Chinese Annals of Mathematics, Series B

This paper is a survey of some recent developments concerning extremal Kähler metrics on Toric Manifolds.

In this paper the authors investigate the boundedness and almost periodicity of solutions of semilinear parabolic equations with boundary degeneracy. The equations may be weakly degenerate or strongly degenerate on the lateral boundary. The authors prove the existence, uniqueness and global exponential stability of bounded entire solutions, and also establish the existence theorem of almost periodic solutions if the data are almost periodic.

As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and the oceans, the time discretization of these equations by an implicit Euler scheme is studied. From the deterministic point of view, the 3D primitive equations are studied in their full form on a general domain and with physically realistic boundary conditions. From the probabilistic viewpoint, this paper deals with a wide class of nonlinear, state dependent, white noise forcings which may be interpreted in either the Itô or the Stratonovich sense. The proof of convergence of the Euler scheme, which is carried out within an abstract framework, covers the equations for the oceans, the atmosphere, the coupled oceanic-atmospheric system as well as other related geophysical equations. The authors obtain the existence of solutions which are weak in both the PDE and probabilistic sense, a result which is new by itself to the best of our knowledge.

A mathematical theory of time-dependent dislocation mechanics of unrestricted geometric and material nonlinearity is reviewed. Within a “small deformation” setting, a suite of simplified and interesting models consisting of a nonlocal Ginzburg Landau equation, a nonlocal level set equation, and a nonlocal generalized Burgers equation is derived. In the finite deformation setting, it is shown that an additive decomposition of the total velocity gradient into elastic and plastic parts emerges naturally from a micromechanical starting point that involves no notion of plastic deformation but only the elastic distortion, material velocity, dislocation density and the dislocation velocity. Moreover, a plastic spin tensor emerges naturally as well.

Let f: ℂ → ℙn be a holomorphic curve of order zero. The authors establish a Jackson difference analogue of Cartan’s second main theorem for the Jackson q-Casorati determinant and introduce a truncated second main theorem of Jackson difference operator for holomorphic curves. In addition, a Jackson difference Mason’s theorem is proved by using a Jackson difference radical of a polynomial. Furthermore, they extend the Mason’s theorem for m + 1 polynomials. Some examples are constructed to show that their results are accurate.

This paper develops further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. This theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. The authors extend the theory for the so-called restricted fractional Laplacian defined on a bounded domain Ω of ℝ N with zero Dirichlet conditions outside of Ω. As an application, an original proof of the corresponding fractional Faber-Krahn inequality is derived. A more classical variational proof of the inequality is also provided.