A Sobolev Inequality and Neumann Heat Kernel Estimate for Unbounded Domains
A Sobolev Inequality and Neumann Heat Kernel Estimate for Unbounded Domains
复制标题
无界域的索博列夫不等式和诺伊曼热核估计
DOI：
10.4310/mrl.1994.v1.n2.a5
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发表时间：
1994
影响因子：
1
通讯作者：
Z. Zhao
中科院分区：
文献类型：
--
作者：
Zhen;Ruth J. Williams;Z. Zhao
设D是RD(d≥2)中具有紧边界的无界区域，且D满足一致内锥性质.证明了对于1≤p<；d，存在一个常数c=c(D，p)，使得对于每个f∈W1，p(D)都成立：�fq≤c�∇Fp，其中1/q=1/p−1/d，且对于r=p，q，�·�r表示L r(D)中的范数.作为Soblev不等式的一个应用，另外，假设D是Rd中的Lipschitz域，且d≥为3，我们得到了D上具有零Neumann边界条件的热核的高斯上界估计.
Suppose D is an unbounded domain in R d (d ≥ 2) with com- pact 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: � fq ≤ c �∇ fp, 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.