Harnack inequalities with dusts

2015 Fiscal Year Final Research Report

• Project/Area Number: 25610017
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Hokkaido University
• Principal Investigator: Aikawa Hiroaki 北海道大学, 理学(系)研究科(研究院), 教授 (20137889)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: Harnack不等式 / Harnack連鎖 / 除外集合 / 容量 / 境界Harnack原理 / 調和測度 / 擬双曲距離 / 熱核

Outline of Final Research Achievements

Consider concentric two balls. If h is a positive harmonic function in the larger ball, then its values on the smaller ball is comparable to the value of h at the center (Harnack inequality). A Harnack chain is the union of balls such that consecutive balls have sufficiently large intersection. Applying the Harnack inequality to each constituent ball yields that a positive harmonic function on the Harnack chain assumes comparable values at the centers of the first and last balls (Harnack principle). We have shown that the same Harnack principle holds even if there exists an exceptional set in the Harnack chain, provided the capacity of the exceptional set in each ball is sufficiently small.

Free Research Field

解析学