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

Description

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. 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