Sobolev inequalities on sets with irregular boundaries
The paper deals with Sobolev-Poincaré inequalities on \(s\)-John domains. A bounded domain \(\Omega\) is called an \(s\)-John domain (\(s\geq 1\)), if there is a point \(x_0\in \Omega\) and a constant \(c_0\geq 1\) such that each point \(x\in\Omega\) can be joined to \(x_0\) in \(\Omega\) by a rectifiable curve (called an \(s\)-John core) \(\gamma :[0,l]\to \Omega \) such that \(\gamma\) is parametrized by the arc length, \(\gamma (0)=x\), \(\gamma (1)=x_0\), and dist\((\gamma (t), \partial\Omega)\geq c_0^{-1}t^s\) for all \(t\in [0,l]\). An example of an \(s\)-John domain is an \(s\)-cusp domain. The authors prove that if \(\Omega\) is an \(s\)-John domain in \(\mathbb{R}^n\) and if \(a\geq 0\), \(b\geq 1-n\), \(1\leq p<q<\infty\), \[ \frac{1}{q}\geq \frac{1}{p}-\frac{1}{n} \quad\text{and}\quad \frac{1}{q}\geq \frac{s(n+b-1)-p+1}{p(n+a)} , \] then there is a constant \(C=C(n,p,q,a,b,\Omega)>0\) such that \[ \left ( \int _{\Omega}|u-\overline{u}_a|^q \rho ^a dx\right)^{1/q} \leq C \left ( \int _{\Omega} |\nabla u|^p \rho ^b dx \right)^{1/p} \] for each \(u\in C^1(\Omega)\). Here \(\rho (x)= \text{dist}(x, \mathbb{R}^n \backslash \Omega)\). This is an improvement of the main result from \textit{P. Hajłasz} and \textit{P. Koskela}, J. Lond. Math. Soc., II. Ser. 58, No.~2, 425-450 (1998; Zbl 0922.46034).
scientific article; zbMATH DE number 1501862
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