论文标题
关于平均曲率的两相无边界问题的本地分析
Local analysis of a two phase free boundary problem concerning mean curvature
论文作者
论文摘要
我们考虑了具有分段恒定系数的两相椭圆运算符的过度确定问题。我们寻找域，以使dirichlet边界值问题的解决方案$ u $也满足其正常派生$ \ partial_n u $是边界上每个点的曲率半径的倍数。当系数满足某些“非批判性”条件时，我们使用依靠形状衍生物和隐式函数定理的扰动参数构建了这个过度确定的问题的非平凡解决方案。此外，在关键情况下，我们使用Crandall-Rabinowitz定理的使用来显示对称性破坏解决方案的分支，从琐碎的解决方案分叉。最后，给出了关于一个相的情况的一些评论，并给出了类似的锯齿蛋白类型问题。
We consider an overdetermined problem for a two phase elliptic operator in divergence form with piecewise constant coefficients. We look for domains such that the solution $u$ of a Dirichlet boundary value problem also satisfies the additional property that its normal derivative $\partial_n u$ is a multiple of the radius of curvature at each point on the boundary. When the coefficients satisfy some "non-criticality" condition, we construct nontrivial solutions to this overdetermined problem employing a perturbation argument relying on shape derivatives and the implicit function theorem. Moreover, in the critical case, we employ the use of the Crandall-Rabinowitz theorem to show the existence of a branch of symmetry breaking solutions bifurcating from trivial ones. Finally, some remarks on the one phase case and a similar overdetermined problem of Serrin type are given.
