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Quasihyperbolic growth conditions and compact embeddings of Sobolev spaces. (English) Zbl 1135.46015

Let \(\Omega\) be an arbitrary connected domain in \(\mathbb R^n\) and let
\[ k_\Omega (x,y) = \inf_\gamma \int_\gamma \frac{ds}{d(z, \partial \Omega)}, \quad x \in \Omega, \quad y \in \Omega, \]
where \(d(z, \partial \Omega)\) is the distance of \(z\in \gamma \subset \Omega\) to the boundary and \(\gamma\) is a curve connecting \(x\) and \(y\). Let \(k_\Omega (x,x_0) \leq c \, d(x, \partial \Omega)^{- \alpha}\) for \(x \in \Omega\) and a fixed off-point \(x_0 \in \Omega\). If \(n \geq 2\) and \(\alpha < \frac{n}{2n-1}\), then the embedding \(W^1_n (\Omega)\) into \(L_n (\Omega)\) is compact. The exponent \(\frac{n}{2n-1}\) is sharp. This is the main assertion of this paper (Theorem 1.1), improving earlier results.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

Sobolev spaces
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References:

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