2013 Fiscal Year Final Research Report
Various functional inequalities and their applications to the variational problems
| Project/Area Number |
24840024
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Single-year Grants |
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| Research Field |
Basic analysis
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| Research Institution | Gifu University |
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Principal Investigator |
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| Project Period (FY) |
2012-08-31 – 2014-03-31
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| Keywords | 関数不等式 / 楕円型偏微分方程式 / 変分問題 |
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| Research Abstract |
Our main purpose of the research lies in studying Sobolev type inequalities and the corresponding variational problems. Especially, we concern the critical embeddings which appear in Trudinger-Moser type inequalities, Gagliardo-Nirenberg type interpolation inequalities and so on. In a period of this fund, we investigated the existence of maximizer associated with the Trudinger-Moser type inequalities of the scaling invariant form obtained by Adachi-Tanaka in 1999, and actually proved the existence of a maximizer. On the other hand, it was known that the similar type Trudinger-Moser inequalities on the whole space which do not satisfy the scaling invariance. Ishiwata (2010) proved that the existence of maximizers for this inequality heavily depends on the dimension. Based on this fact, we considered the variational structure and clarified the relation between the scaling invariance and the existence of a maximizer.
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