Boundary Value Problems, Integral Equations and Related Problems

The following sections are included:

• Preface

• Contents

Existence of a generalized solution is proved for a quasilinear parabolic equation with nonlocal boundary conditions using Faedo-Galerkin approximation.

This paper mainly concerns oblique derivative problems for nonlinear nondivergent elliptic systems of several second order equations with measurable coefficients in a multiply connected domain. Firstly, we give a priori estimates of solutions for the above boundary value problems with some conditions, and then by using the above estimates of solutions and the Leray-Schauder theorem, the existence of solutions for the above problems is proved.

We introduce some new classes of time dependent functions whose defining properties take into account of oscillations around singularities. We study properties of solutions to the heat equation with coefficients in these classes which are much more singular than those allowed under the current theory. In the case of L² potentials and L² solutions, we give a characterization of potentials which allow the Schrödinger heat equation to have a positive solution. This provides a new result on the long running problem of identifying potentials permitting a positive solution to the Schrödinger equation. We also establish a nearly necessary and sufficient condition on certain sign changing potentials such that the corresponding heat kernel has Gaussian upper and lower bound. An application to Navier Stokes equation is also given.

In the present paper, we consider the eigenvalue problem for the Sturm-Liouvulle equation

on the interval (0, π) with the boundary conditions where b is a complex number, θ = 0, 1, and the function q(x) is an arbitrary complex-valued function of the class L₂(0, π).

The famous Tricomi equation was established in 1923 by F. G. Tricomi who is the pioneer of parabolic elliptic and hyperbolic boundary value problems and related problems of variable type. In 1945 F. I. Frankl established a generalization of these problems for the well-known Chaplygin equation subject to a certain Frankl condition. In 1953 and 1955 M. H. Protter generalized these problems even further by improving the Frankl condition. In 1977 we generalized these results in several n-dimensional simply connected domains. In 1990 we proposed the exterior Tricomi problem in a doubly connected domain. In 2002 we considered uniqueness of quasi-regular solutions for a bi-parabolic elliptic bi-hyperbolic Tricomi problem. In 2006 G. C. Wen investigated the exterior Tricomi problem for general mixed type equations. In this paper we establish uniqueness of quasi-regular solutions for the exterior Tricomi and Frankl problems for quaterelliptic-quaterhyperbolic mixed type partial differential equations of second order with eight parabolic degenerate lines and propose certain open problems. These mixed type boundary value problems are very important in fluid mechanics.

In this article, we discuss the Riemann-Hilbert Problem for degenerate elliptic systems of first order linear equations in a simply connected domain. We first give the representation of solutions of the boundary value problem for the systems, and then prove the existence and uniqueness of solutions for the problem.

In [1-4], the authors posed and discussed the Tricomi problem of second order equations of mixed type, and in [6], the author discussed the oblique derivative problems of second order equations of mixed type without parabolic degeneracy. The present paper deals with some discontinuous oblique derivative problems for second order linear equations of mixed (elliptic-hyperbolic) type with parabolic degeneracy. Firstly, we give the formulation and representation of solutions of the above boundary value problems, and then prove the existence of solutions for the problems, in which we use the complex analytic method, namely the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used (see [5,6]).

In this paper, we prove a local well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation in one demension.

This paper mainly concerns calibrating volatility from a jump diffusion model to a finite set of observed option pricing. We proposed a regularization algorithm based on Cont and Tankov's relative entropy regularization to solve this problem. We determine the regularization parameter using quasi-optimality criterion with original data error level unknown. Iteratively Guass-Newton method is developed for solving the unconstrained optimization problem. Finally, the theoretical results are illustrated by numerical experiments.

Based on the model of steady-state heat and moisture transfer through textiles, we propose inverse problems of single layer textile material design under low temperature, for example the thickness design or type design. Adopting the idea of regularization method, solving the inverse problems can be formulated into function minimization problems. Combining the finite difference method for ordinary differential equations with direct search method of one dimensional minimization problems, we derive some iteration algorithms of regularized solution for the inverse design problems. Numerical simulation is achieved in order to verify the validity of proposed methods.

The present paper mainly concerns the inverse problem for linear elliptic complex equations of first order with Riemann-Hilbert type map in simply connected domains. Firstly the formulation and the complex form of the problem for the equations are given, and then the existence of solutions for the above problem is proved by a new complex analytic method, where the advantage of the other methods is absorbed, and the used method in this paper is more simple and the obtained result is more general. As an application of the above results, we can derive the corresponding results of the inverse problem for second order elliptic equations from Dirichlet to Neumann map.

In this paper we obtain the numerical inversion formula of the exponential Radon transform, with Chebyshev polynomials and the inverse formula of the exponential Radon transform. Furthermore we deduce the algorithm for the numerical inversion formula.

