Potential theoretic study for elliptic partial differential equations

2020 Fiscal Year Final Research Report

• Project/Area Number: 18K03332
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Hiroshima University
• Principal Investigator: Shimomura Tetsu 広島大学, 人間社会科学研究科(教), 教授 (50294476)
• Project Period (FY): 2018-04-01 – 2021-03-31
• Keywords: ソボレフ関数 / 楕円型偏微分方程式

Outline of Final Research Achievements

Variable exponent Lebesgue spaces and Sobolev spaces have been intensively investigated for the past twenty years to discuss nonlinear partial differential equations with non-standard growth condition. These spaces have attracted more and more attention in connection with the study of elasticity and electrorheological fluids. In this research, we studied the boundedness of the Hardy-Littlewood maximal operator on Musielak-Orlicz-Morrey spaces and Musielak-Orlicz spaces. As an application of the boundedness of the maximal operator, we established a generalization of Sobolev's inequality for Riesz potentials of functions in Musielak-Orlicz-Morrey spaces and Musielak-Orlicz spaces. We also established generalizations of Sobolev's inequality for double phase functionals.

Free Research Field

ポテンシャル論

Academic Significance and Societal Importance of the Research Achievements

ソボレフ関数は、楕円型偏微分方程式の解がもつ解析的な性質をポテンシャル論的方法により研究する上で重要な関数である。距離空間上や測度に２倍条件を仮定しないnon-doubling measure 空間上において、変動指数をもつさまざまな関数空間におけるソボレフ型定理を発展させることができた。実解析学だけでなく、偏微分方程式論、多様体上の微分幾何学やグラフ上の解析学、電気流動学や弾性学などへの応用が期待される。