The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity

In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of Brunn-Minkowski type for a nonlinear capacity, In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: In the second part of this articl...

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Bibliographic Details
Main Authors: Akman, Murat, Gong, Jasun, Hineman, Jay, Lewis, John, Vogel, Andrew
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2022
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Inequalities (Mathematics)
Minkowski geometry
Nonlinear theories
Minkowski problem
The Brunn-Minkowski inequality
<inline-formula content-type="math/mathml"> A \mathcal {A} </inline-formula>-harmonic PDEs
inequalities and extremum problems
variational formula
Hadamard variational formula
nonlinear capacities
potentials and capacities
ISBN:1470450526, 9781470450526
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Table of Contents:
  • The Brunn-Minkowski inequality for nonlinear capacity -- Introduction -- Notation and statement of results -- Basic estimates for <inline-formula content-type="math/mathml"> A \mathcal {A} </inline-formula>-harmonic functions -- Preliminary reductions for the proof of Theorem -- Proof of Theorem -- Final proof of Theorem -- Appendix -- A Minkowski problem for nonlinear capacity -- Introduction and statement of results -- Boundary behavior of <inline-formula content-type="math/mathml"> A \mathcal {A} </inline-formula>-harmonic functions in Lipschitz domains -- Boundary Harnack inequalities -- Weak convergence of certain measures on <inline-formula content-type="math/mathml"> S n −<!-- − --> 1 \mathbb {S}^{n-1} </inline-formula> -- The Hadamard variational formula for nonlinear capacity -- Proof of Theorem
  • Cover -- Title page -- Part 1. The Brunn-Minkowski inequality for nonlinear capacity -- Chapter 1. Introduction -- Chapter 2. Notation and statement of results -- Chapter 3. Basic estimates for -harmonic functions -- Chapter 4. Preliminary reductions for the proof of Theorem A -- Chapter 5. Proof of Theorem A -- 5.1. Proof of (2.7) in Theorem A -- Chapter 6. Final proof of Theorem A -- Chapter 7. Appendix -- 7.1. Construction of a barrier in (4.17) -- 7.2. Curvature estimates for the levels of fundamental solutions -- Part 2. A Minkowski problem for nonlinear capacity -- Chapter 8. Introduction and statement of results -- Chapter 9. Boundary behavior of -harmonic functions in Lipschitz domains -- Chapter 10. Boundary Harnack inequalities -- Chapter 11. Weak convergence of certain measures on ⁿ⁻¹ -- Chapter 12. The Hadamard variational formula for nonlinear capacity -- Chapter 13. Proof of Theorem B -- 13.1. Proof of existence in Theorem B in the discrete case -- 13.2. Existence in Theorem B in the continuous case -- 13.3. Uniqueness of Minkowski problem -- Acknowledgment -- Bibliography -- Back Cover