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Bounds for capacities in terms of asymmetry. (English) Zbl 0870.31001

A condenser \(\Gamma=\Gamma(\Omega,\Omega')\) in \(\mathbb{R}^2\) consists of a compact set \(\Omega\) and a disjoint closed unbounded set \(\Omega'\). The \(p\)-capacity (\(1<p<\infty\)) of the condenser is \(\text{Cap}_p(\Gamma)=\inf\int\int_{\mathbb{R}^2}|Du|^p dxdy\), the infimum being taken over all functions \(u\) absolutely continuous in \(\mathbb{R}^2\), with \(u=0\) on \(\Omega\) and \(u=1\) on \(\Omega'\). Suppose that \(A(\Omega)=1\) and \(A(\mathbb{R}^2\setminus\Omega')=2^2\) (\(A\) denoting area) and let \(\Gamma^\ast=\Gamma(B(0,1/\sqrt\pi),\mathbb{R}^2\setminus B(0,2/\sqrt\pi))\). It is shown that there exists a constant \(K_p\), depending only on \(p\), such that \[ \text{Cap}_p(\Gamma)\geq(1+K_p\alpha(\Omega)^2)\text{Cap}_p(\Gamma^\ast) \] where \(\alpha(\Omega)\) is the asymmetry of \(\Omega\) defined by \(\alpha(\Omega)=\inf_x A(\mathbb{R}^2\setminus B(x,1/\sqrt\pi))\). The exponent \(2\) in this isoperimetric inequality is sharp. The proof is based on a new symmetrization technique leading to a perturbation lemma for \(2\)-capacity. An analogous result is conjectured for arbitrary dimension \(n\).

MSC:

31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
52A40 Inequalities and extremum problems involving convexity in convex geometry
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