Maximal solution of the Liouville equation in doubly connected domains

Abstract

In this paper we consider the Liouville equation ∆u + λ 2 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 λn → 0 and a sequence of solutions un it holds un log 1 λn → H, where H is a harmonic function in Ω \ γ and λ 2 n log 1 λn Ω e un dx → 8πc Ω , where c Ω is a constant depending on the conformal class of Ω only. 1991 Mathematics Subject Classification. 35J25, 35J20, 35B33, 35B40.

Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities

“…We will show in Section 1.2 that ψ(γ) = ∂B R , R = √ R 1 R 2 . Given this the main result of [27] is the following:…”

“…We assume that V > 0 is a given function, λ > 0 is a small parameter and ρ = O(1) or ρ → +∞ as λ → 0. In a recent paper [27] we proved the existence of the maximal solutions for a particular choice V ≡ 1, ρ = 0 when the problem is posed in doubly connected domains under Dirichlet boundary conditions. We related the maximal solutions with a novel free boundary problem.…”

“…From now on we suppose that λ > 0 is a small parameter and Ω is a bounded, smooth domain, which is doubly connected and such that the bounded component of R 2 \ Ω is not a point. To state the existence result proven in [27] we will introduce the notion of the harmonic measure of a closed curve γ ∈ Ω.…”

“…It defines the orientation so that n is the exterior unit normal of Ω − on γ, and the interior unit normal of Ω + on γ. Keeping this in mind, we have [27]:…”

Phase separating solutions for two component systems in general planar domains

“…The method of the proof of the Theorem 1.1 is in part motivated by the the approach in [10] and relies on careful separation of the problem into the outer equation whose solution is approximately w 0 , given in (1.9), and the inner equation whose one dimensional, leading order solution is given by a suitable scaled solution of the ODE system (2.3) below. The challenge is to combine them locally, near Γ in a smooth way.…”