In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below. Our proof does not rely on the uniformization theorem and the Onofri inequality, thus it is essentially needed in the alternative proof of the uniformization theorem via the Calabi flow. Such an analytic approach also sheds light on how to obtain the boundedness for E ₁ energy in the study of general Kähler manifolds.

Assume M is a closed 3-manifold whose universal covering is not S ³. We show that the obstruction to extend the Ricci flow is the boundedness $L^{\frac{3}{2}}$-norm of the scalar curvature R(t), i.e., the Ricci flow can be extended over finite time T if and only if the $\Vert R(t)\Vert_{L^{\frac{3}{2}}}$ is uniformly bounded for 0≤t<T. On the other hand, if the fundamental group of M is finite and the $\Vert R(t)\Vert_{L^{\frac{3}{2}}}$ is bounded for all time under the Ricci flow, then M is diffeomorphic to a 3-dimensional spherical space-form.

In this paper, we present a numerical scheme based on the local discontinuous Galerkin (LDG) method for the wave propagation of phase transition in a slender cylinder by introducing new temporal auxiliary variables. The stability for the LDG scheme is presented. In order to verify the validity of the LDG scheme, we give the errors and accuracy order of a numerical example. Due to the interaction between the dispersion and the material nonlinearity, some interesting wave patterns occur for different pre-strains and impacts, such as the pattern with transformation front and solitary wave and the pattern with rarefaction wave and solitary wave. We also investigate the interaction of the transformation fronts and rarefaction waves, and demonstrate this interesting wave phenomena.

This paper studies singular optimal control problems for systems described by nonlinear-controlled stochastic differential equations of mean-field type (MFSDEs in short), in which the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. The control variable has two components, the first being absolutely continuous and the second singular. We establish necessary as well as sufficient conditions for optimal singular stochastic control where the system evolves according to MFSDEs. These conditions of optimality differs from the classical one in the sense that here the adjoint equation turns out to be a linear mean-field backward stochastic differential equation. The proof of our result is based on convex perturbation method of a given optimal control. The control domain is assumed to be convex. A linear quadratic stochastic optimal control problem of mean-field type is discussed as an illustrated example.

Let $M$ be a noncompact complete Riemannian manifold. In this paper, we consider the following nonlinear parabolic equation on $M$

We prove a Li–Yau type gradient estimate for positive solutions to the above equation; as an application, we also derive the corresponding Harnack inequality. These results generalize the corresponding ones proved by Li (J Funct Anal 100:233–256, 1991).

For the $p$-harmonic function with strictly convex level sets, we find an auxiliary function which comes from the combination of the norm of gradient of the $p$-harmonic function and the Gaussian curvature of the level sets of $p$-harmonic function. We prove that this curvature function is concave with respect to the height of the $p$-harmonic function. This auxiliary function is an affine function of the height when the $p$-harmonic function is the $p$-Green function on ball.