Isoperimetric Inequalities in Potential Theory

Given a non-empty bounded domain G in ℝn, n ⩾ 2, let r o (G) denote the radius of the ball G o having center 0 and the same volume as G. The exterior deficiency d e (G) is defined by $$ {d_e}(G) = {r_e}(G)/{r_0}(G) - 1 $$ where r e (G) denotes the circumradius of G. Similarly $$ {d_i}(G) = 1 - {r_i}(G)/{r_0}(G) $$ where r i (G) is the inradius of G. Various isoperimetric inequalities for the capacity and the first eigenvalue of G are shown. The main results are of the form $$ Cap{G_0} \geqslant (1 + cf({d_e}(G))) \geqslant Cap{G_0} $$ and $$ {\lambda_1}(G) \geqslant (1 + cf({d_i}(G))){\lambda_1}({G_0}),f(t) = {t^3} $$ if $$ n = 2,f(t) = {t^3}/(\ln 1/t) $$ if $$ n = 3,f(t) = {t^{(n + 3)/2}} $$ if n ⩾ 4 (for convex G and small deficiencies if n ≥ 3).