On C1, C2, and weak type-(1, 1) estimates for linear elliptic operators. Issue 3 (4th March 2017)

• Record Type: Journal Article
• Authors: Dong, Hongjie; Kim, Seick
• Is Part Of: Communications in partial differential equations. Volume 42:Issue 3(2017)
• Page Start: 417
• Page End: 435
• Publication Date: 2017-03-04
• Subjects: Elliptic equations -- Dini mean oscillation condition -- C1 and C2 estimates -- weak type-(1, 1) estimate; Primary: 35B45 -- 35B65 -- Secondary: 35J47; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2017.1278773
• Languages: English
• ISSNs: 0360-5302

Abstract

ABSTRACT: We show that any weak solution to elliptic equations in divergence form is continuously differentiable provided that the modulus of continuity of coefficients in the L 1 -mean sense satisfies the Dini condition. This in particular answers a question recently raised by Yanyan Li and allows us to improve a result of Haïm Brezis. We also prove a weak type-(1, 1) estimate under a stronger assumption on the modulus of continuity. The corresponding results for nondivergence form equations are also established.