Dirichlet’s Problem for an Equation with Periodic Coefficients Depending on a Small Parameter
Dirichlet’s Problem for an Equation with Periodic Coefficients Depending on a Small Parameter
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周期系数取决于小参数的方程的狄利克雷问题
DOI：
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发表时间：
1964
期刊：
影响因子：
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通讯作者：
M. Freidlin
中科院分区：
文献类型：
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作者：
M. Freidlin
This paper studies the limiting behavior of the solution $u^varepsilon (x)$ of Dirichlet’s problem for [ L^varepsilon u^varepsilon = frac{1}{2}sum {a_{ij} } left( {frac{x}{varepsilon }}
ight)frac{{partial ^2 u^varepsilon }}{{partial x^i partial x^j }} + sum {b_i } left( {frac{x}{varepsilon }}
ight)frac{{partial u^varepsilon }}{{partial x^i }} - cleft( {frac{x}{varepsilon }}
ight)u^varepsilon = 0, ] when $varepsilon o 0$. The coefficients of the operator $L^1 $ are assumed to be periodic.It is proved that $lim _{varepsilon o 0} u^varepsilon (x) = u(x)$ exists. The function $u(x)$ is a solution of Dirichlet’s problem for the equation $ar Lu = 0$, where the coefficients of the operator $ar L$ are obtained by averaging the coefficients of the operator $L^varepsilon $.
This paper studies the limiting behavior of the solution $u^varepsilon (x)$ of Dirichlet’s problem for [ L^varepsilon u^varepsilon = frac{1}{2}sum {a_{ij} } left( {frac{x}{varepsilon }}
ight)frac{{partial ^2 u^varepsilon }}{{partial x^i partial x^j }} + sum {b_i } left( {frac{x}{varepsilon }}
ight)frac{{partial u^varepsilon }}{{partial x^i }} - cleft( {frac{x}{varepsilon }}
ight)u^varepsilon = 0, ] when $varepsilon o 0$. The coefficients of the operator $L^1 $ are assumed to be periodic.It is proved that $lim _{varepsilon o 0} u^varepsilon (x) = u(x)$ exists. The function $u(x)$ is a solution of Dirichlet’s problem for the equation $ar Lu = 0$, where the coefficients of the operator $ar L$ are obtained by averaging the coefficients of the operator $L^varepsilon $.