Local Lipschitz bounds for solutions to certain singular elliptic equations involving the one-Laplacian

Abstract

In this paper we study the local Lipschitz regularity of weak solutions to certain singular elliptic equations involving the one-Laplacian. Equations treated here also contains another well-behaving elliptic operator such as the $$p$$ -Laplacian with $$1<p<\infty $$ . The problem is that the one-Laplacian is too singular on degenerate points, what is often called a 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.