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A Sobolev inequality and Neumann heat kernel estimate for unbounded domains. (English) Zbl 0847.46014

Summary: Suppose \(D\) is an unbounded domain in \(\mathbb{R}^d\) \((d\geq 2)\) with compact boundary and that \(D\) satisfies a uniform interior cone property. We show that for \(1\leq p< d\), there exists a constant \(c= c(D, p)\) such that for each \(f\in W^{1, p}(D)\) the following Sobolev inequality holds: \[ |f|_q\leq c |\nabla f |_p, \] where \(1/q= 1/p- 1/d\) and for \(r= p\), \(q\), \(|\cdot |_r\) denotes the norm in \(L^r(D)\). As an application of this Sobolev inequality, assuming in addition that \(D\) is a Lipschitz domain in \(\mathbb{R}^d\) with \(d\geq 3\), we obtain a Gaussian upper bound estimate for the heat kernel on \(D\) with zero Neumann boundary condition.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35K05 Heat equation
60J35 Transition functions, generators and resolvents
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