Asymptotic estimate of the sum of a Dirichlet series on curves

Under quite general conditions, we obtain an asymptotic estimate of the sum of an entire Dirichlet series on curves. The result generalizes the well-known Pavlov and Kövari theorems     

Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem

Let Ω be a C1,1-bounded domain in Rn for n?2. In this paper, we are concerned with the asymptotic behavior of the unique positive classical solution to the singular boundary-value problem Δu+a(x)u−σ=0 in Ω, u|∂Ω=0, where σ?0, a is a nonnegative function in , 0<α<1 and there exists c>0 such that . Here λ?2, μk∈R, ω is a positive constant and δ(x)=dist(x,∂Ω).     

Asymptotic behavior of positive solutions of a Dirichlet problem involving combined nonlinearities

We study the behavior of positive solutions of the following Dirichlet problem

Asymptotic behavior of solutions of a biharmonic Dirichlet problem with large exponents

We analyze blow-up phenomena of bounded integrable solutions of a semilinear fourth order elliptic problem with a large exponent under Dirichlet boundary conditions. We extend the results obtained by Ren-Wei in [26] and [27] to the biharmonic case.     

关于渐近Dirichlet级数的存在定理(英)

本文对带形中有界解析函数可用渐近Dirichlet级数估计给出一个充分必要条件．     

Asymptotic growth on curves of a Dirichlet series with regular dominant on a sequence of points

We study the classes of entire Dirichlet series defined by convex growth dominants. Also, we obtain estimates for the growth and decay of the functions of a given class.     

The growth of Dirichlet series

We define Knopp-Kojima maximum modulus and the Knopp-Kojima maximum term of Dirichlet series on the right half plane by the method of Knopp-Kojima, and discuss the relation between them. Then we discuss the relation between the Knopp-Kojima coefficients of Dirichlet series and its Knopp-Kojima order defined by Knopp-Kojima maximum modulus. Finally, using the above results, we obtain a relation between the coefficients of the Dirichlet series and its Ritt order. This improves one of Yu Jia-Rong’s results, published in Acta Mathematica Sinica 21 (1978), 97–118. We also give two examples to show that the condition under which the main result holds can not be weakened.     

Asymptotic behavior of solutions to the Dirichlet problem for parabolic equations in domains with singularities

We study solutions of the Dirichlet problem for a second-order parabolic equation with variable coefficients in domains with nonsmooth lateral surface. The asymptotic expansion of the solution in powers of the parabolic distance is obtained in a neighborhood of a singular point of the boundary. The exponents in this expansion are poles of the resolvent of an operator pencil associated with the model problem obtained by freezing the coefficients at the singular point. The main point of the paper is in proving that the resolvent is meromorphic and in estimating it. In the one-dimensional case, the poles of the resolvent satisfy a transcendental equation and can be expressed via parabolic cylinder functions.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 12–23, January, 1996.This work was partially supported by the Russian Foundation for Basic Research under grant No. 242 93-01-16035.     

Dirichlet series of Rankin-Cohen brackets

Given modular forms f and g of weights k and ?, respectively, their Rankin-Cohen bracket corresponding to a nonnegative integer n is a modular form of weight k+?+2n, and it is given as a linear combination of the products of the form f(r)g(n−r) for 0?r?n. We use a correspondence between quasimodular forms and sequences of modular forms to express the Dirichlet series of a product of derivatives of modular forms as a linear combination of the Dirichlet series of Rankin-Cohen brackets.