Uniqueness of solutions for a mean field equation on torus
نویسندگان
چکیده
We consider on a two-dimensional flat torus T the following equation ∆u + ρ ( eu ∫ T e u − 1 |T | ) = 0. When the fundamental domain of the torus is (0, a) × (0, b) (a ≥ b), we establish that the constants are the unique solutions whenever ρ ≤ { 8π if b a ≥ π4 , 32 b a if b a ≤ π4 , and this result is sharp if b a ≥ π4 . A similar conclusion is obtained for general two-dimensional torus by considering the length of the shortest closed geodesic. These results are derived by comparing the isoperimetric profile of the torus T with the one of the two-dimensional canonical sphere which has same volume as T .
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