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AN ISOPERIMETRIC INEQUALITY WITH REMAINDER TERM

2006
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Communications in Contemporary Mathematics
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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

doi:10.1142/s0219199706002167
fatcat:w43hshry2zb3jakzp5uqgp47ku