The Asymptotics of the Heat Semigroup for a General Bounded Domain with Mixed Boundary Conditions

Small-time asymptotics of the trace of the heat semigroup .(t)=...=1exp(.t..), where {. . } are the eigenvalues of the negative Laplacian ..=..2.=1(..x.)2 in the (x 1, x 2)-plane, is studied for a general bunded domain . with a smooth boundary .., where a finite number of Dirichlet, Neumann and Robin boundary conditions, on the piecewise smooth parts . i (i = 1, ..., n) of .. such that ..=.ni=1.i, are considered. Some geometrical properties associated with . are determined.