2017/09/01 by Murat Akman, Jasun Gong, Akman, Murat +8 · 1 citation
Computer Science · Decision Sciences · Mathematics · #31B15 #35J20 #35J60 #35J92 #39B62 #52A20 #52A40 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Probabilistic and Robust Engineering Design #math.AP #msc:31B15 #msc:35J20 #msc:35J60 #msc:35J92 #msc:39B62 #msc:52A20 #msc:52A40
paper · pdf · doi:10.48550/arxiv.1709.00447
108 pages. Some small typos were corrected, some minor changes and a new lemma (Lemma 10.2) in section 10
openalex publication_date 2017/09/01 · arxiv created 2018/10/08 · arxiv updated 2018/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we study two classical potential-theoretic problems in convex geometry corresponding to a nonlinear capacity, CapA, 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. In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: [CapA(λE1 +(1-λ)E2)](1)/((n-p))≥λ[CapA(E1)](1)/((n-p))+(1-λ)[CapA(E2 )](1)/((n-p)) when 1<p<n, 0<λ<1, and E1, E2 are convex compact sets with positive A-capacity. Moreover, if equality holds in the above inequality for some E1 and E2, then under certain regularity and structural assumptions on 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 A-harmonic capacitary function in the complement of E. If μE denotes this measure, then the Minkowski problem we consider in this setting is that; for a given finite Borel measure μ on \mathbbSn-1, find necessary and sufficient conditions for which there exists E as above with μE =μ. 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 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≠ n- 1 and translation and dilation when p = n-1.