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The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity
About this Title
Murat Akman, Jasun Gong, Jay Hineman, John Lewis and Andrew Vogel
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 275, Number 1348
ISBNs: 978-1-4704-5052-6 (print); 978-1-4704-7014-2 (online)
DOI: https://doi.org/10.1090/memo/1348
Published electronically: December 13, 2021
Keywords: The Brunn-Minkowski inequality,
nonlinear capacities,
inequalities and extremum problems,
potentials and capacities,
$\mathcal {A}$-harmonic PDEs,
Minkowski problem,
variational formula,
Hadamard variational formula
Table of Contents
1. The Brunn-Minkowski inequality for nonlinear capacity
- 1. Introduction
- 2. Notation and statement of results
- 3. Basic estimates for $\mathcal {A}$-harmonic functions
- 4. Preliminary reductions for the proof of Theorem
- 5. Proof of Theorem
- 6. Final proof of Theorem
- 7. Appendix
2. A Minkowski problem for nonlinear capacity
- 8. Introduction and statement of results
- 9. Boundary behavior of $\mathcal {A}$-harmonic functions in Lipschitz domains
- 10. Boundary Harnack inequalities
- 11. Weak convergence of certain measures on $\mathbb {S}^{n-1}$
- 12. The Hadamard variational formula for nonlinear capacity
- 13. Proof of Theorem
Abstract
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, $\operatorname {Cap}_{\mathcal {A}},$ where $\mathcal {A}$-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 $\mathcal {A}$-harmonic.
In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: \[ \left [\operatorname {Cap}_\mathcal {A} ( \lambda E_1 + (1-\lambda ) E_2 )\right ]^{\frac {1}{(n-p)}} \geq \lambda \, \left [\operatorname {Cap}_\mathcal {A} ( E_1 )\right ]^{\frac {1}{(n-p)}} + (1-\lambda ) \left [\operatorname {Cap}_\mathcal {A} (E_2 )\right ]^{\frac {1}{(n-p)}} \] when $1<p<n, 0 < \lambda < 1,$ and $E_1, E_2$ are convex compact sets with positive $\mathcal {A}$-capacity. Moreover, if equality holds in the above inequality for some $E_1$ and $E_2,$ then under certain regularity and structural assumptions on $\mathcal {A},$ we show that these two sets are homothetic.
In the second part of this article we study a Minkowski problem for a certain measure associated with a compact convex set $E$ with nonempty interior and its $\mathcal {A}$-harmonic capacitary function in the complement of $E$. If $\mu _E$ denotes this measure, then the Minkowski problem we consider in this setting is that; for a given finite Borel measure $\mu$ on $\mathbb {S}^{n-1}$, find necessary and sufficient conditions for which there exists $E$ as above with $\mu _E = \mu .$ We show that necessary and sufficient conditions for existence under this setting are exactly the same conditions as in the classical Minkowski problem for volume as well as in the work of Jerison in \cite{J} for electrostatic capacity. Using the Brunn-Minkowski inequality result from the first part, we also show that this problem has a unique solution up to translation when $p\neq n- 1$ and translation and dilation when $p = n-1$.
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