Nonlocal regularity for a geometric heat flow with fractional integral operator

2023 Fiscal Year Final Research Report

• Project/Area Number: 21K03330
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Kumamoto University
• Principal Investigator: Misawa Masashi 熊本大学, 大学院先端科学研究部(理), 教授 (40242672)
• Project Period (FY): 2021-04-01 – 2024-03-31
• Keywords: ソボレフ不等式 / 条件付き変分問題 / 変分問題に関わる熱流の方法 / ソボレフ流 / 正則性特異性

Outline of Final Research Achievements

We had the following results for the fractional p-Sobolev flow and the doubly nonlinear fractional p-Laplace type equations: (1) Derivation of the scaling transformation intrinsic to the fractional p-Sobolev flow. (2) The global in time existence for Cauchy-Dirichlet problem. The results for the doubly nonlinear parabolic type equations are the followings: (3) The expansion of positivity for nonnegative weak solutions (4) Regularity by the expansion of positivity for the evolutionary p-Laplace type equations (5) A finite type extinction for the fast diffusive doubly nonlinear parabolic type equations (6) The global in time existence for Cauchy-Dirichlet problem.

Free Research Field

Partial differential Equations

Academic Significance and Societal Importance of the Research Achievements

二重非線形拡散方程式の新たな幾何学的変分学的応用を見出した: (1)コンパクトリーマン多様体上の山辺問題に関わる熱流,山辺流,の結果を含み,解の定義域あるいは初期値の凸性の条件を緩和した.(2)偏微分p-ソボレフ流型方程式の正則性評価は,解の族のエネルギークラスにおける弱コンパクト性を導き,集中コンパクト性の分数階p-ラプラス方程式への一般化を与える.(3)二重非線形分数階拡散方程式の弱解の大域存在を非常に一般的条件のもと証明した. この結果は,分数階p-ソボレフ流の弱解の大域存在に応用できる.(4)二重非線形分数階拡散方程式の非負弱解の正値性伝播とヘルダー評価