2026/07/20 by Hongge Chen, Juncheng Wei, Haicheng Yan +1
#math.AP #math-ph #math.MP
We prove that every smooth entire solution u\colonℝ2→ℝ2 of the Ginzburg--Landau equation -Δu=u(1-|u|2) with |u(x)|→1 as |x|→∞ has finite potential energy, i.e., ∫ℝ2(1-|u|2)2 dx<+∞, thereby resolving Brezis' Open Problem 2.5 in [4]. The main difficulty stems from the possible presence of a curl-free mode that carries nonzero circulation and decays only like |x|-1 ; such a mode lies outside L2 and does not admit a single-valued potential. By minimizing over L2 gradient corrections, we construct a comparison field that solves the homogeneous equation and inherits the same circulation. The Kelvin inversion, combined with the De Giorgi--Nash--Moser theory for quasilinear elliptic equations, then produces the optimal decay O(|x|-1) . For a Ginzburg--Landau solution, the Bernstein estimate and the coercivity of the Jacobi form produce an L2 forcing term in the exterior phase equation. The resulting L4 bound on the phase field implies 1-|u|2∈ L2(ℝ2), and therefore the potential energy is finite.