A solution of the abstract Dirichlet problem for Baire-one functions

Let X be a compact convex subset of a locally convex space. We show that any bounded Baire-one function defined on can be extended to an affine Baire-one function on X if and only if X is a Choquet simplex and satisfies a certain topological property.     

The weak Dirichlet problem for Baire functions

Let be a simplex and a compact subset of the set of all extreme points of . We show that any bounded function of Baire class on can be extended to a function of affine class on . Moreover, can be chosen in such a way that .

     

A Dirichlet problem for polyharmonic functions

In this article, the Dirichlet problem of polyharmonic functions is considered. As well the explicit expression of the unique solution to the simple Dirichlet problem for polyharmonic functions is obtained by using the decomposition of polyharmonic functions and turning the problem into an equivalent Riemann boundary value problem for polyanalytic functions, as the approach to find the kernel functions of the solution for the general Dirichlet problem is given. Project supported by NNSF of China.     

A Limit Theorem for Dirichlet L-Functions

We prove a limit theorem on the weak convergence of probability measures in the space of continuous functions for Dirichlet L-functions. The result generalizes a similar theorem for the Riemann zeta-function.     

Discrete limit theorems for general Dirichlet series. III

Here we prove a limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for a general Dirichlet series. The explicit form of the limit measure in this theorem is given. Partially supported by Lithuanian Foundation of Studies and Science     

On the two-point boundary-value problem for the Riccati matrix differential equation

Constructive sufficient conditions for univalent resolvability of a two-point boundary value problem for nonlinear Riccati equation are obtained. An illustrative example is given.     

On special values of certain Dirichlet L-functions

Let r _k (n) denote the number of ways n can be expressed as a sum of k squares. Recently, S. Cooper (Ramanujan J. 6:469–490, [2002]), conjectured a formula for r ₉(t), t≡5 (mod 8), r ₁₁(t), t≡7 (mod 8), where t is a square-free positive integer. In this note we observe that these conjectures follow from the works of Lomadze (Akad. Nauk Gruz. Tr. Tbil. Mat. Inst. Razmadze 17:281–314, [1949]; Acta Arith. 68(3):245–253, [1994]). Further we express r ₉(t), r ₁₁(t) in terms of certain special values of Dirichlet L-functions. Combining these two results we get expressions for these special values of Dirichlet L-functions involving Jacobi symbols.      

Dirichlet Series Associated with Strongly q-Multiplicative Functions

In this work the Dirichlet series associated with real strongly q-multiplicative functions f(n) are studied. We will confine ourselves to the case _i=0 ^q–1 f(i) = 0. It is known that in this case the function _f (s) has an analytic continuation to the whole complex plane as an entire function with trivial zeros on the negative real line. The real function _f (t) satisfying the integral equation with delayed argument for some nonzero real _f naturally appears in the representation of the function _f (s). In this article we find some asymptotic properties of the function _f (s), prove that _f (s) is an entire function of order 2, and also prove that in the region the function _f (s) has only trivial zeros which are simple.     

Dirichlet problem for some hypoelliptic operators

In this paper, the Dirichlet problem for hypoelliptic operators verifying Hörmander condition and the maximum principle is considered.     

Positive solutions for a Dirichlet problem

1. IntroductionSince the work of Ambrosetti and Rabinowitz[l], the problems similar to{;t2:<::l">, (l.l)have been studied extensively But it is well known that, for applying the Moulltain PassTheOrem, we atway8 assum that g(x, 8) is suPerlineax in s at indnity; moeove) a strongercondition like (AR) (see later on) is required. If these conditions are not satisfied, can wealso get solutions for problem (1.1) by a Mountain Pass Theorem? So, ill this paPer, westudy the following Dirichlet pr…     

Explicit Approximate Functional Equations for Various Classes of Dirichlet Series

We give explicit approximate functional equations for various classes of Dirichlet series, such as Lerch zeta function or Dirichlet L-series.Dedicated to Jean-Louis Nicolas on the occasion of his sixtieth birthday2000 Mathematics Subject Classification: Primary—11M06, 11M35, 11Y60     

