Abstract. Suppose D is an unbounded domain in R d (d ≥ 2) with compact boundary and that D satisfies a uniform interior cone property. We show that for 1 ≤ p < d, there exists a constant c = c(D, p) such that for each f ∈ W 1,p (D) the following Sobolev inequality holds:where 1/q = 1/p − 1/d and for r = p, q, · r denotes the norm in L r (D). As an application of this Sobolev inequality, assuming in addition that D is a Lipschitz domain in R d with d ≥ 3, we obtain a Gaussian upper bound estimate for the heat kernel on D with zero Neumann boundary condition.

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