Quasihyperbolic boundary conditions and Poincaré domains (Q1849482)

A domain \(\Omega\subset \mathbb{R}^n\) of finite volume is said to be a \(p\)-Poincaré domain \((p\geq 1)\) if there exists a constant \(M_p\) such that \[ \int_\Omega|u(x)- u_\Omega|^p dx\leq M_p \int_\Omega|\nabla u(x)|^p dx \] for all \(u\in C^\infty(\Omega)\); here \(u_\Omega\) is the mean value of \(u\) on \(\Omega\). A general characterization for Poincaré domains in terms of a capacity-type estimate was given by Maz'ya, and sufficient conditions in terms of integrability of the quasihyperbolic distance \[ k_\Omega(x,y)= \inf_\gamma \int{ds\over \text{dist}(z,\partial\Omega)} \] were obtained by Jerison, Murri, Smith and Stegenya. In particular, \(\Omega\) is an \(n\)-Poincaré domain if \(k_\Omega\) satisfies the condition \[ k_\Omega(x_0, x)\leq{1\over\beta} \log{\text{dist}(x_0, \partial\Omega)\over \text{dist}(x,\partial\Omega)}+ C_0,\quad \forall x\in\Omega,\tag{1} \] where \(x_0\) is a fixed basepoint, \(C_0= C_0(x_0)< \infty\), and \(\beta> 0\). The main result of the present paper is as follows. If \(\Omega\) satisfies (1) with some \(\beta\leq 1\), then it is a \(p\)-Poincaré domain for all \(p\in [1,\infty)\cap (n- n\beta, n)\). For each \(p< n-n\beta\), there exist domains \(\Omega\) satisfying (1) and which are not \(p\)-Poincaré domains. Similar statements are proved for the so-called \((q,p)\)-Poincaré domains as well. The proof relies on the description of Poincaré domains due to Maz'ya. An application to the Neumann problem on domains with irregular boundaries is given, and relations between condition (1) and the \(s\)-John (``twisted cusp'') condition is discussed.