A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations

TL;DR

This paper introduces a singular Sphere Covering Inequality to analyze singular Liouville equations, leading to new uniqueness and symmetry results for solutions on spheres and bounded domains.

Contribution

It develops a singular version of the Sphere Covering Inequality and applies it to establish novel uniqueness and symmetry results for Liouville-type equations.

Findings

New uniqueness results for singular mean field equations

Symmetry results for spherical Onsager vortex equation

Alternative proofs for convex polytope uniqueness

Abstract

We derive a singular version of the Sphere Covering Inequality which was recently introduced in [42], suitable for treating singular Liouville-type problems with superharmonic weights. As an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains, as well as new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in [56]. Furthermore, we derive new symmetry results for the spherical Onsager vortex equation.

Equations

$$\Delta u+h(x)e^u=f(x)\text{mbox}in\Omega ,$$

$$\Delta u_i+h(x)e^{u_i}=f_i(x)\text{mbox}in\Omega ,$$

$$f_2\ge f_1\ge -\Delta H\text{mbox}in\Omega .$$

$$\left\{\begin{array}[]{ll}u_{2}\geq u_{1},\quad u_{2}\not\equiv u_{1}&\mbox{ in }\Omega,\vspace{0.2cm}\\ u_{2}=u_{1}&\mbox{ on }\partial\Omega.\end{array}\right.$$

$${\int }_{\Omega }\left(h(x)e^{u_1}+h(x)e^{u_2}\right)dx\ge 8\pi .$$

$$U_{\lambda ,\alpha }(x)=\ln {\left(\frac{\lambda (1-\alpha )}{1+\frac{{\lambda }^2}{8}|x{|}^{2(1-\alpha )}}\right)}^2,$$

$$\Delta U_{\lambda ,\alpha }+|x{|}^{-2\alpha }{e}^{U_{\lambda ,\alpha }}=0\text{mbox}in{\mathbb{R}}^2\setminus \{0\}.$$

$$\Delta F=f\mbox{ in $\Omega$},\quad F=0\mbox{ on $\partial\Omega$},$$

$$\Delta v+e^{H+F}{e}^v=0\text{mbox}in\Omega ,$$

$$H+F={\mathcal{H}}_{+}-{\mathcal{H}}_{-},$$

$${\mathcal{H}}_{\pm }(x)={\mathfrak{h}}_{\pm }(x)+{\int }_{\Omega }{G}_x(y)d{\mu }_{\pm }(y),$$

$$\left\{\begin{array}[]{rll}-\Delta G_{p}(y)=&\delta_{p}&\mbox{in }\Omega,\\ G_{p}(y)=&0&\mbox{on }\partial\Omega.\end{array}\right.$$

$$\alpha (\omega )=\frac{1}{4\pi }{\mu }_{+}(\tilde{\omega }).$$

$$\alpha(\omega)=\dfrac{1}{4\pi}\int_{E_{+}}-\Delta\bigr{(}H(x)+F(x)\bigr{)}\,dx=\dfrac{1}{4\pi}\int_{E_{+}}-\bigr{(}\Delta H(x)+f(x)\bigr{)}\,dx,$$

$$E_{+}=\bigr{\{}x\in\omega\,:\,-\bigr{(}\Delta H(x)+f(x)\bigr{)}>0\bigr{\}},$$

$$H(x)=4\pi \sum _{i=1}^m{\alpha }_i{G}_{p_i}(x)-4\pi \sum _{j=1}^n{\beta }_j{G}_{q_j}(x),$$

$$-\Delta H=4\pi \sum _{i=1}^m{\alpha }_i{\delta }_{p_i}-4\pi \sum _{j=1}^n{\beta }_j{\delta }_{q_j}.$$

$$\alpha (\omega )=\sum _{i\in I}{\alpha }_i,$$

$$\alpha (\Omega )<1\text{mbox}i.e.{\mu }_{+}(\Omega )<4\pi .$$

$$\Delta u_i+h(x)e^{u_i}=f_i(x)\text{mbox}in\Omega ,$$

$$f_2\ge f_1\text{mbox}a.e.in\Omega ,$$

$$\left\{\begin{array}[]{ll}u_{2}\geq u_{1},\quad u_{2}\not\equiv u_{1}&\mbox{in }\omega,\vspace{0.2cm}\\ u_{2}=u_{1}&\mbox{on }\partial\omega.\end{array}\right.$$

$$\int_{\omega}\bigr{(}h(x)e^{u_{1}}+h(x)e^{u_{2}}\bigr{)}\,dx\geq 8\pi(1-\alpha(\omega)).$$

$$\Delta u_i+h(x)e^{u_i}=f_i(x)\text{mbox}in\Omega ,$$

$$f_2\ge f_1\text{mbox}a.e.in\Omega ,$$

$$\left\{\begin{array}[]{ll}u_{1}\not\equiv u_{2}&\mbox{in }\Omega,\vspace{0.2cm}\\ u_{2}-u_{1}=c&\mbox{on }\partial\Omega,\end{array}\right.$$

