Parabolic equations on general bounded domains are studied. Usingthe refined maximum principle, existence and the semigroup property of solutions are obtained. It is also shown that the solution obtained by PDE's method has the Feynmann_Kac representation for any bounded domains.
A result due to Mather on the existence of Aubry-Mather sets for superlinear positive definite Lagrangian systems is generalized in one-dimensional case. Applications to existence of Aubry-Mather sets of planar Hamiltonian systems are given.
