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Isoperimetric inequalities in potential theory. (English) Zbl 0796.31003
Seien \(G\) ein Gebiet in \(\mathbb{R}^ n\), \(r_ e(G)\) sein Umradius, \(r_ i(G)\) sein Inradius, \(r_ 0(G)\) der Radius der Kugel \(G_ 0\) mit demselben Volumen. Das “äußere Defizit” \(d_ e(G):=(r_ e/r_ 0)-1\) und das “innere Defizit” \(d_ i(G):=1-(r_ i/r_ 0)\) sind Maße für die Asymmetrie des Gebietes \(G\). Es werden untere Schranken für die Kapazität von \(G\) bzw. für den ersten Eigenwert des Laplace-Operators bei Dirichletschen Randbedingungen hergeleitet. Im Fall \(n=2\) lauten sie so: \[ \text{Cap} G \geq \biggl\{ 1+0.02 \bigl[ d_ e/(1 + d_ e) \bigr]^ 3 \biggr\} \text{Cap} G_ 0\quad \text{ bzw. }\quad \lambda_ 1(G) \geq (1+0.004 d^ 3_ i) \lambda_ 1 (G_ 0). \] Diese Ungleichungen sind zwar isoperimetrisch, die Koeffizienten aber scheinbar wenig scharf: die untere Schranke für \(\text{Cap} G/ \text{Cap} G_ 0\) liegt unterhalb 1.02, diejenige für \(\lambda_ 1(G)/ \lambda_ 1(G_ 0)\) unterhalb 1.004.
Die Flächeninhalte der \((n-1)\)-dimensionalen Niveauflächen von Potentialfunktionen spielen in diesen Überlegungen eine wichtige Rolle. In der Ebene \((n=2)\) wird eine Ungleichung von Bonnesen benutzt.
Reviewer: J.Hersch (Zürich)

MSC:
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
52A40 Inequalities and extremum problems involving convexity in convex geometry
35P15 Estimates of eigenvalues in context of PDEs
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