Maximal solution of the Liouville equation in doubly connected domains

2018/08/01 by Kowalczyk, Michal, Pistoia, Angela, Vaira, Giusi · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1808.00127

Abstract

In this paper we consider the Liouville equation Δu +λ² e ^u=0 with Dirichlet boundary conditions in a two dimensional, doubly connected domain Ω. We show that there exists a simple, closed curve γ⊂ Ω such that for a sequence λ_n→ 0 and a sequence of solutions u_n it holds \fracu_nlog(1)/(λ_n)→ H, where H is a harmonic function in Ω∖γ and (λ_n²)/(log(1)/(λ_n))∫_Ωe ^u_n dx→ 8πc_Ω, where c_Ω is a constant depending on the conformal class of Ω only.

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