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

关于这个Kakenhi的研究，我已经获得了一些与Aubry-Mather理论相关的结果，并在几次会议和研讨会上讨论了这些结果。这些结果发表在一些期刊上。第一个结果是阐明了Hamilton-Jacobi方程最小化公式的商Aubry集与唯一性集之间的关系。第二部分是利用奥布里哈密顿-雅可比方程集的比较定理，给出经典不等式的一个新的证明。第三步是导出一个具有Lipschitz常数的最优对数Sobolev不等式。在证明这个不等式时，使用了Hamilton-Jacobi方程的Aubry-Mather理论的渐近解。第四部分研究了二阶梯度项Hamilton-Jacobi方程Cauchy问题黏性解的渐近性中出现的收敛速率。我证明了这个哈密顿函数的半不变性是决定这个速率的一个重要因素。这里，奥布里集与这个哈密顿函数的半不完整性密切相关。作为结论，我认为我已经用奥布里-马瑟理论完成了对这个Kakenhi的研究。

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.