Towards the $C^{p^\prime}$-Regularity Conjecture in Higher Dimensions

• Autor(a) principal: Araújo, Damião J.
• Data de Publicação: 2017
• Outros Autores: Teixeira, Eduardo V., Urbano, José Miguel
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10316/44406; https://doi.org/10.1093/imrn/rnx068

A longstanding conjecture in elliptic regularity theory inquires whether a W^{1,p} function whose p-laplacian is bounded is locally of class C^{1,\frac{1}{p-1}}. While it is well known that such functions are of class C^{1,\alpha} for some unknown 0 < α < 1, establishing the sharp estimate turns out to be a rather delicate problem. Quite recently, the authors managed to establish the conjecture in the plane. In this article, we address the conjecture in higher dimensions and confirm its validity in a number of other meaningful cases.