The problem of finding the temperature u satisfying

from the final data u(x,y,T) ≡ φ(x,y), is discussed. Due to nonlinear ill-posedness for the inverse problem, the Fourier transforms was used to derive a nonlinear integral equation, from which a regularized solution was deduced by perturbing directly the integral equation. It was also mentioned that one should give the appropriate choice of the time T, after which the solution u(x, y, t) will blow up. For the backward parabolic problem, the uniqueness and stability estimates are achieved for the proposed algorithm.

This paper investigates the inverse problem of calibrating the volatility function from given option price data. This is an ill-posed problem because of at least one of three well-posed conditions violating. We start with a simplified model of pure price-dependent volatility to gather insight on the nature of ill-posedness of the problem. We formulate the problem by the, operator equations and establish a Tikhonov regularization model. Projected gradient methods are developed for solving the regularizing problem.

This paper gives a local regularity result for solutions to obstacle problems of the -harmonic equation div with where w(x) is a Muckenhoupt A_p weight with 1 < p < ∞.

This paper deals with the Beltrami system with two characteristic matrices and variable coefficients

where the matrices H(x),D(x) ∈ S(n) satisfy some conditions. A homogeneous elliptic equation of divergence type is derived from the Beltrami system by using the energy and variational methods. A regularity property is obtained by using the Div-Curl fields.

Using the estimates of so-called Hodge decomposition of disturbed vector fields, a Caccioppoli type estimate is established for very weak Solutions of a class of nonlinear equations involved nonhomogeneous items.

The purpose of this paper is to discuss the relations between (K₁,K₂)- quasiregular mappings on Riemannian manifolds and differential forms. Two classes of differential forms are introduced and it is shown that some diferential expressions connected in a natural way to (K₁,K₂)-quasiregular mapping.

This paper deals with the Beltrami system with three characteristic matrices in even dimensions

An elliptic equation of divergence type is derived from (*) under the uniformly elliptic conditions on the matrices H(x),G(x) ∈ S(n) and K(x) of diagonal and positive.

This paper mainly concerns double obstacle problems for second order divergence type elliptic equation div A(x,u,∇u) = 0. Firstly, we give local boundedness of solutions for double obstacle problems, then by using the similarly method, the local regularity of solutions for the above problems is proved.

In this paper, we study the following quasilinear problem:

with Dirichlet boundary condition, where N ≥ 3, 1 < p < N, , 0 ≤ t < p, 1 < q < p, and p*(t) ≡ p(N − t)/(N − p) is the critical Sobolev-Hardy exponent. Via the variational method, we get the the multiplicity of solutions for the quasilinear problem.

A p times continuously differentiable complex-valued function F in a domain D ⊆ C is p-harmonic, if F satisfies the p-harmonic equation , where p(≥ 1) is an integer and Δ represents the complex Laplacian operator. In this paper, the main aim is to introduce two classes and of p-harmonic mappings together with their subclasses and , and investigate the properties of mappings in these classes. First, we obtain characterizations for mappings and in terms of S-Inequality-I and S-Inequality-II, respectively. And then we prove that the image domains of the unit disk D under the mappings in (resp. ) satisfying Inequality-I (resp. Inequality-II) are starlike (resp. convex) of certain order.

In this paper, we list some holomorphic function spaces in Clifford analysis and give their properties. Firstly, we give some properties of regular function space. Next we study hypermonogenic function space and its properties. Finally we study k-holomorphic function and give some of its properties in unbounded domain.

In this paper, we discussed the boundary behavior of some integral operators in Clifford analysis, which are very important for representing harmonic and bi-harmonic functions. By using the standard techniques, we obtained some Privalov theorems and Plemelj formulae for these integrals.

In this paper we give a new proof of certain identities for the fundamental solutions of iterated Dunkl-Dirac operators by using Dunkl transform.

In this paper, we consider a Hilbert boundary value problem in ℝ_0,m Clifford analysis. Find a ℝ_0,m-valued left monogenic function u(x) in which can extend continuously to , whose positive boundary value u⁺ satisfies

where c(t) is a ℝ_{0,m − 1}-valued function, a(x) is a given ℝ_0;m-valued function, whose inverse a⁻¹(x) exsists. We may transform the Hilbert problem into a Riemann boundary value problem, in [1] the Riemann problem may be solved by the successive approximation, if a(x) satisfies some conditions.

Let , where λ is a positive real constant. In this paper, by using the methods from quaternion calculus, we investigate the complex vector solution of the equation Du = 0, and work out a systematic theory analogous to the quaternionic regular function. Differing from that the component functions of a quaternionic regular function are harmonic, the component functions of the solution satisfy the Modified Helmholtz equation, that is (λ² − △)u_i = f, i = 1,2. In addition, we give out a distribution solution of the inhomogeneous equation Du = f and study some properties of the solution.

The changes that have occurred and advances that have been achieved in the behavior of fracture for functionally graded materials (FGMs) subjected to a mechanical and/or temperature change. This paper mainly reviews the research of functionally graded composites material involving the static and dynamic crack problem, thermal elastic fracture analysis, the elastic wave propagation and contact problem. Development of analytical methods to obtain the solution of the transient thermal/mechanical fields in FGMs are introduced.

