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NONLINEAR HILBERT BOUNDARY PROBLEM IN THE EUCLIDEAN SPACE E3

NONLINEAR HILBERT BOUNDARY PROBLEM IN THE EUCLIDEAN SPACE E3 DEMONSTRATIOVol. I I IMATHEMATICANo 2M71Wanda SznitkoNONLINEAR HILBERT BOUNDARY PROBLEMIN THE EUCLIDEAN SPACE E^1. INTRODUCTIONLetFjSbe a closed Lyapunov surface dividinginto a bounded domainD+thespaceand unbounded domain D~.The linear Hilbert boundary problem in the spaceF^ (see[1]) consists in finding a piecewise holomorphic column 4>(P),vanishing at infinity andsatisfyingforeachPQ £ Stheboundary condition<t>+(PQ) = G<t>-(P0) + g(P 0 ),where the elements(1)g^(P 0 ) of the column62(p0)s W=(2)6,(*0)satisfy the Hfilder condition, and- 71 -Gis the constant matrixW.Sznitko2hhHh- hhhh- h- hhh- h(5)The only solution of the problem (1), vanishing atnity is the columnH?)= WLinfi-j M(P,Q)(x + )~ 1 g^)dS Q ,(4)where(5)<f =denotes the distance between the points P and Q,N(a,|3,y) - the unit vector of the normal to the surface S atthe point Q, oriented externally with respect to the domain D + .The matrices D (X, Y, Z) and D*(X, Y, Z) are defined asfollowsD(X,Y,Z) =0XYZX0-ZYYZ0-XZ-YX0,D*(X,Y,Z) =0XYzX0Z-YY-z0XZY-X0and the matrixX(P) =G,PcD+E,PeD",where E is the ujiit matrix, satisfies for eachboundary conditionX + (P Q ) = GX"(Po)- 72 -(6)PQe Sthe(7)Nonlinear Hilbert boundary problem3In this paper we present a solution of the nonlinear Hilbert boundary problem for a finite number of closed Lyapunovsurfaces in the space E^. Using the Schauder fixed pointtheorem and the theorem proved by W.Zakowski in http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

NONLINEAR HILBERT BOUNDARY PROBLEM IN THE EUCLIDEAN SPACE E3

Sznitko, Wanda
Demonstratio Mathematica , Volume 3 (2): 18 – Apr 1, 1971
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Publisher
de Gruyter
Copyright
© by Wanda Sznitko
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1971-0203
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIOVol. I I IMATHEMATICANo 2M71Wanda SznitkoNONLINEAR HILBERT BOUNDARY PROBLEMIN THE EUCLIDEAN SPACE E^1. INTRODUCTIONLetFjSbe a closed Lyapunov surface dividinginto a bounded domainD+thespaceand unbounded domain D~.The linear Hilbert boundary problem in the spaceF^ (see[1]) consists in finding a piecewise holomorphic column 4>(P),vanishing at infinity andsatisfyingforeachPQ £ Stheboundary condition<t>+(PQ) = G<t>-(P0) + g(P 0 ),where the elements(1)g^(P 0 ) of the column62(p0)s W=(2)6,(*0)satisfy the Hfilder condition, and- 71 -Gis the constant matrixW.Sznitko2hhHh- hhhh- h- hhh- h(5)The only solution of the problem (1), vanishing atnity is the columnH?)= WLinfi-j M(P,Q)(x + )~ 1 g^)dS Q ,(4)where(5)<f =denotes the distance between the points P and Q,N(a,|3,y) - the unit vector of the normal to the surface S atthe point Q, oriented externally with respect to the domain D + .The matrices D (X, Y, Z) and D*(X, Y, Z) are defined asfollowsD(X,Y,Z) =0XYZX0-ZYYZ0-XZ-YX0,D*(X,Y,Z) =0XYzX0Z-YY-z0XZY-X0and the matrixX(P) =G,PcD+E,PeD",where E is the ujiit matrix, satisfies for eachboundary conditionX + (P Q ) = GX"(Po)- 72 -(6)PQe Sthe(7)Nonlinear Hilbert boundary problem3In this paper we present a solution of the nonlinear Hilbert boundary problem for a finite number of closed Lyapunovsurfaces in the space E^. Using the Schauder fixed pointtheorem and the theorem proved by W.Zakowski in

Journal

Demonstratio Mathematica – de Gruyter

Published: Apr 1, 1971

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