论文标题
非线性椭圆形库奇问题的平均值迭代
Mean value iterations for nonlinear elliptic Cauchy problems
论文作者
论文摘要
我们调查了一类带有$ c^\ infty $ cefficients的非线性椭圆运算符的凯奇问题，该类别是常规套装$ω\ subset r^n $。 Cauchy数据以歧管$γ\ subset \partialΩ$给出，我们的目标是重建非线性椭圆方程的$ h^1（ω）$解决方案，该方程在$ \ partialω/γ$上。 我们根据应用于固定点方程的分段MANN迭代提出了两种迭代方法，这些方法与原始问题密切相关。第一种方法包括获得相应的线性库奇问题并分析线性固定点方程。给出了收敛的证明，并获得了收敛速率。在第二种方法上，考虑了非线性固定点方程，并研究了完全非线性的迭代方法。证明了一些初步收敛结果，并提供了数值分析。
We investigate the Cauchy problem for a class of nonlinear elliptic operators with $C^\infty$-coefficients at a regular set $Ω\subset R^n$. The Cauchy data are given at a manifold $Γ\subset \partialΩ$ and our goal is to reconstruct the trace of the $H^1(Ω)$ solution of a nonlinear elliptic equation at $\partial Ω/ Γ$. We propose two iterative methods based on the segmenting Mann iteration applied to fixed point equations, which are closely related to the original problem. The first approach consists in obtaining a corresponding linear Cauchy problem and analyzing a linear fixed point equation; a convergence proof is given and convergence rates are obtained. On the second approach a nonlinear fixed point equation is considered and a fully nonlinear iterative method is investigated; some preliminary convergence results are proven and a numerical analysis is provided.