$${\int }_{\Omega }h(x)e^{u_1}dx={\int }_{\Omega }h(x)e^{u_2}dx=\rho ,$$

$$\begin{array}{rl}& {\Delta }_{g}v+\rho \left(\frac{e^v}{{\int }_{{\mathbb{S}}^2}{e}^{v}d{V}_g}-\frac{1}{4\pi }\right)=4\pi \sum _{j=1}^N{\alpha }_j\left({\delta }_{p_j}-\frac{1}{4\pi }\right)\text{mbox}on{\mathbb{S}}^2,\\ & {\int }_{{\mathbb{S}}^2}vd{V}_g=0,\end{array}$$

$$\Pi (x_1,x_2,x_3)=\left(\frac{x_1}{1-x_3},\frac{x_2}{1-x_3}\right),$$

$$w(x)=v({\Pi }^{-1}(x))-\ln \left({\int }_{{\mathbb{S}}^2}{e}^{v}d{V}_g\right)x\in {\mathbb{R}}^2.$$

Full text

D.B. and A.J. are partially supported by FIRB project ”Analysis and Beyond”, by PRIN project 2012, ERC PE1_11, ”Variational and perturbative aspects in nonlinear differential problems”, and by the Consolidate the Foundations project 2015 (sponsored by Univ. of Rome ”Tor Vergata”), ”Nonlinear Differential Problems and their Applications”. C.G. is partially supported by NSF grant DMS-1601885 and NSFC grant No 11371128, and A.M. is supported in part by NSF grant DMS-1715850.

1. Introduction

We are concerned with a class of elliptic equations with exponential nonlinearities, namely the following Liouville-type equation,

[TABLE]

where $\Omega\subset\mathbb{R}^{2}$ is a smooth bounded domain and $h(x)$ is a positive function. The latter equation (and its counterpart on manifolds, see (15) below) has been widely discussed in the last decades since it arises in several problems of mathematics and physics, such as Electroweak and Chern-Simons self-dual vortices [65, 67, 75], conformal geometry on surfaces [71, 46, 24, 25], statistical mechanics of two-dimensional turbulence [20] and of self-gravitating systems [74] and cosmic strings [61], theory of hyperelliptic curves [22], Painlevé equations [27] and Moser-Trudinger inequalities [18, 37, 40, 45, 60]. There are by now many results concerning existence and multiplicity [3, 7, 8, 9, 11, 15, 16, 21, 30, 32, 34, 35, 36, 54, 57, 58], uniqueness [12, 13, 14, 42, 44, 50, 52, 66], blow-up phenomena [6, 10, 17, 19, 29, 31, 47, 48, 69, 76] and entire solutions [2, 28, 63].

1.1. Singular Sphere Covering Inequality

A basic inequality related to (1) was recently introduced in [42], see Theorem A below, which yields sharp Moser-Trudinger inequalities and symmetry properties of Liouville type equations in $\mathbb{R}^{2}$ [42], symmetry properties of mean field equations on flat tori [43], uniqueness of solutions of mean field equation in bounded domains [44] and symmetry and uniqueness properties of Sinh-Gordon equation and Toda systems in bounded domains [41]. The inequality can be stated in the following form, see Theorem 3.1 in [42] and Theorem 1.1 in [44].

Theorem A ([42]). Let $\Omega\subset\mathbb{R}^{2}$ be a smooth, bounded, simply-connected domain and let $u_{i}\in C^{2}(\Omega)\cap C(\overline{\Omega})$, $i=1,2$, satisfy,

[TABLE]

where $h=e^{H}\in C^{2}(\Omega)\cap C(\overline{\Omega})$ is such that,

[TABLE]

Suppose that,

[TABLE]

Then it holds,

[TABLE]

The latter result is based on symmetric rearrangements and the Alexandrov-Bol inequality, see the discussion in the sequel. Suppose for simplicity $h\equiv 1$ and $f_{1}\equiv f_{2}\equiv 0$ in (2). Then, by means of the substitution $\widetilde{u}_{i}=\dfrac{u_{i}}{2}-\ln(2)$, Theorem A roughly asserts that the total area of two distinct neighbourhoods $M_{1},M_{2}$, with Gaussian curvature equal to $1$, of possibly distinct surfaces, such that $M_{1}$ and $M_{2}$ admits local conformal charts $\Phi_{i}:M_{i}\to B_{1}$, $i=1,2$ where $B_{1}$ is the Euclidean unit disk, with the same conformal factor on the boundary, is greater than that of the whole unit sphere, which is why one refers to the latter result as the Sphere Covering Inequality.

We point out that condition (3) on the weight $h(x)$ can not be dropped and indeed the result is false in general if we remove such assumption. Suppose for a moment $f_{1}\equiv f_{2}\equiv 0$ in Theorem A. With a small abuse of terminology, a weight $h$ satisfying (3) with $f_{1}\equiv f_{2}\equiv 0$, i.e. $\Delta H\geq 0$, will be referred as a subharmonic weight. Analogously, we will refer to superharmonic weights whenever we have the reverse inequality $\Delta H\leq 0$. Such restriction on the weights, which in this case need to be subharmonic, prevents the application of the Sphere Covering Inequality in a large class of problems which are rather interesting and challenging, some of which we will address later on.

