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Additional resources for Communications in Mathematical Physics - Volume 229
In view of the periodic boundary conditions, we may set (160) over the flat two dimensional torus M = R2 /ZZa1 + ZZa2 . By means of this identification we are reduced to seek solutions for the elliptic system: m nj δpj on M, − w = 4g 2 ew + g 2 eu − 4π j =1 (163) 2 2 2 ϕ g g 2 w u 0 − u = −2g e − e + on M. 2 cos2 θ 2 cos2 θ Integrating over M, we find the following constraints for the solvability of (163): 2 ew + g 2 eu = 4π N, 4g M M 2g 2 M ew + 2 2 g2 u = g ϕ0 |M| . e 2 cos2 θ M 2 cos2 θ (164) Consequently, M M 4π N − g 2 ϕ02 |M| , 4g 2 sin2 θ g 2 ϕ02 |M| − 4π N cos2 θ eu = , g 2 sin2 θ ew = (165) which imply the following necessary condition for the sovability of (163): g 2 ϕ02 < g 2 ϕ02 4π N < .
115, 344–358 (1993) 11. : Uniform estimates and blow-up behaviour for solutions of − u = V (x)eu in two dimensions. Comm. E. 16 (8,9), 1223–1253 (1991) 12. : Vortex condensation in the Chern–Simons–Higgs model: an existence theorem. Commun. Math. Phys. 168, 321–336 (1995) 13. : A special class of stationary flows for two dimensional Euler equations: A statistical mechanics description. Commun. Math. Phys. 143, 501–525 (1992) 14. : A special class of stationary flows for two dimensional Euler equations: A statistical mechanics description, part II.
Thus, for any λ ∈ , we can choose a sequence λn λ such that 0≤ cλn − cλ ≤ C, λ − λn for some constant C independent of n. (139) Liouville Type Equations with Singular Data 37 Using (139), we see that, for hn ∈ Dλn ⊂ Dλ satisfying: sup w∈hn (D) Jλn (w) ≤ cλn + λ − λn , (140) and for all w ∈ hn (D) such that Jλ (w) ≥ cλ − (λ − λn ) , we have: (141) 1 ln |M| V0 ew dτg = 2cotan2 θ M ≤ 2cotan2 θ Jλn (w) − Jλ (w) λ − λn cλn − cλ λ − λn ≤ 2cotan2 θ (C + 1) ≡ C1 , and consequently, ||∇w||2L2 (M) 1 ≤ 4cotan θJλn (w) + 2λn ln |M| V0 ew dτg 2 M ≤ 4cotan θ cλn + λ − λn + 2λn C1 .