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

✍ Scribed by Murat Akman, Jasun Gong, Jay Hineman, John Lewis, Andrew Vogel

• Publisher: American Mathematical Society
• Year: 2022
• Tongue: English
• Leaves: 128
• Series: Memoirs of the American Mathematical Society, 1348
• Category: Library

✦ Synopsis

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, Cap A, where A -capacity is associated with a nonlinear elliptic PDE whose structure is modeled on the p-Laplace equation and whose solutions in an open set are called A -harmonic.

✦ Table of Contents

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 𝒜-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 𝒜-harmonic functions in Lipschitz domains
Chapter 10. Boundary Harnack inequalities
Chapter 11. Weak convergence of certain measures on 𝕊ⁿ⁻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