On the other hand, Theorem A is obtained in a smooth setting and thus is not suitable in treating singular problems. However, the presence of singular sources in (1) naturally arises both in geometry and in mathematical physics, typically as a sum of Dirac deltas which represent either conic points of the metric or vortex points of gauge or vorticity fields (see the references above). One of our aims is to address such kind of problems, see subsections 1.2, 1.3. However, the generalization of Theorem A to a weak setting is not straightforward and one needs to carry out a delicate argument.

Therefore, our first goal in this paper is to provide a singular version of the Sphere Covering Inequality suitable for treating both singular problems and superharmonic weights, see Theorem 1.2 below. In order to state the result let us introduce some notation first. Given $\alpha\in[0,1)$, $\lambda>0$ we set

[TABLE]

which satisfies

[TABLE]

Let $\Omega\subset\mathbb{R}^{2}$ be a smooth and bounded domain. Let $f\in L^{q}(\Omega)$ for some $q>1$ and $h=e^{H}$ be given. Let $F$ be the solution of

[TABLE]

so that (1) can be equivalently formulated as

[TABLE]

where $v=u-F$. By the Riesz decomposition we have,

[TABLE]

where $\mathcal{H}_{+},\mathcal{H}_{-}$ are two superharmonic functions taking the form

[TABLE]

where $\mathfrak{h}_{\pm}$ are harmonic functions in $\Omega$, $\mu_{\pm}$ are non negative and mutually orthogonal measures of bounded total variation in $\Omega$ and $G_{p}$, $p\in\Omega$, is the Green’s function,

[TABLE]

Definition 1.1**.**

Let $\omega\subseteq\Omega$ be any nonempty subdomain. We will denote by $\widetilde{\omega}$ the interior of the closure of the union of $\omega$ with its “holes”, that is, with the bounded component of the complement of $\omega$ in $\mathbb{R}^{2}$. If $\omega$ is simply-connected or if it is the union of simply connected domains, then $\widetilde{\omega}\equiv\omega$, while if the bounded component of the complement of $\omega$ in $\mathbb{R}^{2}$ is not empty, then $\omega\subset\widetilde{\omega}$.

Definition 1.2**.**

Let $f,h$ be given as above and let $\mu_{+}$ be defined as in (6). Let $\omega\subseteq\Omega$ be any nonempty subdomain and let $\widetilde{\omega}$ be given as in Definition 1.1. We define $\alpha(\omega)=\alpha(\omega,h,f)\geq 0$ to be

[TABLE]

For reader’s convenience let us show how the above construction applies to two model cases.

Example 1.1**.**

Let us consider a smooth setting where $f,h\in C^{2}(\Omega)\cap C(\overline{\Omega})$ and suppose for simplicity $\widetilde{\omega}=\omega$. Then we have,

[TABLE]

where

[TABLE]

*while clearly $\alpha(\omega)=0$ whenever $E_{+}=\emptyset$.

---

Example 1.2**.**

Let us consider a singular weight $h=e^{H}$ in the form

[TABLE]

where $G_{p}(x)$ is given in (7), $p_{i},q_{j}$ are distinct points in $\Omega$ and $\alpha_{i},\beta_{j}>0$. We have,

[TABLE]

Suppose for simplicity $f=0$ and $\widetilde{\omega}=\omega$. Then we have $\mu_{+}=4\pi\sum_{i=1}^{m}\alpha_{i}\delta_{p_{i}}$ and

[TABLE]

*where I=\bigr{\{}i\in\{1,\dots,m\}\,:\,p_{i}\in\omega\bigr{\}}.

---

Finally, from now on we will suppose that,

[TABLE]

This is motivated by the case when all the measure $\mu_{+}$ is collapsed to a singular Dirac delta, i.e. $\mu_{+}=4\pi\alpha_{p}\delta_{p}$, where we need $\alpha(\Omega)=\alpha_{p}<1$ to ensure the integrability of $e^{H+F}$. From the geometric point of view this means that $p$ is not a ”cusp” but just a conical singularity, see [64].

Remark 1.1**.**

In particular, since we are assuming $\mu_{+}(\Omega)<4\pi$, then we deduce the following regularity property. Observe that there exists at most one point $x_{0}\in\Omega$ such that $\mu_{+}(x_{0})\geq 2\pi$. Thus, if we start from $he^{u}\in L^{1}(\Omega)$ where $u\in L^{1}(\Omega)$ is a solution of (1) in the sense of distributions, then it can be shown that $u\in W^{2,q}(\Omega)$ for some $q>1$ and that for each $r>0$ small enough there exists $s_{r}>2$ such that $u\in W^{2,s_{r}}(\Omega\setminus B_{r}(x_{0}))$, see [5]. We will quote this regularity property by saying that $u\in W_{\mbox{\rm\scriptsize loc}}^{2,s,\mbox{\rm\scriptsize loc}}(\Omega\setminus\{x_{0}\})$ for some $s>2$. In particular we conclude that $u$ is a strong solution of (1).

Our first main result is the following singular Sphere Covering Inequality.

