论文标题
对于一阶段无边界问题的稳定解决方案的非修饰性
Nondegeneracy for stable solutions to the one-phase free boundary problem
论文作者
论文摘要
我们证明了对单相自由边界问题稳定解决方案的非高素质条件。证明是通过de giorgi的迭代，我们需要迈克尔和西蒙的Sobolev不平等，因此，对自由边界的平均曲率进行了不可或缺的估计。然后，如果伯恩斯坦类型定理用于稳定的稳定定理，则应用非平稳估计以获取稳定的自由边界的局部曲率边界，相同维度的整个解决方案是有效的。特别是，我们以$ n = 2 $尺寸获得此曲率估计。
We prove the nondegeneracy condition for stable solutions to the one-phase free boundary problem. The proof is by a De Giorgi iteration, where we need the Sobolev inequality of Michael and Simon and, consequently, an integral estimate for the mean curvature of the free boundary. We then apply the nondegeneracy estimate to obtain local curvature bounds for stable free boundaries in dimension $n$, provided the Bernstein type theorem for stable, entire solutions in the same dimension is valid. In particular, we obtain this curvature estimate in $n=2$ dimensions.
