Congruence Criteria for Finite Subsets of Quaternionic Elliptic and Quaternionic Hyperbolic Spaces

We investigate congruence classes of m-tuples of points in the quaternionic elliptic space ?P ⁿ . We establish a canonical bijection between the set of congruence classes of m-tuples of points in ?P ⁿ and the set of equivalence classes of positive semidefinite Hermitian m×m matrices of rank at most n+1 with the 1's on the diagonal. We show that with each m-tuple of points in ?P ⁿ is associated a tuple of points on the real unit sphere S ². Then we get that the congruence class of an m-tuple of points in ?P ⁿ is determined by the congruence classes of all its triangles and by the direct congruence class of the associated tuple on the sphere S ² provided that no pair of points of the m-tuple has distance π/2. Finally we carry out the same kind of investigation for the quaternionic hyperbolic space ?H ⁿ . Most of the results are completely analogous, although there are also some interesting differences.     

Fundaments of Quaternionic Clifford Analysis I: Quaternionic Structure

Introducing a quaternionic structure on Euclidean space, the fundaments for quaternionic and symplectic Clifford analysis are studied in detail from the viewpoint of invariance for the symplectic group action.     

四元数分析中的T算子与两类边值问题

本文研究四元数分析中的非齐次 Ｄｉｒａｃ方程．引入了这类方程的分布解即 Ｔ算子,证明了Ｔ算子的一些性质并考察了非齐次Ｄｉｒａｃ方程的Ｄｉｒｉｃｈｌｅｔ边值问题,并将结果推广到高阶非齐次Ｄｉｒａｃ方程及这种方程的一类边值问题的情况．     

Poisson Kernel for a Degenerate Elliptic Equation and Application to Boundary Value Problems

In this paper we consider the Dirichlet problem to a degenerate elliptic equation in a domain whose interior contains a degenerate surface. By means of the method of expansion of Poisson kernel and applying the properties of special functions, we obtain the twice continuously differentiable solution of the problem on the entire space including infinity.     

On a Class of Elliptic Problems and Its Application to Heat Transfer in Nonconvex Bodies

This work presents a procedure for constructing the solution to a class of problems with application to heat transfer processes in which the energy reemission is not negligible. Such problems are characterized by a Poisson equation subjected to certain nonlinear boundary conditions. The solution is constructed from a sequence whose elements may be obtained from a minimum principle. Some practical situations are presented.     

On the Energy of Distributions, with Application to the Quaternionic Hopf Fibrations

 The energy of an oriented q-distribution ? in a compact oriented manifold M is defined to be the energy of the section of the Grassmannian manifold of oriented q-planes in M induced by ?. In the Grassmannian, the Sasaki metric is considered. We show here a condition for a distribution to be a critical point of the energy functional. In the spheres, we see that Hopf fibrations are critical points. Later, we prove the instability for these fibrations.     

On the Energy of Distributions, with Application to the Quaternionic Hopf Fibrations

 The energy of an oriented q-distribution ? in a compact oriented manifold M is defined to be the energy of the section of the Grassmannian manifold of oriented q-planes in M induced by ?. In the Grassmannian, the Sasaki metric is considered. We show here a condition for a distribution to be a critical point of the energy functional. In the spheres, we see that Hopf fibrations are critical points. Later, we prove the instability for these fibrations. (Received 30 December 2000; in revised form 11 April 2001)     

Nonlinear Elliptic Problems of Neumann-Type

In this paper we study a nonlinear elliptic differential equation driven by the p-Laplacian with a multivalued boundary condition of the Neumann type. Using techniques from the theory of maximal monotone operators and a theorem on the range of the sum of monotone operators, we prove the existence of a (strong) solution.     

On a Class of Domain Optimization Problems in Elliptic Boundary Value Problems

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｄｏｍａｉｎｏｐｔｉｍｉｚａｔｉｏｎｐｒｏｂｌｅｍｉｓａｋｉｎｄｏｆｓｈａｐｅｏｐｔｉｍｉｚａｔｉｏｎｐｒｏｂｌｅｍ．Ｓｐｅｃｉｆｉｃａｌ－ｌｙ,ｗｅｗａｎｔｔｏｆｉｎｄａｄｏｍａｉｎ（ｗｈｉｃｈｉｓｃａｌｅｄｏｐ...     

