论文标题
本地Lipschitz的界限，用于某些涉及一拉普拉斯的奇异椭圆方程的解决方案
Local Lipschitz bounds for solutions to certain singular elliptic equations involving one-Laplacian
论文作者
论文摘要
在本文中，研究了涉及单拉乳胶质的某些单一椭圆方程的弱解决方案的规律性。此处处理的方程式还包含另一个行为良好的椭圆操作员，例如$ p $ -laplacian，$ 1 <p <\ infty $。问题在于，单拉平式在退化点上太奇异了，通常所谓的方面，这使得甚至很难获得弱解决方案的Lipschitz规律性。通过制定合适的近似方案，并避免在近似溶液的方面进行分析来克服这一困难。关键估计是当地的先验统一Lipschitz估算了正规方程的经典解决方案，这是Moser的迭代证明的。 De Giorgi的截断也可以获得另一个局部先验统一的Lipschitz边界。本文中本地Lipschitz估计的证明是相当古典的，并且是基本的，因为根本不使用非线性潜在估计。
In this paper local Lipschitz regularity of weak solutions to certain singular elliptic equations involving one-Laplacian is studied. Equations treated here also contains another well-behaving elliptic operator such as $p$-Laplacian with $1<p<\infty$. The problem is that one-Laplacian is too singular on degenerate points, what is often called facet, which makes it difficult to obtain even Lipschitz regularity of weak solutions. This difficulty is overcome by making suitable approximation schemes, and by avoiding analysis on facet for approximated solutions. The key estimate is a local a priori uniform Lipschitz estimate for classical solutions to regularized equations, which is proved by Moser's iteration. Another local a priori uniform Lipschitz bounds can also be obtained by De Giorgi's truncation. Proofs of local Lipschitz estimates in this paper are rather classical and elementary in the sense that nonlinear potential estimates are not used at all.
