A Sobolev Inequality and Neumann Heat Kernel Estimate for Unbounded Domains

Abstract

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.

A general blow-up result on nonlinear boundary-value problems on exterior domains

“…The proof is similar to that of the previous proposition, except that we need the following estimates for the Neumann Green's function on unbounded domains, which is given in [3].…”

“…where N is the Neumann Green's function of the Laplacian on D c (see [3]). Combining (4.12) and (4.13), we have, for any large R > 0, Z…”

On the existence of positive solutions for semilinear elliptic equations with Neumann boundary conditions

“…The latter estimate can be derived by proving a Sobolev inequality in a similar manner to that in [5] (see especially Lemma 5 and Theorem 1 there). Set Ii = OD \A and 12 = OD AA.…”

“…Definition 1.2. Chen et alThis process is called normally reflecting Brownian motion on D. To state our main theorem, we need the following result which is proved in[5]. )/lx _ .Jd--2 X E OD} is uniformly integrable with respect to the surface measure ~r on OD, iae.,In (1.12) the limit is uniform in sets A C OD such that ~r(A) ---+ 0.…”

“…By Lernma 5 of[5], there is a Thus the inner product ~a is equivalent to ~ on W2'2(Da) and therefore WaL2(Da) is a Hilbert space with respect to ~'~. Vf 9 Vgdx + fo f gdx.…”