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Article overview
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A general existence result for the Toda system on compact surfaces | Luca Battaglia
; Aleks Jevnikar
; Andrea Malchiodi
; David Ruiz
; | Date: |
23 Jun 2013 | Abstract: | In this paper we consider the following Toda system of equations on a compact
surface:
{{array}{ll} - Delta u_1 = 2
ho_1 (frac{h_1 e^{u_1}}{int_Sigma h_1
e^{u_1} dV_g} - 1) -
ho_2 (frac{h_2 e^{u_2}}{int_Sigma h_2 e^{u_2} dV_g} -
1) - 4 pi sum_{j=1}^{m} alpha_{1,j} (delta_{p_j}-1), - Delta u_2 = 2
ho_2 (frac{h_2 e^{u_2}}{int_Sigma h_2 e^{u_2} dV_g} - 1) -
ho_1
(frac{h_1 e^{u_1}}{int_Sigma h_1 e^{u_1} dV_g} - 1) - 4 pi sum_{j=1}^{m}
alpha_{2,j} (delta_{p_j}-1), &
{array}
. which is motivated by the study of models in non-abelian Chern-Simons
theory. Here h_1, h_2 are smooth positive functions,
ho_1,
ho_2 two
positive parameters, p_i points of the surface and alpha_{1,i}, alpha_{2,j}
non-negative numbers. We prove a general existence result using variational
methods. The same analysis applies to the following mean field equation -
Delta u =
ho_1 (frac{h e^{u}}{int_Sigma h e^{u} dV_g} - 1) -
ho_2
(frac{h e^{-u}}{int_Sigma h e^{-u} dV_g} - 1), which arises in fluid
dynamics. | Source: | arXiv, 1306.5404 | Services: | Forum | Review | PDF | Favorites |
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