An Upper Estimate for a Heat Kernel With Neumann Boundary Condition

Using an upper solution we obtain a bound from above for the heat kernel ψ( x,y,t ) for a region Ω which is star‐shaped with respect to one of the points, say y . The estimate is for the Neumann problem and holds for short times. The form of the bound is ψ ( x ,   y ,   t ) ⩽( 4 π t )− N / 2 exp [ − |x − y | 2 / 4 t] + exp [ − d( x ,   y ) 2 / 4 t + O ( t − 1 / 2) ] ; moreover, for x ∈Ω\ Y ( y ), ψ ( x ,   y ,   t ) ⩽( 4 π t )− N / 2{ exp [ − |x − y | 2 / 4 t] + f ( x ,   y ) exp [ − d( x ,   y ) 2 / 4 t ] ( 1 + O ( t ) ) } . Here Y ( y ) is a closed subset of Ω ⋐ R N with measure zero, d(x,y) is the minimum distance between x and y via the boundary ∂Ω: d(x,y) = inf Z∈∂Ω (|x‐z| + |y‐z|), and f(.,y) is a positive function, continuous away from Y , and equal to unity on δΩ.