Coexistence of periodic-2 and periodic-3 caustics for nearly circular analytic billiard maps

For symmetrically analytic deformation of the circle (with certain Fourier decaying rate), the necessary condition for the corresponding billiard map to keep the coexistence of periodic $2,3$ caustics is that the deformation has to be an isometric transformation

Limit of solutions for semilinear Hamilton-Jacobi equations with degenerate viscosity

In the paper we prove the convergence of viscosity solutions $u_{\lambda}$ as $\lambda\rightarrow0_+$ for the parametrized degenerate viscous Hamilton-Jacobi equation $$H(x,d_x u, \lambda u)=\alpha(x)\Delta u,\quad \alpha(x)\geq 0,\quad x\in \mathbb T^n$$ under suitable convex and monotonic conditions on $H: T^*M\times\mathbb R\rightarrow\mathbb R$. Such a limit can be characterized in terms of stochastic Mather measures associated with the critical equation \[ H(x,d_x u,0)=\alpha(x)\Delta u. \