Theorem 1.2**.**

Let $\Omega\subset\mathbb{R}^{2}$ be a smooth, bounded, simply-connected domain. Let $x_{0}\in\Omega$ be fixed as in Remark 1.1 and $u_{i}\in W_{\mbox{\rm\scriptsize loc}}^{2,s,\mbox{\rm\scriptsize loc}}(\Omega\setminus\{x_{0}\})\cap W^{2,q}(\Omega)\cap C(\overline{\Omega})$ for some $s>2$ and some $q>1$, $i=1,2$, satisfy,

[TABLE]

where $h=e^{H}$ and $f_{1},f_{2}\in L^{q}(\Omega)$ are such that $\alpha(\Omega,h,f_{i})<1$, $i=1,2$, with $\alpha(\Omega,h,f_{i})$ defined as in (8). Suppose that,

[TABLE]

and that there exists a smooth subdomain $\omega\subseteq\Omega$ such that,

[TABLE]

Let $\alpha(\omega)=\alpha(\omega,h,f_{1})$. Then, it holds

[TABLE]

Moreover, the equality holds if and only if (modulo conformal transformations) $\omega=B_{\delta}(0)$ for some $\delta>0$, $f_{1}\equiv f_{2}:=f$, $h(x)e^{u_{i}}\equiv|x|^{-2\alpha}e^{U_{\lambda_{i},\alpha}}$, $i=1,2$, for some $\lambda_{2}>\lambda_{1}$ where $U_{\lambda_{i},\alpha}$ are defined as in (4), $\mu_{+}=-\Delta H-f=4\pi\alpha\delta_{p=0}$ in $\omega$ and $\alpha=\alpha(\omega)$.

Remark 1.3**.**

*We point out that if $f_{1}\equiv f_{2}$, the condition $u_{2}\geq u_{1},u_{2}\not\equiv u_{1}$ in the above theorem can be replaced just by $u_{2}\not\equiv u_{1}$.

We also observe that if $\alpha(\omega)=0$ we recover the standard Sphere Covering Inequality of Theorem A, which however holds now in a weak setting.*

Remark 1.4**.**

The smoothness assumption in Theorem 1.2 about $\omega$ is not necessary and can be remarkably weakened as far as $\omega\Subset\Omega$, see [5].

The latter result is obtained in the spirit of Theorem A with a non trivial adaptation to the weak setting by exploiting weighted symmetric rearrangements and a singular Alexandrov-Bol’s inequality, see section 2 for full details. We stress that, according to the terminology introduced right after Theorem A, we are now able to cover the case of superharmonic weights.

Suppose for the moment that $f_{1}\equiv f_{2}\equiv 0$ in Theorem 1.2. Observe that the case $h(x)e^{u_{i}}\equiv|x|^{-2\alpha}e^{U_{\lambda_{i},\alpha}}$, $i=1,2$, corresponding to the equality in Theorem 1.2 is a limiting case in which all the measure $\mu_{+}$ is concentrated in a Dirac delta, i.e. $\mu_{+}=4\pi\alpha\delta_{p=0}$. In particular, the latter Dirac delta corresponds to a conical singularity of order $-\alpha$ on the surface related to (5), see for example [5]. In other words, by means of the substitution $\widetilde{u}_{i}=\dfrac{u_{i}}{2}-\ln(2)$, in the terminology of singular surfaces (in the sense of Alexandrov [64]), Theorem 1.2 roughly asserts that the total area of two distinct neighbourhoods $M_{1},M_{2}$, with regular Gaussian curvature equal to $1$, of possibly distinct singular surfaces, such that $M_{1}$ and $M_{2}$ admits local conformal charts $\Phi_{i}:M_{i}\to B_{1}$, $i=1,2$ where $B_{1}$ is the Euclidean unit disk, with the same conformal factor on the boundary, and the same local total (singular) curvature (which is $\frac{1}{4\pi}\mu=-\frac{1}{4\pi}\Delta H$ in $B_{1}$), is greater than that of a whole unit sphere with two antipodal conical singularities of order $-\alpha$ (where $\alpha$ is $\frac{1}{4\pi}\mu_{+}(B_{1})$), namely an ’American football’. We refer to [5] for more details concerning this geometric interpretation. Therefore we allude to Theorem 1.2 as the singular Sphere Covering Inequality.

Let us return to equation (10). If we further assume the two solutions $u_{1},u_{2}$ have the same total mass in $\Omega$, i.e. that (14) below holds, then we can argue as in [44] and improve Theorem 1.2. More precisely, we can relax the boundary condition in (11) and treat more general situations, see (13). This will be crucially used in proving uniqueness of solutions of singular Liouville equation (21) on bounded domains, see Theorem 1.8. Even though Theorem 1.5 shares some similarities with Theorem 1.2, its proof, based on a reversed Alexandrov-Bol’s inequality, substantially differs from the proof of Theorem 1.2, see section 3. Our second main result is the following.

Theorem 1.5**.**

Let $\Omega\subset\mathbb{R}^{2}$ be a smooth, bounded, simply-connected domain. Let $x_{0}\in\Omega$ be fixed as in Remark 1.1 and $u_{i}\in W_{\mbox{\rm\scriptsize loc}}^{2,s,\mbox{\rm\scriptsize loc}}(\Omega\setminus\{x_{0}\})\cap W^{2,q}(\Omega)\cap C(\overline{\Omega})$ for some $s>2$ and some $q>1$, $i=1,2$, satisfy,

[TABLE]

where $h=e^{H}$ and $f_{1},f_{2}\in L^{q}(\Omega)$ are such that $\alpha(\Omega,h,f_{i})<1$, $i=1,2$, where $\alpha(\Omega,h,f_{i})$ are defined in (8). Suppose that,

[TABLE]

and that,

[TABLE]

for some $c\in\mathbb{R}$. Suppose moreover that,

[TABLE]

and set $\alpha(\Omega)=\alpha(\Omega,h,f_{1})$. Then, it holds $\rho>8\pi(1-\alpha(\Omega))$.

