Abstract We investigate the existence of weak solutions of a mixed boundary value problem for second order semilinear elliptic equation. The result is obtained by using regularity estimates for mixed linear elliptic problems and an appropriate fixed point theorem. For the homogeneous problem, we also get a result on the dependence of the solution on the small perturbations of the boundary with the Dirichlet and the Neumann data. Based on the fixed point iteration from the proof of the main result, we propose a numerical scheme and provide numerical examples. Parallelisation via the domain decomposition method is also given.

中文翻译：

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关于半线性椭圆方程的混合边值问题

摘要 我们研究了二阶半线性椭圆方程混合边值问题弱解的存在性。结果是通过使用混合线性椭圆问题的正则估计和适当的不动点定理获得的。对于齐次问题，我们还得到了解对边界小扰动的依赖关系的结果，Dirichlet 和 Neumann 数据。基于主要结果证明的不动点迭代，我们提出了一个数值方案并提供了数值例子。还给出了通过域分解方法的并行化。