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

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。第一个结果是阐明了Hamilton-Jacobi方程极小化公式的商Aubry集与唯一性集之间的关系。第二部分是利用Hamilton-Jacobi方程的Aubry集的比较定理给出了经典不等式的一个新的证明。第三个是导出一个最优的对数Sobolev不等式。在这个不等式的证明中，使用了Hamilton-Jacobi方程的Aubry-Mather理论的渐近解。第四部分研究了具有二次梯度项的Hamilton-Jacobi方程Cauchy问题粘性解的渐近性态的收敛速度。我表明，该哈密顿量的不稳定性是决定该速率的重要因素。在这里，Aubry集与这个Hamilton量的不稳定性密切相关，因此，我认为我用Aubry-Mather理论对这个Kakenhi的研究是完整的。

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.