Analytic aspects of the Toda system: I. A Moser‐Trudinger inequality
Analytic aspects of the Toda system: I. A Moser‐Trudinger inequality
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DOI：
10.1002/cpa.10004
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发表时间：
2000-11
影响因子：
3
通讯作者：
J. Jost;Guofang Wang
中科院分区：
文献类型：
--
作者：
J. Jost;Guofang Wang
In this paper, we analyze solutions of the open Toda system and establish an optimal Moser‐Trudinger type inequality for this system. Let Σ be a closed surface with area 1 and K = (aij)N × N the Cartan matrix for SU(N + 1), i.e., $\left ( \matrix{ TTT \phantom{-}2 & -1 & \phantom{-}0 & \cdots & \cdots & \phantom{-}0\cr -1& \phantom{-}2 &-1& \ddots &{0} & \phantom{-}\vdots\cr \phantom{-}0&-1&\ddots&\ddots& \ddots & \phantom{-}\vdots\cr \phantom{-}\vdots&\ddots&\ddots&\ddots&-1&\phantom{-}0\cr \phantom{-}\vdots &{0}&\ddots&-1&2&-1\cr \phantom{-}0&\cdots&\cdots & \phantom{-}0 &-1&\phantom{-} 2nd\cr} \right ) .$ We show that $$\eqalign{&\Phi_{M}(u)=\cr &{{1}\over{2}} \sum_{i,j=1}^N\int_\Sigma a_{ij} \left(\nabla u_i \nabla u_j+2M_iu_j \right) -\sum_{i=1}^N M_i\log \int_\Sigma \exp \left(\sum_{j=1}^N a_{ij}u_j \right)\cr}$$ has a lower bound in (H1(Σ))N if and only if $M_j \le 4\pi \quad\hbox{for } j=1,2, \ldots, N.$ This inequality is optimal. As a direct consequence, if Mj < for 4π for j = 1, 2, …, N, ΦM has a minimizer u that satisfies $$-\Delta u_i = M_i \left({\exp \left(\sum_{j=1}^N a_{ij} u_j \right)}\over{\int_\Sigma \exp \left(\sum_{j=1}^N a_{ij} u_j \right)} -1 \right) \quad\hbox{for } 1\le i \le N\,.$$ © 2001 John Wiley & Sons, Inc.
In this paper, we analyze solutions of the open Toda system and establish an optimal Moser‐Trudinger type inequality for this system. Let Σ be a closed surface with area 1 and K = (aij)N × N the Cartan matrix for SU(N + 1), i.e., $\left ( \matrix{ TTT \phantom{-}2 & -1 & \phantom{-}0 & \cdots & \cdots & \phantom{-}0\cr -1& \phantom{-}2 &-1& \ddots &{0} & \phantom{-}\vdots\cr \phantom{-}0&-1&\ddots&\ddots& \ddots & \phantom{-}\vdots\cr \phantom{-}\vdots&\ddots&\ddots&\ddots&-1&\phantom{-}0\cr \phantom{-}\vdots &{0}&\ddots&-1&2&-1\cr \phantom{-}0&\cdots&\cdots & \phantom{-}0 &-1&\phantom{-} 2nd\cr} \right ) .$ We show that $$\eqalign{&\Phi_{M}(u)=\cr &{{1}\over{2}} \sum_{i,j=1}^N\int_\Sigma a_{ij} \left(\nabla u_i \nabla u_j+2M_iu_j \right) -\sum_{i=1}^N M_i\log \int_\Sigma \exp \left(\sum_{j=1}^N a_{ij}u_j \right)\cr}$$ has a lower bound in (H1(Σ))N if and only if $M_j \le 4\pi \quad\hbox{for } j=1,2, \ldots, N.$ This inequality is optimal. As a direct consequence, if Mj < for 4π for j = 1, 2, …, N, ΦM has a minimizer u that satisfies $$-\Delta u_i = M_i \left({\exp \left(\sum_{j=1}^N a_{ij} u_j \right)}\over{\int_\Sigma \exp \left(\sum_{j=1}^N a_{ij} u_j \right)} -1 \right) \quad\hbox{for } 1\le i \le N\,.$$ © 2001 John Wiley & Sons, Inc.