The Hardy constant. (English) Zbl 0857.26005

Davies, E. B.

Q. J. Math., Oxf. II. Ser. 46, No. 184, 417-431 (1995).

Let \(\Omega\) be either a bounded region in \(\mathbb{R}^N\) or an incomplete \(N\)-dimensional Riemannian manifold of finite diameter. Suppose that the completion \(\overline\Omega\) of \(\Omega\) is compact and let \(\partial\Omega:=\overline\Omega\backslash\Omega\). Let \(\rho(x,y)\) denote the extension of the Riemannian distance function to \(\overline\Omega\times\overline\Omega\) and \(d(x):=\min\{\rho(x,y);y\in \partial\Omega\}\), \(x\in\Omega\).
The author assumes that the inequality \[ \int_\Omega{|f(x)|^2\over d(x)^2} d\text{ vol}\leq c\int_\Omega |\nabla f(x)|^2 d\text{ vol}+ a\int_\Omega|f(x)|^2 d\text{ vol}\tag{\(*\)} \] holds for all \(f\in W^{1,2}_0(\Omega)\) with some \(a,c\in[0,\infty)\) and defines the weak Hardy constant of \(\Omega\) to be the infimum of all \(c>0\) such that there exists \(a<\infty\) for which \((*)\) holds. Then he shows that every point of \(\Omega\) has a local weak Hardy constant associated to it and that the weak Hardy constant is given by \(c=\max\{h(x);x\in \partial\Omega\}\).
Moreover, the paper contains a number of methods of computing and estimating \(h(x)\) for various types of boundary points.

Reviewer: B.Opic (Praha)

MSC:

26D10

Inequalities involving derivatives and differential and integral operators

Keywords:

Hardy’s inequality; weak Hardy constant