AN ISOPERIMETRIC INEQUALITY WITH REMAINDER TERM

FUTOSHI TAKAHASHI
2006 Communications in Contemporary Mathematics  
We prove a version of the isoperimetric inequality for mappings with remainder term Let $S$ $=(32\pi)^{1/3}$ and $Q(u)= \int_{\mathrm{R}^{2}}u\cdot u_{x_{1}}\Lambda uX2dx$ for a mapping $u:\mathrm{R}^{2}arrow \mathrm{R}^{3}$ in afunction space $\overline{\mathcal{W}}$ defined below. Then the classical isoperimetric inequality for mappings says that $S|Q(u)|^{2/3}\leq f_{\mathrm{R}^{2}}|\nabla u|^{2}dx$ holds for any $u\in\overline{\mathcal{W}}$ . Let $\mathcal{M}$ be a manifold of functions in
more » ... ld of functions in $\overline{\mathcal{W}}$ for which we have equality in the above isoperimetric inequality. Following an argument by Bianchi and Egnell [2] for the case of Sobolev ineq uality and using a crucial estimate proved by Isobe [8], we show that for some positive constant $C>0$ , $J_{\mathrm{R}^{2}}^{[}|\nabla u|^{2}dx-S|Q(u)|^{2[3}\geq Cd(u, \mathrm{A}4)^{2}$
doi:10.1142/s0219199706002167 fatcat:w43hshry2zb3jakzp5uqgp47ku