2020/07/31 by Shuntaro Tsubouchi · 13 citations
Computer Science · Mathematics · #A priori and a posteriori #Advanced Mathematical Modeling in Engineering #Applied mathematics #Degenerate energy levels #Elliptic curve #Geometry #Laplace operator #Lipschitz continuity #Lipschitz domain #Mathematical analysis #Mathematical proof #Mathematics #Nonlinear Partial Differential Equations #Nonlinear system #Numerical methods in inverse problems #Pure mathematics #Truncation (statistics) #math.AP #msc:35A15 #msc:35B65 #msc:35J92
paper · pdf · doi:10.1007/s00526-020-01889-0
published in Calculus of Variations and Partial Differential Equations 60(1) (Springer Science+Business Media) · 29 pages, including Appendix
openalex created_date 2020/07/16 · arxiv created 2020/07/31 · openalex publication_date 2021/01/18 · arxiv updated 2021/01/20 · openalex updated_date 2026/08/05
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<∞. 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.