In this paper, we establish some new sharp Sobolev inequalities on any smooth bounded domain \documentclass[12pt]{minimal}
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$\Omega \subset{\Bbb R}^n$\end{document}. Let \documentclass[12pt]{minimal}
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$S_1$\end{document} and S be the sharp constants corresponding to the Sobolev embedding and trace inequalities respectively. If \documentclass[12pt]{minimal}
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$n\ge 4$\end{document}, there exist constants \documentclass[12pt]{minimal}
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$A(\Omega)$\end{document}, \documentclass[12pt]{minimal}
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$A_1(\Omega)>0$\end{document} such that \documentclass[12pt]{minimal}
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$\forall u \in H^1(\Omega)$\end{document}\documentclass[12pt]{minimal}
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\[ \|u\|_{2n/(n-2), \Omega}^2 \le 2^{2/n}{S_1} \|\nabla u\|_{2, \Omega}^2 + A(\Omega) \|u\|_{2n/(n-1), \Omega}^2 \] \end{document} and \documentclass[12pt]{minimal}
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\[ \|u\|^2_{2(n-1)/(n-2),\partial \Omega } \le S \|\nabla u\|_{2, \Omega}^2 + A_1(\Omega) \|u\|_{2n/(n-1),\Omega}^2\,; \] \end{document} If \documentclass[12pt]{minimal}
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$n=3$\end{document}, for any \documentclass[12pt]{minimal}
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$k_3 >3$\end{document}, there exist constants \documentclass[12pt]{minimal}
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$A(\Omega, k_3), A_1(\Omega, k_3)>0 $\end{document} such that \documentclass[12pt]{minimal}
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$\forall u \in H^1(\Omega)$\end{document}\documentclass[12pt]{minimal}
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\[ \|u\|^2_{2n/(n-2), \Omega} \le 2^{2/n}{S_1} \cdot \|\nabla u\|_{2, \Omega}^2+ A(\Omega, k_3) \|u\|^2_{k_3, \Omega} \] \end{document} and \documentclass[12pt]{minimal}
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\[ \|u\|_{2(n-1)/(n-2),\partial \Omega}^2 \le S \|\nabla u\|_{2, \Omega}^2+ A_1(\Omega, k_3) \|u\|^2_{k_3, \Omega}\,. \] \end{document}
