The authors prove a new Carleman estimate for general linear second order parabolic equation with nonhomogeneous boundary conditions. On the basis of this estimate, improved Carleman estimates for the Stokes system and for a system of parabolic equations with a penalty term are obtained. This system can be viewed as an approximation of the Stokes system.
The authors derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau’s results and Ecker-Huisken’s results are generalized to higher codimension. In this way, Hildebrandt-Jost-Widman’s result for the Bernstein type theorem is improved.
The authors study the p(x)-Laplacian equations with nonlinear boundary condition. By using the variational method, under appropriate assumptions on the perturbation terms f 1(x, u), f 2(x, u) and h 1(x), h 2(x), such that the associated functional satisfies the “mountain pass lemma” and “fountain theorem” respectively, the existence and multiplicity of solutions are obtained. The discussion is based on the theory of variable exponent Lebesgue and Sobolev spaces.
The author studies the Green correspondence and quasi-Green correspondence for indecomposable modules over strongly graded rings. The motivation is to investigate the influence of induction and restriction processes on indecomposability of graded modules.
The authors study the problem of uniqueness of meromorphic mappings and obtain two results which partially improve two theorems of Yan and Chen in 2006.
Let G be a finite group with a non-central Sylow r-subgroup R, Z(G) the center of G, and N a normal subgroup of G. The purpose of this paper is to determine the structure of N under the hypotheses that N contains R and the G-conjugacy class size of every element of N is either 1 or m. Particularly, it is shown that N is Abelian if N ∩ Z(G) = 1 and the G-conjugacy class size of every element of N is either 1 or m.
The authors study radial solutions to a model equation for the Navier-Stokes equations. It is shown that the model equation has self-similar singular solution if 5 ≤ n ≤ 9. It is also shown that the solution will blow up if the initial data is radial, large enough and n ≥ 5.