It is worth to point out the following fact concerning the assumption about $\Omega$ being simply-connected.

Remark 1.6**.**

Actually, both Theorems 1.2, 1.5 hold for multiply-connected domains $\Omega$ provided the solutions of (10) take constant values on $\partial\Omega$, more precisely $u_{i}+H=c_{i}$ on $\partial\Omega$ for some $c_{1},c_{2}\in\mathbb{R}$. This follows from the fact that in the latter situation an Alexandrov-Bol’s inequality related to (10) with $\Omega$ multiply-connected is available, see [13].

To motivate our studies and to see some applications of Theorems 1.2 and 1.5 we will address uniqueness issues concerning singular Liouville-type equations both on spheres and on bounded domains, and symmetry properties for the spherical Onsager vortex equation, see respectively subsections 1.2, 1.3, 1.4 below. The argument will yield both to new results as well as to new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in [56]. We believe that Theorems 1.2 and 1.5 will have several other applications as it was for Theorem A (see the discussion before Theorem A). This will be the topic of our forthcoming papers.

1.2. The singular Liouville equation on $\mathbb{S}^{2}$.

Let us start by considering the following equation

[TABLE]

where $\rho$ is a positive parameter, $\{p_{1},\dots,p_{N}\}\subset\mathbb{S}^{2}$, $\alpha_{j}>-1$ for $j=1,\dots,N$ and $\mathbb{S}^{2}\subset\mathbb{R}^{3}$ is the unit sphere with $|\mathbb{S}^{2}|=4\pi$ equipped with its standard Riemannian metric $g$, $\Delta_{g}$ is the Laplace-Beltrami operator and $dV_{g}$ is the volume form. Since the equation in (15) is invariant under translations $v\mapsto v+c$, then we can normalize solutions to have zero mean value.

Problem (15) is related to both mean field equations with vortex points and spherical metrics with conic singularities. We refer to the references in the beginning of the introduction for more details and some of the known results.

Equation (15) can be equivalently considered on the plane $\mathbb{R}^{2}$ via the stereographic projection: suppose without loss of generality that no one of the points $p_{i}$’s coincides with the north pole $\mathcal{N}=(0,0,1)\in\mathbb{R}^{3}$ and let $\Pi:\mathbb{S}^{2}\setminus\{\mathcal{N}\}\to\mathbb{R}^{2}$ be the stereographic projection with respect to $\mathcal{N}$, i.e.

[TABLE]

and define

[TABLE]

With a small abuse of notation we will write $\sum_{j}\alpha_{j}$ to denote $\sum_{j=1}^{N}\alpha_{j}$. Then, $w$ satisfies

[TABLE]

for some $\{q_{1},\dots,q_{N}\}\subset\mathbb{R}^{2}$. Letting further,

[TABLE]

we get

[TABLE]

where

[TABLE]

and

[TABLE]

Next, let us discuss what is the measure $\mu_{+}$ defined in (6) corresponding to equation (17). To make the presentation simpler let us consider the regular case, i.e. $N=0$, postponing the general case to section 4. Recall the notation $h=e^{H}$. In order to apply either Theorem A or Theorem 1.2 we need first to consider the sign of

[TABLE]

Observe that the sign of this term depends on whether $\rho<8\pi$ or $\rho\geq 8\pi$. The value $\rho=8\pi$ plays an important role in Liouville-type problems alike (15): for example, it is related to the sharp Moser-Trudinger inequality which yields boundedness from below and coercivity of the energy functional associated to (15) for $\rho<8\pi$ (we refer to the survey [57] for full details on this matter). For the latter range of the parameter $\rho$ one expects the solution of (15) to be unique: this indeed holds true and it was first obtained in [49] and [50]. The argument used is mainly based on the deep results derived in [33]: by the asymptotic behavior of solutions to equations alike (17) one carries out the moving plane method to show that all the solutions to (17), (18) with $\rho<8\pi$ are radially symmetric with respect to the origin.

However Theorem A does not apply in this framework since the term in (19) is negative and thus the weight $h(x)$ is superharmonic according to the terminology introduced right after Theorem A.

