A variational approach to the Cauchy problem for nonlinear elliptic differential equations
A variational approach to the Cauchy problem for nonlinear elliptic differential equations
复制标题
非线性椭圆微分方程柯西问题的变分法
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
N. Tarkhanov
中科院分区：
文献类型：
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作者：
I. Ly;N. Tarkhanov
本文用变分方法讨论了一类非线性椭圆柯西问题的松弛问题，该问题的数据在边界曲面的一段S上，在最优控制文献中称为“方程误差法”。柯西问题是指在区域χ中未知函数y的任何边值问题，其性质是S上的数据如果与χ中的微分方程结合，则允许通过函数方程确定y在S上的所有导数。在柯西问题的真实的解析数据的情况下，柯西-科瓦列夫斯卡娅定理保证了局部解在S附近的存在性。我们也承认超定椭圆系统，在这种情况下，这些柯西数据集S的柯西问题是可解的是非常“薄”。出于这个原因，我们讨论的Cauchy问题的变分设置总是拥有一个广义的解决方案。
Abstract We discuss the relaxation of a class of nonlinear elliptic Cauchy problems with data on a piece S of the boundary surface by means of a variational approach known in the optimal control literature as “equation error method”. By the Cauchy problem is meant any boundary value problem for an unknown function y in a domain χ with the property that the data on S, if combined with the differential equations in χ, allow one to determine all derivatives of y on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy–Kovalevskaya theorem. We also admit overdetermined elliptic systems, in which case the set of those Cauchy data on S for which the Cauchy problem is solvable is very “thin”. For this reason we discuss a variational setting of the Cauchy problem which always possesses a generalised solution.