The sphere covering inequality and its applications

Abstract

We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a byproduct we find another sphere covering inequality which can be viewed as the dual of the original one. We also prove sphere covering inequalities on surfaces satisfying general isoperimetric inequalities, and discuss their applications to elliptic equations with exponential nonlinearities in dimension two. The approach in this paper extends, improves, and unifies several inequalities about solutions of elliptic equations with exponential nonlinearities.

Uniqueness and convergence on equilibria of the Keller–Segel system with subcritical mass

“…For equation ( * ) with Neumann boundary condition, we establish an integral inequality and prove that the solution of ( * ) is unique if 0 < λ ≤ 8π and u satisfies some symmetric properties. While for ( * ) with Dirichlet boundary condition, the same uniqueness result is obtained without symmetric condition by a different approach inspired by some recent works [19,21]. As an application of the uniqueness results, we prove that the radially symmetric solution of the classical Keller-Segel system with subcritical mass subject to Neumann boundary conditions will converge to the unique constant equilibrium as time tends to infinity if Ω is a disc in two dimensions.…”

“…[18,19,20,21,45] for details). Based on their results in [19,21], we shall derive the uniqueness of (1.12) in the subcritical mass cases. Remark 1.2: We remark that the the Dirichlet problem (1.12) no longer has a constant solution provided that λ > 0.…”

“…Next we shall recall some facts about the rearrangement with the measures. Such discussions have been detailed in [1,2,3,11,19,47], we shall sketch the process here only. For any function φ ∈ C 2 (Ω) which is constant on ∂Ω can be equimeasurably rearranged with respect to the measures e u dx and e U θ dx, where u is a function satisfying that ∆u + e u ≥ 0 and U θ is defined in (5.2).…”

“…Proposition 5.B. [19] Let u ∈ C 0,1 (Ω) satisfy ∆u + e u ≥ 0 in Ω, and let U θ be given in (5.2). Suppose φ ∈ C 2 (Ω) and φ ≡ C on ∂Ω.…”

Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions

“…To simplify our notation, we shall always assume |M | = 1. Equation (1.3) and its counterpart on bounded planar domains arise in several areas of mathematics and physics and there are by now many results concerning existence ( [2,15,9,10,11,24,29,30,41,42]), uniqueness of solutions ( [4,12,13,14,26,47,48,49,60,65]) and blow-up analysis ( [3,5,16,17,19,28,31,57,58]). On one hand, they are derived as a mean field limit in the statistical mechanics description of two dimensional turbulent Euler flows ( [20,21]) and selfgravitating systems ( [54,56,72]).…”