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

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.