The Hilbert boundary value problem for nonlinear elliptic systems

1983 ◽  
Vol 94 (1-2) ◽  
pp. 97-112 ◽  
Author(s):  
Heinrich Begehr ◽  
George C. Hsiao
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
A Priori ◽  
Elliptic Systems ◽  
Representation Formula ◽  
Constructive Method ◽  
Nonlinear Elliptic System ◽  
Approximation Procedure ◽  
Nonlinear Elliptic ◽  
Hilbert Boundary Value Problem

SynopsisThe Hilbert boundary value problem for a first order nonlinear elliptic system in the plane with linear boundary conditions of nonnegative index is (under suitable side conditions uniquely) solved by use of the Newton imbedding method. This constructive method is based on an a priori estimate which arises from an integral representation formula for C1-functions first developed by Haack and Wendland. The approximation procedure yields an error estimate too.

Remark on Hilbert's boundary value problem for Beltrami systems

1984 ◽  
Vol 98 (3-4) ◽  
pp. 305-310 ◽  
Author(s):  
Heinrich Begehr
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
A Priori ◽  
Continuation Method ◽  
Elliptic Systems ◽  
Constructive Proof ◽  
Nonlinear Elliptic Systems ◽  
Nonlinear Elliptic ◽  
Hilbert Boundary Value Problem ◽  
Positive Index

SynopsisThe Schauder continuation method for nonlinear problems is based on appropriate a priori estimates for related linear equations. Recently, in a paper by the present author and G. C. Hsiao, the Hilbert boundary value problem with positive index for nonlinear elliptic systems in the plane was solved by this method but the constructive derivation of the a priori estimate necessarily required a restriction on the ellipticity condition. This is because the norm of the generalized Hilbert transform in the case of positive index is too big. Here, as in a forthcoming paper by G.C. Wen, an indirect and therefore non-constructive proof of the a priori estimate is given which does not require any further restrictions and allows the Hilbert boundary value problem to be solved for nonlinear elliptic systems in general.

scholarly journals ON THE REGULARIZATION OF EQUATIONS OF THE MECHANICS OF MIXTURES OF VISCOUS COMPRESSIBLE FLUIDS

2017 ◽  
pp. 54-71
Author(s):  
Nikolay Kucher ◽  
Nikolay Kucher ◽  
Aleksandra Zhalnina ◽  
Aleksandra Zhalnina
Keyword(s):  
Boundary Value Problem ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Compressible Fluids ◽  
Time Interval ◽  
Approximation Procedure ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation

Mathematical models of multi-velocity continua, through which the motion of multicomponent mixtures are described, represent a rather extensive area of modern mechanics and mathematics. Mathematical results (statements of problems, theorems on the existence and uniqueness, properties of solutions, etc.) for such models are rather modest in comparison with the results for classical single-phase media. The present paper aims to fill this gap in some extent and is devoted to investigating the global correctness of the boundary value problem for a nonlinear system of differential equations, which is some regularity of the mathematical model of nonstationary spatial flows of a mixture of viscous compressible fluids. Construction of the solution of the problem considered in this article is a key step for the mathematical analysis of the initial model of the mixture, since it allows to obtain globally defined solutions of the latter by means of a limiting transition and, in addition, the proposed algorithm for constructing solutions to the regularized problem is practical. This algorithm is based on the finite-dimensional approximation procedure for an infinite-dimensional problem, and therefore a mathematically grounded algorithm for the numerical solution of the boundary value problem of the motion of a mixture of viscous compressible fluids in a region bounded by solid walls can be constructed on this basis. The local in time solvability of finite- dimensional problems is proved by applying the principle of contracting mappings and the local solution can be extended to an arbitrary time interval with the help of a priori estimates.

scholarly journals The Riemann – Hilbert boundary value problem for elliptic systems of the orthogonal type in R3

2020 ◽  
Vol 56 (1) ◽  
pp. 7-16
Author(s):  
A. I. Basik ◽  
E. V. Hrytsuk ◽  
T. A. Hrytsuk
Keyword(s):  
Vector Field ◽  
Boundary Value Problem ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Elliptic Systems ◽  
Tangent Plane ◽  
The Matrix ◽  
Hilbert Boundary Value Problem ◽  
Simply Connected Region ◽  
Simply Connected

In this paper, a class of elliptic systems of four 1st order differential equations of the orthogonal type in R3 is considered. For such systems we study the issue of regularizability of the Riemann – Hilbert boundary value problem in an arbitrary limited simply-connected region with a smooth boundary in R3. Using the coefficients of the elliptic system and the matrix of the boundary operator, a special vector field is constructed, and its not entering the tangent plane in any point of the boundary provides the Lopatinski condition of the regularizability of the boundary value problem. The obtained condition permits to prove that the set of regularizable Riemann – Hilbert boundary value problems for the considered class of systems has two components of homotopic connectedness, and the index of an arbitrary regularizable problem equals to minus one.

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