The argument in [49] applies also to the singular case $N=1$ with singular source $4\pi\alpha_{q}\delta_{q}$, $\alpha_{q}>-1$. Then, letting $\alpha_{-}=\min\{\alpha_{q},0\}$ one deduces uniqueness of solution for $\rho<8\pi(1+\alpha_{-})$. On the other hand, the case of multiple singular sources $N\geq 3$ is almost completely open due to the fact that we can not rely on radial properties any more. The only exceptions concern metrics of constant Gaussian curvature with $N$-conical singularities at $p_{j}$ of negative order $\alpha_{j}\in(-1,0)$ on $\mathbb{S}^{2}$ (convex polytopes), that is, solutions of (15) with,

[TABLE]

Based on an algebraic geometric approach, the authors in [56] showed (among other things) the uniqueness of such metrics in the subcritcal case [71] (i.e. 4\pi\bigr{(}2+\sum_{j}\alpha_{j}\bigr{)}<8\pi\bigr{(}1+\min\limits_{j}\{\alpha_{j},0\}\bigr{)}), when $\alpha_{j}\in(-1,0)$ for all $j=1,\dots,N$ and $N\geq 3$. Here we will exploit the singular Sphere Covering Inequality of Theorem 1.2 to handle both superharmonic weights and the multiple singular sources in (17), see section 4. As a consequence we obtain uniqueness results relevant for both the mean field theory of 2D turbulence [20, 26] and the uniqueness of convex polytopes [56]. Our third main result is the following,

Theorem 1.7**.**

Let $\rho>0$ and $\alpha_{j}\in(-1,0)$ for $j=1,\dots,N$, $N\geq 0$. Then we have:

$(i)$* If $N\geq 0$ and \rho<4\pi\bigr{(}2+\sum_{j}\alpha_{j}\bigr{)}, then (15) admits at most one solution;*

$(ii)$* If $N\geq 3$ and \rho=4\pi\bigr{(}2+\sum_{j}\alpha_{j}\bigr{)}, then (15) admits at most one solution.*

The proof of Theorem 1.7 is rather delicate since, among other facts (see Remark 1.10), it requires a careful use of the characterization of the equality sign in the singular Sphere Covering Inequality (12). After all this is not surprising since part $(ii)$ is somehow sharp. Indeed, for $N=1$ solutions to (15) with $\rho=4\pi(2+\alpha_{1})$ do not exist, as it is well known that the ’tear drop’ (which is $\mathbb{S}^{2}$ with one conical singularity) does not admit constant curvature, see for example [2, 38]. On the other hand, for $N=0$ and $N=2$ (still with $-1<\alpha_{1}\leq\alpha_{2}<0$ and \rho=4\pi\bigr{(}2+\sum_{j}\alpha_{j}\bigr{)}), solutions to (15) are classified and uniqueness does not hold, see [28, 63, 72]. In particular, uniqueness fails in general if some $\alpha_{j}$ is positive, see for example [56].

Moreover, Theorem 1.7 also covers most of the previously known results and gives a new self-contained proof of them. Indeed, for $N=0$ and $\rho<8\pi$, we get the sharp uniqueness result of [49, 50], for $N=1$ and $\rho<8\pi(1+\alpha_{1})<4\pi(2+\alpha_{1})$, we obtain the sharp result of [49], while part $(ii)$ covers the uniqueness of convex polytopes [56].

Finally, we remark that whenever the subcriticality condition \rho<8\pi\bigr{(}1+\min\limits_{j}\{\alpha_{j},0\}\bigr{)} is also satisfied, then we have existence [71] and uniqueness in $(i)$ and $(ii)$. The fact that the uniqueness threshold 4\pi\bigr{(}2+\sum_{j}\alpha_{j}\bigr{)} may be larger than the subcritical threshold seems to suggest another possible application of Theorem 1.7 to the non existence issue for (15) in the supercritical regime \rho\in\bigr{(}8\pi\bigr{(}1+\min\limits_{j}\{\alpha_{j},0\}\bigr{)},4\pi\bigr{(}2+\sum_{j}\alpha_{j}\bigr{)}\bigr{)}. Indeed, the recent evaluation of the topological degree $d_{\rho}$ associated to (15) in [32] shows that if $N\geq 3$ and $-1<\alpha_{1}<\dots<\alpha_{N}<0$, then $d_{\rho}=0$ for \rho\in\bigr{(}8\pi(1+\alpha_{1}),4\pi\bigr{(}2+\sum_{j}\alpha_{j}\bigr{)}\bigr{)}. If we knew that any such a solution is non degenerate, then we would conclude by Theorem 1.7 that solutions do not exist in this supercritical region. This motivates the following:

Open problem. Is it true that if $N\geq 3$,  $-1<\alpha_{1}<\dots<\alpha_{N}<0$ and \rho\in\bigr{(}8\pi(1+\alpha_{1}),4\pi\bigr{(}2+\sum_{j}\alpha_{j}\bigr{)}\bigr{)}, then (15) has no solutions?

We point out that some non existence results in this direction were obtained in [2, 15, 68] only for the case $N\leq 2$ by using Pohozaev-type identities.

In concluding this part we mention the nondegeneracy result for solutions of (15) for $N\geq 3$ with only one or two negative $\alpha_{j}$ recently obtained in [73].

1.3. The singular Liouville equation on bounded domains.

Next let us consider the counterpart of (15) on bounded domains, that is,

[TABLE]

where $\Omega\subset\mathbb{R}^{2}$ is a smooth open bounded domain, $\rho$ is a positive parameter, $\{p_{1},\dots,p_{N}\}\subset\Omega$ and $\alpha_{j}>-1$ for $j=1,\dots,N$. The latter equation is related to mean field equations of turbulent Euler flows and we refer to the references above for more details about this point.

