On a singular Liouville-type equation and the Alexandrov isoperimetric inequlity

Abstract

We obtain a generalized version of an inequality, first derived by C. Bandle in the analytic setting, for weak subsolutions of a singular Liouville-type equation. As an application we obtain a new proof of the Alexandrov isoperimetric inequality on singular abstract surfaces. Interestingly enough, motivated by this geometric problem, we obtain a seemingly new characterization of local metrics on Alexandrov's surfaces of bounded curvature. At least to our knowledge, the characterization of the equality case in the isoperimetric inequality in such a weak framework is new as well.

The sphere covering inequality and its applications

“…In particular, if we assume further that µ (M ) ≤ 4π(1−Θ) a , then (M, g) satisfies (1 − Θ, a)-isoperimetric inequality. In fact this is even true when g is singular, see [BC1,BC2,BGJM] and the references therein.…”

Non-degeneracy and uniqueness of solutions to singular mean field equations on bounded domains

“…We will refer to the latter property by saying that u ∈ W 2,s,loc (Ω \ {x 0 }) for some s > 2. Clearly u is a strong solution of (5) and similar integrability properties are deduced also on the weight h, see Proposition 1.4 in [5].…”

“…To this end, the first tool we need is an Alexandrov-Bol's inequality for solutions of (5) suitable for our setting. Such an inequality was first proved in the analytical framework in [1] and more recently generalized to the weak setting in [4,5]. However what we need here is a more general statement which allows one to push the inequality, still in this weak setting, up to the (non-smooth) boundary of the domain.…”

“…Therefore, the first aim of this paper is to complete the above program and to prove non-degeneracy of solutions for (1) for general singular data on nonsmooth domains. Actually, we will develop the strategy for a much more general problem, see (5) below, in which the sum of Dirac deltas may be replaced by a general measure of bounded variation. This is somehow the more general form of the singularities which one can attach to the Dirichlet problem for (3) as it naturally arises in the analysis of Alexandrov surfaces with bounded integral curvature, see [5] and references therein.…”

“…Next, since µ + (Ω) < 4π, then there exists at most one point x 0 ∈ Ω such that µ + (x 0 ) ≥ 2π. As a consequence, by arguing as in [5], if he u ∈ L 1 (Ω) where u ∈ L 1 (Ω) is a solution of (5) just in the sense of distributions, then we gain u ∈ W 2,q (Ω) for some q > 1 and in particular, for each r > 0 small enough, there exists s r > 2 such that u ∈ W 2,sr (Ω \ B r (x 0 )). We will refer to the latter property by saying that u ∈ W 2,s,loc (Ω \ {x 0 }) for some s > 2.…”