential

KW - Regularity

KW - Uniqueness

KW - Variational formula

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JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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