In this paper, the Fourier integral transform-singular integral equation method is presented for the Mode I crack problem of the functionally graded orthotropic coating-substrate structure. The elastic property of the material is assumed vary continuously along the thickness direction. The principal directions of orthotropy are parallel and perpendicular to the boundaries of the strip. Numerical examples are presented to illustrate the effects of the crack length, the material nonhomogeneity and the thickness of coating on the stress intensity factors.

Concerning Nöether's theory of some two-dimensional singular integral operator with continuous coeffcients, K. Ch. Boimatov and G. Dzangibekov carried out a series of further effective research and then gave the complete effective Nöether's condition and index calculation formula. Besides, with the relevant results, we give the representation of solutions and the index formula of the non-homogeneous Dirichlet problem and non-homogeneous Neumann problem for general elliptic systems of second order equations.

New classes of analytic functions defined by using the Salagean operator are introduced and studied. We provide coefficient inequalities, integral means inequalities and subordination relationships of these classes.

Let the elastic domain be a disk, and its boundary the unit circle. By dint of the stability of Cauchy-type integral with respect to the perturbation of integral curve, the stability of the first fundamental problem and the second fundamental problem in plane elasticity will be discussed under the smooth perturbation for the boundary curve.

In this paper, we consider the Dirichlet boundary value problem of Poisson-type equations on a disk. We assume that the exact solution performs singular properties that its derivatives go to infinity at the boundary of the disk. A stretching polynomial-like function with a parameter is used to construct local grid refinements and the Swartztrauber-Sweet scheme is considered over the non-uniform partition. The effects of the parameter are analyzed completely by numerical experiments, which show that there exists an optimal value for the parameter to have a best approximate solution. Moreover, we show that the discrete system can be considered as a stable one by exploring the concept of the effective condition number.

We develop a fast multiscale Galerkin method to solve the linear ill-posed integral equations with approximately known right-hand sides and operators via Lavrentiev regularization. This method leads to fast solutions of discrete Lavrentiev regularization. The convergence rates of Lavrentiev regularization are achieved by using a modified discrepancy principle.

In this paper, approximate solutions of an inverse boundary value problem in gas dynamics are discussed. We mainly deal with a two dimensional compressible subsonic gas flow over an uneven bottom, which can be transformed into a mixed boundary value problem for nonlinear elliptic complex equations by means of the theory of generalized analytic functions and complex boundary value problems. By using Newton imbedding method, approximate solutions of the mixed boundary value problem are obtained. And under suitable conditions, we could also give error estimates of the approximate solutions.

This paper deals with a numerical method for the solution of the fuzzy heat equation. First we express the necessary materials and definitions, then consider a difference scheme for the one dimensional heat equation. In forth section we express the necessary conditions for stability and check stability of our scheme. In final part we give an example for considering numerical results. In this example we obtain the Hausdorff distance between exact solution and approximate solution.

This paper deals with a numerical method for the solution of the heat equation with non-linear nonlocal boundary conditions. Here non-linear terms are approximated by Richtmyer's linearization method. The integrals in the boundary equations are approximated by the composite Simpson rule. A difference scheme is considered for the one-dimensional heat equation. In final part the numerical results produced by this method is compared.

Subdiffusion is a fractional Brownian motion described by linear equation with fractional Riemann–Liouville time derivative. We present the procedure of solving the subdiffusion equation for the system with infinitely thin membrane. The procedure exploits the Green's function and the boundary conditions at the membrane, which are taken in the general form as a linear combination of particles concentration and flux.

Growth at infinity is given for modified Riesz potential in the half space.

In this paper, the problem of smooth solutions for a general second order parabolic type complex equation

is discussed, and the result-there are C^∞-functions f for which the equation above has no C²-solutions is proved by using a method of function construction, moreover, the dependent relations between the solution u and the diffusion term f is given.

In this paper, a new theorem to the range conditions for the exponential Radon transform is established, their equivalent relationship theorem is proved with a new method.

In Carnot-Carathéodory space, we prove the existence of an exponent r₁ (1< r₁ < p), such that every very weak solution of the equation div(∣∇u∣^{p − 2}∇u) = 0 is a Q-quasiminima of functional ∫_Ω∣Xu∣^rdx, and Q is independent of r.

Local regularity results for the minima of variational integrals with the integrand satisfies the growth conditions of Muckenhoupt weight has been obtained by choosing the suitable cut-off function and the reverse Hölder inequality.

In this paper, the solutions of the Riemann boundary value problems on infinite dimensional locally convex spaces into other infinite dimensional locally convex spaces are presented, via piecewise holomorphic mappings on infinite dimensional locally convex spaces, except for infinite dimensional locally convex manifolds as a boundary, where a key concept-the infinite dimensional index of a mapping is defined.