| Project/Area Number |
17K05337
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Mathematical analysis
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| Research Institution | Tokyo University of Science |
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Principal Investigator |
Tachikawa Atsushi 東京理科大学, 理工学部数学科, 嘱託教授(非常勤) (50188257)
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| Project Period (FY) |
2017-04-01 – 2021-03-31
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| Project Status |
Discontinued (Fiscal Year 2020)
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| Budget Amount *help |
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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| Keywords | 変分問題 / 弱解の正則性 / p(x)-growth / Φ-growth / double phase functional / Double phase / 変動指数を持つ汎関数 / 解析学 / double phase / Orlicz空間 / 解の正則性 / 部分正則性 / 非標準的増大度 / 関数方程式論 / Non-standard growth / 変動指数 |
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| Outline of Final Research Achievements |
The aim of this research is to obtain new results about regularity of solutions for variational problems. We call the problems to obtain minimum point (or more generally, critical points and stationary points) as variational problems. Many familiar shapes such as soap films are the solution to the variational problem.
When dealing with variational problems mathematically, it is generally difficult to directly show the existence of a solution in the class of continuous or even differentiable functions. So, in many cases, we follow the process showing the existence of "weak solutions" which solve the problem in some weak sense in Sobolev spaces; spaces of "weakly differentiable" functions, and then showing the differentiability of the weak solutions. In this research, we considered the second process and have obtain several new results.
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| Academic Significance and Societal Importance of the Research Achievements |
一般に非線型偏微分方程式に対して，ソボレフ空間（弱い意味での導関数がp-乘可積分である関数の空間）における解，すなわち弱解の存在は比較的容易に示されることが多く，むしろ存在が保証された弱解が，そもそもの問題にとって適切な滑らかさ（連続性，微分可能性）を持つことを示す点に難しさがある場合が多い．このような弱解の滑らかさに関数問題を「正則性の問題」と呼んでいる．本研究では，変分問題の解に対してこの「滑らかさの問題」を扱い，新たな結果を得た．
本研究で扱った問題は，近年ヨーロッパを中心に，その応用も含めて盛んに研究されている分野であり，この分野で新たな結果を得た意義は小さくない．
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