二阶椭圆型方程斜导数问题的奇异摄动*
Singularly Perturbed Nonlinear Second Order Elliptic Equation
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摘要: 本文研究一类带有非线性边值条件的半线性二阶椭圆型方程的奇异摄动问题: 这里,ε是小参数,(∂)u/(∂)l表示沿着和边界不相切方向(x,ε)的方向导数。给出了上述边值问题的解的渐近展开式,利用压缩映象原理证明了渐近解的一致有效性。Abstract: In this paper,we study singular perturbation problems of some semi-linear second order elliptic equations with nonlinear boundary value conditions: where ε is a small positive parameter and (∂)u/(∂)l is a directional derivative,which lies on an oblique vector (x,ε).We have given a construction of the asymptotic solutions and proof of their asymptotic correctness,which is based on the principle of contraction mapping.
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[1] Fife,P.C.,Semiilinear elliptic boundary value problems with small parameters,Arch.Rat.Mech.Anal.,52,2(1973),205-232. [2] O’Malley,R.E.,Introduction to Singular Perturbations,Academic Press,New York(1974). [3] Олейник О.А.,О свойствах решений некоторых краевых задач для уравнений бллиптического типа,Мам.Сб.Т.,30,72(1952),695-702. [4] Gilbary,D.and N.S.Trudinger,Elliptic Partial Differential Equations of Second Order Berlin,Springer-Verlag(1977).(中译本,上海科技出版社(1981). [5] Miranda,C.,Partial Differential Equations of Elliptic Type,2nd.Rev.Ed.,Berlin,Springer-Verlag(1970). [6] 江福汝,关于常微分方程非线性边值问题的奇异摄动,中国科学,A辑,3(1985),232-242. -
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