As for (15) in subsection 1.2 one expects uniqueness of solutions to (20) below a certain level of $\rho$. Indeed, for the regular case (i.e. $N=0$), uniqueness was first proved in [66] for $\Omega$ simply-connected and $\rho<8\pi$, then improved in [23] for $\rho=8\pi$ and finally generalized to the case of $\Omega$ multiply-connected in [13] and [44] for the case of more general boundary conditions. The argument is mainly based on the Alexandrov-Bol inequality and the study of the linearized equation, and it was generalized in [12] to cover the singular case where $\rho\leq 8\pi$ and $\alpha_{j}>0$ for all $j=1,\dots,N$, $N>0$. More recently, in [73] the authors considered the case of one negative singularity, i.e. $\alpha_{1}\in(-1,0)$ and $\alpha_{j}>0$ for all $j=2,\dots,N$, proving uniqueness of solutions provided $\rho\leq 8\pi(1+\alpha_{1})$.

On the other hand, the case of multiple negative singular sources is completely open and we will apply the singular Sphere Covering Inequality of Theorem 1.5 to handle this situation. Indeed, suppose that there exist $u_{1},u_{2}$ satisfying (20). We set,

[TABLE]

so that, for $i=1,2$, we have,

[TABLE]

where

[TABLE]

and

[TABLE]

Next, let us define,

[TABLE]

and let $\alpha=\alpha(\Omega,h,f)>0$, satisfying (9), be the measure associated to (21) as defined in (8), that is

[TABLE]

Then, Theorem 1.5 readily yields our fourth main result.

Theorem 1.8**.**

Let $\Omega\subset\mathbb{R}^{2}$ be a smooth, bounded and simply-connected domain and fix $\rho\leq 8\pi(1-\alpha)$. Then (20) admits at most one solution.

We remark that (20) admits a variational formulation and that the corresponding functional is well known to be coercive for \rho<8\pi\bigr{(}1+\min\limits_{j}\{\alpha_{j},0\}\bigr{)}. Therefore we have existence and uniqueness in Theorem 1.8 if either $|J|\geq 2$ and $\rho\leq 8\pi(1-\alpha)$ or if $|J|=1$ and $\rho<8\pi(1-\alpha)\equiv 8\pi(1+\alpha_{1})$ or if $|J|=0$ and $\rho<8\pi$.

Remark 1.9**.**

Since Theorem 1.5 applies to multiply-connected domains, see Remark 1.6 for more details, we conclude that Theorem 1.8 holds for $\Omega$ multiply-connected as well. Moreover, we can also treat the case where $e^{u}$ is replaced by $e^{H}e^{u}$ with $H$ subharmonic and where $u=f$ on $\partial\Omega$ with $f\in C(\partial\Omega)$.

We point out that Theorem 1.8 covers all the previously known results. Indeed, for $N=0$ and $\rho\leq 8\pi$, we get the sharp uniqueness result of [23] and [13] (i.e. the regular case). For $N>0$, $\alpha_{j}>0$ for all $j=1,\dots,N$ and $\rho\leq 8\pi$, we get the sharp result of [12], while in the case $N\geq 2$ of only one negative singularity $\alpha_{1}\in(-1,0)$ with $\rho\leq 8\pi(1+\alpha_{1})$ we recover the sharp result of [73].

Remark 1.10**.**

It is worth to make a remark about the discrepancy of the uniqueness thresholds as obtained in Theorems 1.7 and 1.8. It turns out that a rather elementary but still crucial point in the proof of Theorem 1.7 is that, since the equation is solved on $\mathbb{S}^{2}$, then one has an upper bound on the total (positive) singular curvature, see (58). This estimate, in turn, allows one to adopt an optimization trick which rules out the case where the singular Sphere Covering Inequality would yield no information, that is when $\alpha(\omega)\geq 1$ in Theorem 1.2. This is not anymore possible on a bounded domain and we come up with a threshold which, for $|J|\geq 2$ and unlike the case of $\mathbb{S}^{2}$, is always lower than the subcritical existence threshold 8\pi\bigr{(}1+\min\limits_{j}\{\alpha_{j},0\}\bigr{)}. Actually, it seems that this uniqueness result could have been obtained also by an adaptation of the argument in [12] and it is a challenging open problem to understand whether or not uniqueness still holds for \rho\in\bigr{(}8\pi(1-\alpha),8\pi\bigr{(}1+\min\limits_{j}\{\alpha_{j},0\}\bigr{)}\bigr{)} with $|J|\geq 2$.

1.4. The Onsager mean field equation on the sphere.

Let us consider the equation,

[TABLE]

where $\langle\cdot,\cdot\rangle$ is the scalar product in $\mathbb{R}^{3}$, $n=\vec{n}\in\mathbb{R}^{3}$ is a unit vector, $\beta\geq 0$ and $\gamma\in\mathbb{R}$. Since $\gamma$ can be changed to $-\gamma$ by replacing the north pole with the south pole, there is no loss of generality in assuming that $\gamma\geq 0$. Observe that the equation in (22) is invariant under the addition of a constants, which is why we can impose the condition of zero mean value. Equation (22) is the mean field equation arising from the spherical Onsager vortex theory, see [26, 59, 62].

Let us briefly list the known results concerning (22). By a moving plane argument, it is shown in [50] that if $\beta<8\pi$, then for any $\gamma\geq 0$ the equation (22) has a unique solution which is axially symmetric with respect to $\vec{n}$. Moreover, the author made the following conjecture.