Half Dirichlet problem for matrix functions on the unit ball in Hermitian Clifford analysis

The simultaneous null solutions of the two complex Hermitian Dirac operators are focused on in Hermitian Clifford analysis, where the Hermitian Cauchy integral was constructed and will play an important role in the framework of circulant (2×2) matrix functions. Under this setting we will present the half Dirichlet problem for circulant (2×2) matrix functions on the unit ball of even dimensional Euclidean space. We will give the unique solution to it merely by using the Hermitian Cauchy transformation, get the solution to the Dirichlet problem on the unit ball for circulant (2×2) matrix functions and the solution to the classical Dirichlet problem as the special case, derive a decomposition of the Poisson kernel for matrix Laplace operator, and further obtain the decomposition theorems of solution space to the Dirichlet problem for circulant (2×2) matrix functions.     

The Dirichlet problem for Hessian quotient equations in exterior domains

In this paper, we obtain the uniqueness and existence of viscosity solutions with prescribed asymptotic behavior at infinity to Hessian quotient equations in exterior domains.     

广义带形区域中的Dirichlet 问题

本文给出广义带形区域中Dirichlet 问题解的积分表示. 如果一类函数在广义带型区域内部调和并在边界上取值为零, 本文给出其需要满足的充要条件.     

The Dirichlet problem on quadratic surfaces

We give a fast, exact algorithm for solving Dirichlet problems with polynomial boundary functions on quadratic surfaces in such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every polynomial in can be uniquely written as the sum of a harmonic function and a polynomial multiple of a quadratic function, thus extending a theorem of Ernst Fischer. We then use this decomposition to reduce the Dirichlet problem to a manageable system of linear equations. The algorithm requires differentiation of the boundary function, but no integration. We also show that the polynomial solution produced by our algorithm is the unique polynomial solution, even on unbounded domains such as elliptic cylinders and paraboloids.

     

Dirichlet problem with indefinite nonlinearities

We consider the following nonlinear elliptic equation in a bounded domain with the Dirichlet boundary condition, and , g₁(u)u and g₂(u)u are positive for |u| > > 1. Some existence results are given for superlinear g₁ and g₂ via the Morse theory.Received: 16 Januray 2003, Accepted: 26 August 2003, Published online: 24 November 2003Mathematics Subject Classification (2000): 35J20, 35J25, 58E05Parts of the work were completed while the authors were visiting the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. The authors thank the hospitality of ICTP. Both authors are supported by NSFC, RFDP, MCME, the second author is also supported by the Foundation for University Key Teacher of the Ministry of Education of China and the 973 project of the Ministry of Science and Technology of China.     

On the n-dimensional Dirichlet problem for isometric maps

We exhibit explicit Lipschitz maps from Rn to Rn which have almost everywhere orthogonal gradient and are equal to zero on the boundary of a cube. We solve the problem by induction on the dimension n.     

The abstract Dirichlet problem for continuous vector-valued functions

Let X be a simplex with the set \({\text {ext}}\, X\) of extreme points being Lindelöf. We show that a continuous bounded mapping defined on \({\text {ext}}\, X\) with values in a Fréchet space F can be extended to a mapping that is a pointwise limit of a bounded sequence of affine continuous mappings from X to F.     

On Dirichlet and Neumann problems for harmonic functions

The aim of the paper is to examine some aspects of the boundary value problems for harmonic functions in half-spaces related to approximation theory. M. V. Keldyshmentioned curious fact on richness in some sense of the solutions of Dirichlet problem in upper half-plane for a fixed continuous boundary data on the real axis. This can be considered as a model version for the Dirichlet problem with continuous boundary data, defined except a single boundary point, with no restrictions imposed on solutions near that point.Some extensions and multi-dimensional versions of Keldysh’s richness are obtained and related questions on existence, representation and richness of solutions for the Dirichlet and Neumann problems discussed.