Isoperimetric inequalities in potential theory |
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Authors: | W Hansen N Nadirashvili |
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Institution: | (1) Department of Mathematics, University of Bielefeld, P.O. Box 8640, 4800 Bielefeld 1, Germany |
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Abstract: | Given a non-empty bounded domainG in
n
,n 2, letr
0(G) denote the radius of the ballG
0 having center 0 and the same volume asG. The exterior deficiencyd
e
(G) is defined byd
e
(G)=r
e
(G)/r
0(G)–1 wherer
e
(G) denotes the circumradius ofG. Similarlyd
i
(G)=1–r
i
(G)/r
0(G) wherer
i
(G) is the inradius ofG. Various isoperimetric inequalities for the capacity and the first eigenvalue ofG are shown. The main results are of the form CapG (1+cf(d
e
(G))) CapG
0 and 1(G) (1+cf(d
i
(G))) 1(G
0),f(t)=t
3 ifn=2,f(t)=t
3/(ln 1/t) ifn=3,f(t)=t
(n+3)/2 ifn 4 (for convex G and small deficiencies ifn 3). |
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Keywords: | 31A15 31B15 35P15 52A40 |
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