Necessary and Sufficient Conditions for Associated Differential Operators in Quaternionic Analysis and Applications to Initial Value Problems

This paper deals with the initial value problem of type $$\begin{array}{ll} \qquad \frac{\partial u}{\partial t} = \mathcal{L} u := \sum \limits^3_{i=0} A^{(i)} (t, x) \frac{\partial u}{\partial x_{i}} + B(t, x)u + C(t, x)\\ u (0, x) = u_{0}(x)\end{array}$$ in the space of generalized regular functions in the sense of Quaternionic Analysis satisfying the differential equation $$\mathcal{D}_{\lambda}u := \mathcal{D} u + \lambda u = 0,$$ where ${t \in [0, T]}$ is the time variable, x runs in a bounded and simply connected domain in ${\mathbb{R}^{4}, \lambda}$ is a real number, and ${\mathcal{D}}$ is the Cauchy-Fueter operator. We prove necessary and sufficient conditions on the coefficients of the operator ${\mathcal{L}}$ under which ${\mathcal{L}}$ is associated with the operator ${\mathcal{D}_{\lambda}}$ , i.e. ${\mathcal{L}}$ transforms the set of all solutions of the differential equation ${\mathcal{D}_{\lambda}u = 0}$ into solutions of the same equation for fixedly chosen t. This criterion makes it possible to construct operators ${\mathcal{L}}$ for which the initial value problem is uniquely soluble for an arbitrary initial generalized regular function u ₀ by the method of associated spaces constructed by W. Tutschke (Teubner Leipzig and Springer Verlag, 1989) and the solution is also generalized regular for each t.     

Resonance Problems for a Class of Singular Elliptic Equations

In this article we study the resonance problems for Hardy-Sobolev operator $ L_{\mu }:= -\Delta _{p}-\fraca {(\mu }{|x|^{p})}, 0\le \mu \lt {\fraca {(N-p}{p} )}\,^{p} $ with unbounded nonlinearity. Existence of weak solutions is established using the minimax theorems in critical point theory.     

Multiplicity Results for a Class of Quasilinear Elliptic Problems

By using variational methods we prove the multiplicity of weak solutions of a class of asymptotically p-linear problems.     

Sign-Changing Solutions for a Class of Nonlinear Elliptic Problems

In the present paper,the author studies the existence of sign-changing solutions for nonlinear elliptic equations,which have jumping nonlinearities,and may or may not be resonant with respect to Fu(?)ik spectrum,via linking methods under Cerami condition.     

Analysis of Optimal Control Problems of Semilinear Elliptic Equations by BV-Functions

Set-Valued and Variational Analysis - Optimal control problems for semilinear elliptic equations with control costs in the space of bounded variations are analysed. BV-based optimal controls favor...     

Quaternionic analysis, representation theory and physics

We develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. We first review the Cauchy-Fueter and Poisson formulas and explain their representation theoretic meaning. The requirement of unitarity of representations leads us to the extensions of these formulas in the Minkowski space, which can be viewed as another real form of quaternions. Representation theory also suggests a quaternionic version of the Cauchy formula for the second order pole. Remarkably, the derivative appearing in the complex case is replaced by the Maxwell equations in the quaternionic counterpart. We also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory, such as the spectrum of the hydrogen atom, polarization of vacuum, one-loop Feynman integrals. We also make some further conjectures. The main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis, representation theory and four-dimensional physics.     

On the Structure of the Taylor Series in Clifford and Quaternionic Analysis

The Fueter variables form a basis of the space of (quaternionic or Cliffordian) hyperholomorphic homogeneous polynomials of degree one, and their symmetrized products give the respective bases of spaces of hyperholomorphic homogeneous polynomials for any degree k. In the present paper we introduce new bases, i.e., new types of hyperholomorphic variables which lead to the Taylor-type series expansions reflecting the structure of the set of all (quaternionic or Cliffordian algebra-valued) hyperholomorphic functions.     

Fold-Completeness of Generalized Eigenvectors of a Class of Elliptic Problems

We consider a class of non regular elliptic problems, which depend on spectral parameter, which may appear in the equation and also in the boundary conditions. Using the coerciveness estimates established previously and the unbounded linear operators pencils via Keldysh we prove the n-totality of the generalized eigenvectors.     

Boundary Value Problems for the Elliptic Sine-Gordon Equation in a Semi-strip

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however, a major difficulty for this problem is the existence of non-integrable singularities of the function q _y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann–Hilbert problem to an equivalent modified Riemann–Hilbert problem, we show that the solution can be expressed in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h(λ). The determination of the function h remains open.