Conjecture B. *Let $\gamma>0$ and $\beta\leq 16\pi$. Then every solution of (22) is axially symmetric with respect to $\vec{n}$. *

In this direction the following results for $\beta>8\pi$ has been proved in [51].

Theorem C ([51]). For every $\gamma>0$, there exists $\beta_{0}=\beta_{0}(\gamma)>8\pi$ such that, for $8\pi<\beta\leq\beta_{0}$, any solution of (22) is axially symmetric with respect to $\vec{n}$.

Theorem D ([51]). Let $\{v_{i}\}_{i}$ be a sequence of solutions of (22) with $\gamma=0$ and $\beta_{i}\rightarrow 16\pi$. Suppose that $\lim_{i\rightarrow\infty}\sup_{\mathbb{S}^{2}}v_{i}(y)=+\infty$. Then $v_{i}$ is axially symmetric with respect to some direction $\vec{n}_{i}$ in $\mathbb{R}^{3}$ for $i$ large enough.

Recently, in [42] the authors applied the standard Sphere Covering inequality, see Theorem A, to prove the following result.

Theorem E ([42]). Suppose $8\pi<\beta\leq 16\pi$ and

[TABLE]

Then every solution of (22) is axially symmetric with respect to $\vec{n}$.

The aim here is to use the singular Sphere Covering Inequality, Theorem 1.2, to get a new symmetry result. Our fifth main result is the following.

Theorem 1.11**.**

Suppose $8\pi<\beta\leq 16\pi$ and

[TABLE]

Then every solution of (22) is evenly symmetric with respect to a plane passing through the origin and containing the vector $\vec{n}$.

Remark 1.12**.**

We point out that for $8\pi<\beta\leq 16\pi$ it holds,

[TABLE]

and thus Theorem 1.11 covers a wider range of parameters compared to Theorem E.

This paper is organized as follows. In section 2 we introduce the argument which yields to the proof of the singular Sphere Covering Inequality of Theorem 1.2, in section 3 we deduce the improved version of it under same total mass condition, see Theorem 1.5, in section 4 we prove the uniqueness result for the singular Liouville equation on $\mathbb{S}^{2}$, see Theorem 1.7, and in section 5 we finally derive symmetry of solutions for the spherical Onsager vortex equation, i.e. Theorem 1.11.

2. The Singular Sphere Covering Inequality.

In this section we derive the singular Sphere Covering Inequality of Theorem 1.2. The argument is mainly based on weighted symmetric rearrangements and a singular Alexandrov-Bol’s inequality in the spirit of Theorem A.

Let us start by recalling the following version of the Alexandrov-Bol inequality, first proved in the analytical framework in [1] and more recently generalized to the weak setting in [4, 5].

Proposition 2.1** ([4, 5]).**

Let $\Omega\subset\mathbb{R}^{2}$ be a smooth, bounded, simply-connected domain. Let $x_{0}\in\Omega$ be fixed as in Remark 1.1 and let $u\in W_{\mbox{\rm\scriptsize loc}}^{2,s,\mbox{\rm\scriptsize loc}}(\Omega\setminus\{x_{0}\})\cap W_{\mbox{\rm\scriptsize loc}}^{2,q}(\Omega)$ for some $s>2$ and some $q>1$, satisfy

[TABLE]

where $h=e^{H}$ and $f\in L_{\mbox{\scriptsize\emph{loc}}}^{q}(\Omega)$ are such that $\alpha(\Omega,h,f)$ (as defined in (8)) satisfies $\alpha(\Omega,h,f)<1$. Let $\omega\subseteq\Omega$ be a smooth subdomain and let $\alpha(\omega)=\alpha(\omega,h,f)$. Then it holds,

[TABLE]

Moreover, the equality holds if and only if (modulo conformal transformations) $\omega=B_{\delta}(0)$ for some $\delta>0$, $h(x)e^{u}\equiv|x|^{-2\alpha}e^{U_{\lambda,\alpha}}$ for some $\lambda_{2}>\lambda_{1}$ where $U_{\lambda,\alpha}$ is defined in (4), $\mu_{+}=-\Delta H-f=4\pi\alpha\delta_{p=0}$ in $\omega$ and $\alpha=\alpha(\omega)$. In particular, if $\omega$ is not simply-connected, then the inequality is always strict.

Remark 2.2**.**

The smoothness assumption about $\omega$ is not necessary and can be remarkably weakened as far as $\omega\Subset\Omega$, see [5]. Moreover, we point out that if $\mu_{+}=0$ then $\alpha(\omega,h,f)=0$ and we recover the standard Alexandrov-Bol inequality.

We will need in the sequel the following counterpart of the singular Alexandrov-Bol inequality in the radial setting which is derived in the spirit of [42, 66].

Proposition 2.3**.**

Let $\alpha\in[0,1)$, $R>0$ and $\psi\in C(\overline{B_{R}(0)})\cap W^{1,p}(B_{R}(0))$ for some $p>2$, be a strictly decreasing radial function satisfying,

[TABLE]

Then,

[TABLE]

Moreover, if $\int_{\partial B_{r}(0)}|\nabla\psi|\,d\sigma\not\equiv\int_{B_{r}(0)}|x|^{-2\alpha}e^{\psi}\,dx$