Solutions Of Mean-field Equations Research Articles

Uniqueness of bubbling solutions of mean field equations with non-quantized singularities

For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions if some blowup points coincide with the singularities of the Dirac data. If the strength of the Dirac mass at each blowup point is not a multiple of [Formula: see text], we prove that bubbling solutions are unique. This paper extends previous results of Lin-Yan [C. S. Lin and S. S. Yan, On the mean field type bubbling solutions for Chern–Simons–Higgs equation, Adv. Math. 338 (2018) 1141–1188] and Bartolucci et al. [D. Bartolucci, A. Jevnikar, Y. Lee and W. Yang, Uniqueness of bubbling solutions of mean field equations, J. Math. Pures Appl. (9) 123 (2019) 78–126].

Local uniqueness and non-degeneracy of blow up solutions of mean field equations with singular data

We are concerned with the mean field equation with singular data on bounded domains. By assuming a singular point to be a critical point of the 1-vortex Kirchhoff-Routh function, we prove local uniqueness and non-degeneracy of bubbling solutions blowing up at a singular point. The proof is based on sharp estimates for bubbling solutions of singular mean field equations and a suitably defined Pohozaev-type identity.

Uniqueness of bubbling solutions of mean field equations

We prove the uniqueness of blow up solutions of the mean field equation as ρn→8πm, m∈N. If un,1 and un,2 are two sequences of bubbling solutions with the same ρn and the same (non degenerate) blow up set, then un,1=un,2 for sufficiently large n. The proof of the uniqueness requires a careful use of some sharp estimates for bubbling solutions of mean field equations [22] and a rather involved analysis of suitably defined Pohozaev-type identities as recently developed in [51] in the context of the Chern–Simons–Higgs equations. Moreover, motivated by the Onsager statistical description of two dimensional turbulence, we are bound to obtain a refined version of an estimate about ρn−8πm in case the first order evaluated in [22] vanishes.

Uniqueness of solutions of mean field equations in 𝑅²

In this paper, we prove uniqueness of solutions of mean field equations with general boundary conditions for the critical and subcritical total mass regime, extending the earlier results for null Dirichlet boundary condition. The proof is based on new Bol’s inequalities for weak radial solutions obtained from rearrangement of the solutions.

Sharp estimates for solutions of mean field equations with collapsing singularity

ABSTRACTThe pioneering work of Brezis-Merle [7], Li-Shafrir [27], Li [26], and Bartolucci-Tarantello [3] showed that any sequence of blow-up solutions for (singular) mean field equations of Liouville type must exhibit a “mass concentration” property. A typical situation of blowup occurs when we let the singular (vortex) points involved in the equation (see (1.1) below) collapse together. However in this case, Lin-Tarantello in [30] pointed out that the phenomenon: “bubbling implies mass concentration” might not occur and new scenarios open for investigation. In this paper, we present two explicit examples which illustrate (with mathematical rigor) how a “nonconcentration” situation does happen and its new features. Among other facts, we show that in certain situations, the collapsing rate of the singularities can be used as blow-up parameter to describe the bubbling properties of the solution-sequence. In this way, we are able to establish accurate estimates around the blow-up points which we hope to use toward a degree counting formula for the shadow system (1.34) below.

Mean field equations, hyperelliptic curves and modular forms: II

We develop a theory connecting the following three areas: (a) the mean field equation (MFE) $\triangle u + e^u = \rho\, \delta_0$, $\rho \in \mathbb R_{>0}$ on flat tori $E_\tau = \mathbb C/(\mathbb Z + \mathbb Z\tau)$, (b) the classical Lam\'e equations and (c) modular forms. A major theme in part I is a classification of developing maps $f$ attached to solutions $u$ of the mean field equation according to the type of transformation laws (or monodromy) with respect to $\Lambda$ satisfied by $f$. We are especially interested in the case when the parameter $\rho$ in the mean field equation is an integer multiple of $4\pi$. In the case when $\rho = 4\pi(2n + 1)$ for a non-negative integer $n$, we prove that the number of solutions is $n + 1$ except for a finite number of conformal isomorphism classes of flat tori, and we give a family of polynomials which characterizes the developing maps for solutions of mean field equations through the configuration of their zeros and poles. Modular forms appear naturally already in the simplest situation when $\,\rho=4\pi$. In the case when $\rho = 8\pi n$ for a positive integer $n$, the solvability of the MFE depends on the \emph{moduli} of the flat tori $E_\tau$ and leads naturally to a hyperelliptic curve $\bar X_n=\bar X_{n}(\tau)$ arising from the Hermite-Halphen ansatz solutions of Lam\'e's differential equation $\frac{d^2 w}{dz^2}-(n(n+1)\wp(z;\Lambda_{\tau}) + B) w=0$. We analyse the curve $\bar X_n$ from both the analytic and the algebraic perspective, including its local coordinate near the point at infinity, which turns out to be a smooth point of $\bar{X}_n$. We also specify the role of the branch points of the hyperelliptic projection $\bar X_n \to \mathbb P^1$ when the parameter $\rho$ varies in a neighborhood of $\rho = 8\pi n$.

