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APPLICATION OF CHAPLYGIN'S METHOD TO THE FIRST BOUNDARY VALUE PROBLEM FOR A NON-LINEAR ELLIPTIC EQUATION

APPLICATION OF CHAPLYGIN'S METHOD TO THE FIRST BOUNDARY VALUE PROBLEM FOR A NON-LINEAR ELLIPTIC... DEMONSTRATIO MATHEMATICAVol. IINo 2197·Jerzy ChmajAPPLICATION OF CHAPLYGIN'S METHOD TO THE FIRSTBOUNDARY VALUE PROBLEM FOR A NON-LINEAR ELLIPTIC EQUATION1. INTRODUCTIONLet V denote a domain in E^ bounded bythe closedLapunov surface S. Consider the following boundary valueproblem. Find a function u(A) satisfying in V the equationΔα(Α) = f (A,u(A))and satisfying onSAe V(1)the boundary conditionu(P) = 0Pe S(2)Chaplygin's method [1] was applied to theinvestigationof the analogous problem in Eg by I.P.Mysowskikh [2]'.Heassumed f u » f u u "to be continuous and to satisfyfuu(orfuu«0).In this paper we shall base on the properties of Green'sfunction of the first kind for the Laplace equation. We shallprove the existence and the uniqueness of a solution of theboundary value problem (1), (2) applying Chaplygin'smethodand making the following assumptionsI. f(A,u) is a real function defined for A e V + S , - ~ < u < ~and satisfying the Holder-Lipschitz condition of the formΙ ί ( Α ' , α ' ) - f ( A " , u")|< K|A' A"| h + k| u' - u"|where 0 < h < 1 ,K>0.II. The coefficientk(3)in (3) satisfies the inequality0 < k < 2q- 117 -(4)32J . China jwhereC =Af.V+'SI I I . f ( A , http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

APPLICATION OF CHAPLYGIN'S METHOD TO THE FIRST BOUNDARY VALUE PROBLEM FOR A NON-LINEAR ELLIPTIC EQUATION

Chmaj, Jerzy
Demonstratio Mathematica , Volume 2 (2): 10 – Apr 1, 1970
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References (2)

Publisher
de Gruyter
Copyright
© by Jerzy Chmaj
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1970-0204
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. IINo 2197·Jerzy ChmajAPPLICATION OF CHAPLYGIN'S METHOD TO THE FIRSTBOUNDARY VALUE PROBLEM FOR A NON-LINEAR ELLIPTIC EQUATION1. INTRODUCTIONLet V denote a domain in E^ bounded bythe closedLapunov surface S. Consider the following boundary valueproblem. Find a function u(A) satisfying in V the equationΔα(Α) = f (A,u(A))and satisfying onSAe V(1)the boundary conditionu(P) = 0Pe S(2)Chaplygin's method [1] was applied to theinvestigationof the analogous problem in Eg by I.P.Mysowskikh [2]'.Heassumed f u » f u u "to be continuous and to satisfyfuu(orfuu«0).In this paper we shall base on the properties of Green'sfunction of the first kind for the Laplace equation. We shallprove the existence and the uniqueness of a solution of theboundary value problem (1), (2) applying Chaplygin'smethodand making the following assumptionsI. f(A,u) is a real function defined for A e V + S , - ~ < u < ~and satisfying the Holder-Lipschitz condition of the formΙ ί ( Α ' , α ' ) - f ( A " , u")|< K|A' A"| h + k| u' - u"|where 0 < h < 1 ,K>0.II. The coefficientk(3)in (3) satisfies the inequality0 < k < 2q- 117 -(4)32J . China jwhereC =Af.V+'SI I I . f ( A ,

Journal

Demonstratio Mathematica – de Gruyter

Published: Apr 1, 1970

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