New development and its application of Aubry-Mather theory for Hamilton-Jacobi equations

Hamilton-Jacobi方程Aubry-Mather理论的新进展及其应用

• 批准号: 21540168
• 负责人: YASUHIRO Fujita
• 金额: $ 2.83万
• 依托单位: University of Toyama
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009 至 2011
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-21540168/
• 关键词: 関数方程式; ハミルトン-ヤコビ方程式; オーブリー・マザー理論; 対数型ソボレフの不等式; Hamilton-Jacobi方程式; Aubry-Mather理論; 超縮小性; Lipschitz定数; 極小一意性集; 商Aubry集合の完全非連結性; 不等式

About the study of this Kakenhi, I have obtained several results related with the Aubry-Mather theory and talked about these results in several conferences and seminars. These results are published in some journals.The first result is to clarify the relation between the quotient Aubry sets and uniqueness sets for minimization formula for Hamilton-Jacobi equations. The second one is to provide a new proof of classical inequalities by using a comparison theorem for the Aubry set of Hamilton-Jacobi equations. The third one is to derive an optimal logarithmic Sobolev inequality with Lipschitz constant. In the proof of this inequality, an asymptotic solution of the Aubry-Mather theory for a Hamilton-Jacobi equation is used. The fourth one is to investigate a rate of convergence appearing in the asymptotic behavior of a viscosity solution to the Cauchy problem for the Hamilton-Jacobi equation with quadratic gradient term. I showed that the semiconvexity property of this Hamiltonian is an important factor which determines this rate. Here, the Aubry set is closely related with the semiconvexity property of this Hamiltonian.As a conclusion, I think that I have done a complete job about the study of this Kakenhi by using the Aubry-Mather theory.

关于这个Kakenhi的研究，我得到了一些与Aubry-Mather理论相关的结果，并在几次会议和研讨会上谈到了这些结果。这些结果发表在一些期刊上。第一个结果是阐明Hamilton-Jacobi方程最小化公式的商Aubry集和唯一性集之间的关系。第二个是通过使用汉密尔顿-雅可比方程组的 Aubry 比较定理来提供经典不等式的新证明。第三个是推导具有 Lipschitz 常数的最优对数 Sobolev 不等式。在证明该不等式时，使用了 Hamilton-Jacobi 方程的 Aubry-Mather 理论的渐近解。第四个是研究带有二次梯度项的 Hamilton-Jacobi 方程的柯西问题粘度解的渐近行为中出现的收敛速度。我证明了这个哈密顿量的半凸性质是决定这个速率的一个重要因素。这里，Aubry集与这个哈密顿量的半凸性质密切相关。作为结论，我认为我已经用Aubry-Mather理论完成了这个Kakenhi的研究。

项目成果

An approach to classical inequalities via Hamilton-Jacobi equations

通过哈密尔顿-雅可比方程求解经典不等式

• 发表时间: 2010
• 作者: H.Kaneda;T.Suzuki;T.Onaka;Y.Doi;M.Kawada;B.-C.Koo.S.Makiuti;T.Nakagawa;Y.Okada;S.Sergeant;H.Shibai;M.Shirahata;北川暢子;Y.Fujito
• 通讯作者: Y.Fujito

On Hamilton-Jacobi equations and Euclidean logarithmic Sobolev inequalities

关于 Hamilton-Jacobi 方程和欧几里得对数 Sobolev 不等式

• 发表时间: 2009
• 作者: H. Kozono;T. Yanagisawa;木坂正史;Y.Fujita
• 通讯作者: Y.Fujita

Lipschitz regularizing effect and its application for Hamilton--Jacobi equations with discontinuous initial data

Lipschitz正则化效应及其在具有不连续初始数据的Hamilton--Jacobi方程中的应用

• 发表时间: 2012
• 作者: Shinozaki;Keisuke; Sato;Yoichi; Sawada;Kenichiro; Ando;Makiko; Sugita;Hiroyuki; Yamawaki;Toshihiko; Mizutani;Tadahito; Komatsu;Keiji; Okazaki;Shun; Ogawa;Hiroyuki; Nakagawa;Takao; Matsuhara;Hideo; and 4 coauthors;佐伯修;Yasuhiro Fujita
• 通讯作者: Yasuhiro Fujita

Some topological results for minimal uniqueness sets of Hamilton-Jacobi equations

汉密尔顿-雅可比方程组最小唯一性集的一些拓扑结果

• 发表时间: 2010
• 作者: 生方雅人;大屋幸輔;M.Guest;N. Kawazumi;H.Ishii;Tomoyuki Yoshida;Y.Giga;舟木直久;Y.Fujita
• 通讯作者: Y.Fujita

Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations

不等式和 Hamilton-Jacobi 方程的 Aubry-Mather 理论

• 发表时间: 2009
• 期刊: Communications on Pure and Applied Analysis Vol.8, No.2
• 作者: Fujita;Y.;Ohmori;K
• 通讯作者: K