Integral representations and boundary value problems for a linear third-order elliptic system with an interior supersingular point

For a supersingular elliptic system, we find an integral representation of the solution and the corresponding inversion formula depending on the values of roots of the characteristic equation, which is of interest from both the theoretical and the practical viewpoint. All studies are carried out for the case in which the singular point is an interior point of the domain. Note that this case is most complicated. In the resulting integral representations, we clearly single out the singular part of the solutions, which permits analyzing their asymptotic behavior with respect to r. We study the influence of the supersingular point on the solvability of boundary value problems and find a well-posed statement of a number of Dirichlet and Riemann-Hilbert boundary value problems.     

Uniqueness Results for the Riemann-Hilbert Problem with a Vanishing Coefficient

We study the Riemann-Hilbert problem of finding φ, ψ ∈ H^p such that their nontangential boundary values satisfy the equation

On a singular elliptic system at resonance

This paper is devoted to the study of an elliptic system with singular coefficients. Existence and multiplicity results at resonance are obtained via variational methods.     

A Riemann-Hilbert type problem for second order non-regular elliptic equation in weighted spaces

The paper investigates a Riemann-Hilbert type problem for second order nonregular elliptic equation in weighted spaces. It is established that the number of linearly independent solutions of the homogeneous problem and the number of conditions on the boundary functions depend not only on the order of singularity of the weight function and coefficient indices of the considered problem, but also on the behavior of these functions at singular points.     

Bounds for the singular set of solutions to non linear elliptic systems

We give new estimates for the Hausdorff dimension of the singular set of solutions to elliptic systems If the vector fields a and b are Hölder continuous with respect to the variables (x,u) with exponent , then, under suitable assumptions, the Hausdorff dimension of the singular set of any weak solution is at most . We consider natural growth assumptions on a(x,u,Du) with respect to u and critical ones on the right hand side b(x,u,Du), with respect to Du.Accepted: 12 March 2003, Published online: 16 May 2003     

A Riemann-Hilbert Problem for the Moisil-Teodorescu System

In a bounded domain with smooth boundary in ?³ we consider the stationary Maxwell equations for a function u with values in ?³ subject to a nonhomogeneous condition (u, v)_x = u₀ on the boundary, where v is a given vector field and u₀ a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.     

Constructive Solvability Conditions for the Riemann-Hilbert Problem

Sufficient and necessary conditions for the solvability of the Riemann-Hilbert problem are studied. These conditions consist in the possibility of constructing stable and semistable pairs (of bundles and connections) for a given monodromy. The obtained results make it possible to develop algorithms for testing the solvability conditions for the Riemann-Hilbert problem.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 643–655.Original Russian Text Copyright ©2005 by I. V. V’yugin.I dedicate this work to the blessed memory of my teacher Andrei Andreevich Bolibrukh     

Problems for elliptic singular equations with a gradient term

We investigate the homogeneous Dirichlet problem for a class of second-order nonlinear elliptic partial differential equations with singular data. In particular, we study the asymptotic behaviour of the solution near the boundary up to the second order under various assumptions on the growth of the coefficients of the equation.     

Integral representations and boundary value problems for a second-order elliptic system with a singular point

For a second-order elliptic system with a singular point, we obtain integral representations and inversion formulas for the case in which the singular point is an interior point of the domain. In the integral representations, we clearly extract the singular part of the solutions, which permits one to study the asymptotics of the solutions as r → 0. In addition, we give a well-posed statement of a number of boundary value problems.     

Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem

We study Brezis-Nirenberg type theorems for the equation

     

Problem with shifts in interior characteristics for equations of mixed elliptic-hyperbolic type with a singular coefficient

We consider a problem with local shift conditions on a segment of the degeneration line and with shifts in interior characteristics of the equation. The uniqueness of the solution is proved with the use of the extremum principle. To prove the existence of a solution, we apply the theory of singular integral equations and Fredholm integral equations.     

Cauchy-Riemann方程的Riemann-Hilbert边值问题

本文考虑多柱域上非齐次的Cauchy-Riemann方程的Riemann-Hilbert边值问题.讨论了上述边值问题可解的充分必要条件,并给出了边值问题解的积分表达式.