AN UPPER ESTIMATE FOR A HEAT KERNEL WITH NEUMANN BOUNDARY-CONDITION

Using an upper solution we obtain a bound from above for the heat kern el psi(x, y, t) for a region OMEGA which is star-shaped with respect t o one of the points, say y. The estimate is for the Neumann problem an d holds for short times. The form of the bound is psi(x, y, t) less-th an-or-equal-to (4pit)-N/2 exp[-\x - y\2/4t] + exp [-d(x, y)2/4t + O(t- 1/2)]; moreover, for x is-an-element-of OMEGA\Y(y), psi(x, y, t) less- than-or-equal-to (4pit)-N/2 (exp[-\x - y\2/4t] + f(x, y) exp[-d(x, y)2 /4t] (1 + O(t))). Here Y(y) is a closed subset of OMEGA subset-of R(N) with measure zero, d(x,y) is the minimum distance between x and y via the boundary partial derivative OMEGA:d(x,y) = inf(z is-an-element-of partial derivative OMEGA) (\x - z\ + \y - z\), and f(., y) is a posit ive function, continuous away from Y, and equal to unity on partial de rivative OMEGA.