On a Generalized Riemann-Hilbert Boundary Value Problem for Second Order Elliptic Systems in the Plane

1999 ◽  
pp. 41-55 ◽  
Author(s):  
M. Akal
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Elliptic Systems ◽  
Second Order ◽  
Hilbert Boundary Value Problem

ON THE FIRST BOUNDARY VALUE PROBLEM FOR THE CLASS OF ELLIPTIC SYSTEMS DEGENERATING AT AN INNER POINT

2001 ◽  
Vol 6 (1) ◽  
pp. 147-155 ◽  
Author(s):  
S. Rutkauskas
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Elliptic Systems ◽  
Second Order ◽  
Type Problem ◽  
Dirichlet Type ◽  
Holder Class ◽  
Uniqueness Of The Solution ◽  
Dirichlet Type Problem

The Dirichlet type problem for the weakly related elliptic systems of the second order degenerating at an inner point is discussed. Existence and uniqueness of the solution in the Holder class of the vector‐functions is proved.

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.

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.

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.

Existence of solutions for a second order boundary value problem with the Clarke subdifferential

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2763-2771 ◽  
Author(s):  
Dalila Azzam-Laouir ◽  
Samira Melit
Keyword(s):  
Boundary Value Problem ◽  
Differential Inclusion ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Clarke Subdifferential ◽  
Lipschitzian Function ◽  
The Clarke Subdifferential

In this paper, we prove a theorem on the existence of solutions for a second order differential inclusion governed by the Clarke subdifferential of a Lipschitzian function and by a mixed semicontinuous perturbation.

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