On a Conjecture of Cai-Zhang-Shen for Figurate Primes

A conjecture of Cai-Zhang-Shen for figurate primes says that every integer n > 1 is the sum of two figurate primes. In this paper we give respectively equivalent propositions to the conjecture in the cases of even and odd integers and then confirm the conjecture by considering functions with several variables.Comment: 12page

Some New Inequalities of Dirichlet Eigenvalues for Laplace Operator with any Order

In this paper, we establish several inequalities of Dirichlet eigenvalues for Laplace operator $\Delta$ with any order on \emph{n}-dimensional Euclidean space. These inequalities are more general than known Yang's inequalities and contain new consequences. To obtain them, we borrow the approach of Illias and Makhoul, and use a generalized Chebyshev's inequality

Interior HW^{1,p} estimates for divergence degenerate elliptic systems in Carnot groups

Let X_1,...,X_q be the basis of the space of horizontal vector fields on a homogeneous Carnot group in R^n (q<n). We consider a degenerate elliptic system of N equations, in divergence form, structured on these vector fields, where the coefficients a_{ab}^{ij} (i,j=1,2,...,q, a,b=1,2,...,N) are real valued bounded measurable functions defined in a bounded domain A of R^n, satisfying the strong Legendre condition and belonging to the space VMO_{loc}(A) (defined by the Carnot-Caratheodory distance induced by the X_i's). We prove interior HW^{1,p} estimates (2<p<\infty) for weak solutions to the system