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ON A NONEXISTENCE OF POSITIVE SOLUTION OF LAPLACE EQUATION IN UPPER HALF-SPACE

ON A NONEXISTENCE OF POSITIVE SOLUTION OF LAPLACE EQUATION IN UPPER HALF-SPACE DEMONSTRATIO MATHEMATICAVol. XXXNo 11997Dinh Van Ruy, N g u y e n Thanh Long, Duong Thi Thanh BinhO N A N O N E X I S T E N C E OF P O S I T I V E SOLUTIONOF L A P L A C E E Q U A T I O N IN U P P E R H A L F - S P A C E1. IntroductionConsider the following Laplace equation in the upper half-space(1.1)Au = 0,= {{x,y,z)eR3(x,y,z)£Rl:z>Q}with a nonlinear boundary condition of the form(1.2)-uz(x,y,0)(x,y)eR2.= f(x,y,u(x,y,0)),In [1] there was studied the Laplace equation of axial symmetry form(1.3)urr + -ur + uzz = 0,rr > 0, z > 0,with a nonlinear boundary condition(1.4)-uz(r,0)= I0exp(-r2/r20)+ ua(r,Q),r > 0,where I o , r o , a are given positive constants. The problem (1.3), (1.4) is astationary case of the problem relative to ignition by radiation. In [1] it wasproved that the problem (1.3), (1.4) in the case 0 < a < 2 has no positivesolution. Afterwards, this result has been extended in [2] for more generalnonlinear boundary condition(1-5)~uz{r,0)= g(r,u(r,0)),r>0.In this paper we consider the problem (1.1), (1.2) with a given function /which is continuous, nondecreasing http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

ON A NONEXISTENCE OF POSITIVE SOLUTION OF LAPLACE EQUATION IN UPPER HALF-SPACE

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Publisher
de Gruyter
Copyright
© by Dinh Van Ruy
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1997-0103
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXXNo 11997Dinh Van Ruy, N g u y e n Thanh Long, Duong Thi Thanh BinhO N A N O N E X I S T E N C E OF P O S I T I V E SOLUTIONOF L A P L A C E E Q U A T I O N IN U P P E R H A L F - S P A C E1. IntroductionConsider the following Laplace equation in the upper half-space(1.1)Au = 0,= {{x,y,z)eR3(x,y,z)£Rl:z>Q}with a nonlinear boundary condition of the form(1.2)-uz(x,y,0)(x,y)eR2.= f(x,y,u(x,y,0)),In [1] there was studied the Laplace equation of axial symmetry form(1.3)urr + -ur + uzz = 0,rr > 0, z > 0,with a nonlinear boundary condition(1.4)-uz(r,0)= I0exp(-r2/r20)+ ua(r,Q),r > 0,where I o , r o , a are given positive constants. The problem (1.3), (1.4) is astationary case of the problem relative to ignition by radiation. In [1] it wasproved that the problem (1.3), (1.4) in the case 0 < a < 2 has no positivesolution. Afterwards, this result has been extended in [2] for more generalnonlinear boundary condition(1-5)~uz{r,0)= g(r,u(r,0)),r>0.In this paper we consider the problem (1.1), (1.2) with a given function /which is continuous, nondecreasing

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Demonstratio Mathematica – de Gruyter

Published: Jan 1, 1997

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