A concentration phenomenon around a shrinking hole for solutions of mean field equations

A concentration phenomenon around a shrinking hole for solutions of mean field equations

A series approach to wetting and layering transitions. III. The chiral clock model

For pt.II see ibid., vol.21, p.159-171, (1988). In the third of this triad of papers which study interfacial phase transitions using series expansions the authors treat the wetting transition of an interface in the three-state chiral clock model. Previous work has shown that on a simple cubic lattice at low temperatures the interface wets through a large, possibly infinite, number of layering transitions. They extend the low-temperature series results to an arbitrary number of nearest neighbours and show that, in the mean-field limit of infinite coordination number, only two layering transitions are seen. This is in agreement with numerical solutions of mean-field equations. Hence the mean-field approximation in this case does not provide a correct description of interface behaviour in three dimensions.

Singlet-groundstate magnetism in TbP

Elastic neutron scattering and magnetic susceptibility data are reported for temperatures around the Neel-point,TN=7.3 K, and for zero magnetic field. AboveTN, the temperature dependence of the magnetic central peak intensity can adequately be described within the RPA assuming isotropic exchange between nearest and next-nearest neighbours as the only parameters. This two-parameter model is quantitatively confirmed by the susceptibility data. AtTN, magnetic Bragg-intensities arise almost discontinuously (reaching 70% of the saturation within 0.1 K) accompanied by thermal hysteresis. For all temperatures belowTN the sublattice magnetic moment is explained by solutions of meanfield equations, if an effective quadrupole-quadrupole interaction is included. The quadrupolar coupling parameter is fully consistent with a value determined recently from the softening of thec44-mode. These results show, that earlier suggestions of renormalization group results upon the origin of the first order phase transition in TbP are not needed.

Mean field equations, hyperelliptic curves and modular forms: I

We develop a theory connecting the following three areas: (a) the mean field equation (MFE) $\triangle u + e^u = \rho\, \delta_0$, $\rho \in \mathbb R_{>0}$ on flat tori $E_\tau = \mathbb C/(\mathbb Z + \mathbb Z\tau)$, (b) the classical Lam\'e equations and (c) modular forms. A major theme in part I is a classification of developing maps $f$ attached to solutions $u$ of the mean field equation according to the type of transformation laws (or monodromy) with respect to $\Lambda$ satisfied by $f$. We are especially interested in the case when the parameter $\rho$ in the mean field equation is an integer multiple of $4\pi$. In the case when $\rho = 4\pi(2n + 1)$ for a non-negative integer $n$, we prove that the number of solutions is $n + 1$ except for a finite number of conformal isomorphism classes of flat tori, and we give a family of polynomials which characterizes the developing maps for solutions of mean field equations through the configuration of their zeros and poles. Modular forms appear naturally already in the simplest situation when $\,\rho=4\pi$. In the case when $\rho = 8\pi n$ for a positive integer $n$, the solvability of the MFE depends on the \emph{moduli} of the flat tori $E_\tau$ and leads naturally to a hyperelliptic curve $\bar X_n=\bar X_{n}(\tau)$ arising from the Hermite-Halphen ansatz solutions of Lam\'e's differential equation $\frac{d^2 w}{dz^2}-(n(n+1)\wp(z;\Lambda_{\tau}) + B) w=0$. We analyse the curve $\bar X_n$ from both the analytic and the algebraic perspective, including its local coordinate near the point at infinity, which turns out to be a smooth point of $\bar{X}_n$. We also specify the role of the branch points of the hyperelliptic projection $\bar X_n \to \mathbb P^1$ when the parameter $\rho$ varies in a neighborhood of $\rho = 8\pi n$.