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Register a new userRemarks on a mean field equation on $mathbb{S}^{2}$
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In this note, we study symmetry of solutions of the elliptic equation begin{equation*} -Delta _{mathbb{S}^{2}}u+3=e^{2u} hbox{on} mathbb{S}^{2}, end{equation*} that arises in the study of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only constant solutions, and consequently imply the rigidity of Hawking mass for stable constant mean curvature (CMC) sphere.
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We establish theorems on the existence and compactness of solutions to the $sigma_2$-Nirenberg problem on the standard sphere $mathbb S^2$. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Mobius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Mobius invariant degenerate elliptic equations, a priori estimates of first and second order derivatives of solutions to the $sigma_2$-Nirenberg problem, and a B^ocher type theorem for the associated fully nonlinear Mobius invariant elliptic equations. Given these results, we are able to complete a fine analysis of a sequence of blow-up solutions to the $sigma_2$-Nirenberg problem. In particular, we prove that there can be at most one blow-up point for such a blow-up sequence of solutions. This, together with a Kazdan-Warner type identity, allows us to prove $L^infty$ a priori estimates for solutions of the $sigma_2$-Nirenberg problem under some simple generic hypothesis. The higher derivative estimates then follow from classical estimates of Nirenberg and Schauder. In turn, the existence of solutions to the $sigma_2$-Nirenberg problem is obtained by an application of the by now standard degree theory for second order fully nonlinear elliptic operators.
We study in this paper three aspects of Mean Field Games. The first one is the case when the dynamics of each player depend on the strategies of the other players. The second one concerns the modeling of noise in discrete space models and the formulation of the Master Equation in this case. Finally, we show how Mean Field Games reduce to agent based models when the intertemporal preference rate goes to infinity, i.e. when the anticipation of the players vanishes.
Let $Sigma$ be a closed Riemann surface, $h$ a positive smooth function on $Sigma$, $rho$ and $alpha$ real numbers. In this paper, we study a generalized mean field equation begin{align*} -Delta u=rholeft(dfrac{he^u}{int_Sigma he^u}-dfrac{1}{mathrm{Area}left(Sigmaright)}right)+alphaleft(u-fint_{Sigma}uright), end{align*} where $Delta$ denotes the Laplace-Beltrami operator. We first derive a uniform bound for solutions when $rhoin (8kpi, 8(k+1)pi)$ for some non-negative integer number $kin mathbb{N}$ and $alpha otinmathrm{Spec}left(-Deltaright)setminusset{0}$. Then we obtain existence results for $alpha<lambda_1left(Sigmaright)$ by using the Leray-Schauder degree theory and the minimax method, where $lambda_1left(Sigmaright)$ is the first positive eigenvalue for $-Delta$.
We consider the energy-critical half-wave maps equation $$partial_t mathbf{u} + mathbf{u} wedge | abla| mathbf{u} = 0$$ for $mathbf{u} : [0,T) times mathbb{R} to mathbb{S}^2$. We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterization of these solutions as minimal surfaces with (not necessarily free) boundary on $mathbb{S}^2$. In particular, we discover an explicit Lorentz boost symmetry, which is implemented by the conformal Mobius group on the target $mathbb{S}^2$ applied to half-harmonic maps from $mathbb{R}$ to $mathbb{S}^2$. Complementing our classification result, we carry out a detailed analysis of the linearized operator $L$ around half-harmonic maps $mathbf{Q}$ with arbitrary degree $m geq 1$. Here we explicitly determine the nullspace including the zero-energy resonances; in particular, we prove the nondegeneracy of $mathbf{Q}$. Moreover, we give a full description of the spectrum of $L$ by finding all its $L^2$-eigenvalues and proving their simplicity. Furthermore, we prove a coercivity estimate for $L$ and we rule out embedded eigenvalues inside the essential spectrum. Our spectral analysis is based on a reformulation in terms of certain Jacobi operators (tridiagonal infinite matrices) obtained from a conformal transformation of the spectral problem posed on $mathbb{R}$ to the unit circle $mathbb{S}$. Finally, we construct a unitary map which can be seen as a gauge transform tailored for a future stability and blowup analysis close to half-harmonic maps. Our spectral results also have potential applications to the half-harmonic map heat flow, which is the parabolic counterpart of the half-wave maps equation.
We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain $$left{ begin{array}{ll} -Delta u=lambda_1dfrac{V_1 e^{u}}{ int_{Omega_{boldsymbolepsilon}} V_1 e^{u} dx } - lambda_2tau dfrac{ V_2 e^{-tau u}}{ int_{Omega_{boldsymbolepsilon}}V_2 e^{ - tau u} dx}&text{in $Omega_{boldsymbolepsilon}=Omegasetminus displaystyle bigcup_{i=1}^m overline{B(xi_i,epsilon_i)}$} u=0 &text{on $partial Omega_{boldsymbolepsilon}$}, end{array} right. $$ where $B(xi_i,epsilon_i)$ is a ball centered at $xi_iinOmega$ with radius $epsilon_i$, $tau$ is a positive parameter and $V_1,V_2>0$ are smooth potentials. When $lambda_1>8pi m_1$ and $lambda_2 tau^2>8pi (m-m_1)$ with $m_1 in {0,1,dots,m}$, there exist radii $epsilon_1,dots,epsilon_m$ small enough such that the problem has a solution which blows-up positively and negatively at the points $xi_1,dots,xi_{m_1}$ and $xi_{m_1+1},dots,xi_{m}$, respectively, as the radii approach zero.
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