Regularity of solutions to degenerate p-Laplacian equations

• Published in: Journal of mathematical analysis and applications Vol. 401; no. 1; pp. 458 - 478
• Main Authors: Cruz-Uribe, David, Moen, Kabe, Naibo, Virginia
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.05.2013
• Subjects: [formula omitted] weights; [formula omitted]-Laplacian; Hörmander vector fields; Ap weights; Hörmander vector fields; p-Laplacian
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2012.12.023

We prove regularity results for solutions of the equation div(〈AXu,Xu〉p−22AXu)=0,1<p<∞, where X=(X1,…,Xm) is a family of vector fields satisfying Hörmander’s condition, and A is an m×m symmetric matrix that satisfies degenerate ellipticity conditions. If the degeneracy is of the form λw(x)2/p|ξ|2≤〈A(x)ξ,ξ〉≤Λw(x)2/p|ξ|2,w∈Ap, then we show that solutions are locally Hölder continuous. If the degeneracy is of the form k(x)−2p′|ξ|2≤〈A(x)ξ,ξ〉≤k(x)2p|ξ|2,k∈Ap′∩RHτ, where τ depends on the homogeneous dimension, then the solutions are continuous almost everywhere, and we give examples to show that this is the best result possible. We give an application to maps of finite distortion.