| Budget Amount *help |
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
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| 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.
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