论文标题
$ p $ harmonic功能的Neumann问题，以及在公制空间中引起的非本地运算符
Neumann problems for $p$-harmonic functions, and induced nonlocal operators in metric measure spaces
论文作者
论文摘要
Following ideas of Caffarelli and Silvestre in~\cite{CS}, and using recent progress in hyperbolic fillings, we define fractional $p$-Laplacians $(-Δ_p)^θ$ with $0<θ<1$ on any compact, doubling metric measure space $(Z,d,ν)$, and prove existence, regularity and stability for the non-homogenous non-local等式$（ - Δ_p）^θu= f。$这些结果又取决于新存在，全球Hölder的规律性和稳定性定理，我们为Neumann问题证明了$ p $ -laplacians $Δ_p$，$ 1 <$ 1 <p <p <\ p <\ iftty $，在量子上的界限中，量子上的量子均与一定的量相吻合。我们的工作也超出了紧凑的环境，并且包括特殊情况，欧几里得，里曼尼亚人和卡诺组设置的其他作者以前的结果大部分结果。与公制度量空间环境中的其他最新贡献不同，我们的工作不依赖于$（z，d，ν）$支持庞加莱不平等的假设。
Following ideas of Caffarelli and Silvestre in~\cite{CS}, and using recent progress in hyperbolic fillings, we define fractional $p$-Laplacians $(-Δ_p)^θ$ with $0<θ<1$ on any compact, doubling metric measure space $(Z,d,ν)$, and prove existence, regularity and stability for the non-homogenous non-local equation $(-Δ_p)^θu =f.$ These results, in turn, rest on the new existence, global Hölder regularity and stability theorems that we prove for the Neumann problem for $p$-Laplacians $Δ_p$, $1<p<\infty$, in bounded domains of measure metric spaces endowed with a doubling measure that supports a Poincaré inequality. Our work also extends beyond the compact setting, and includes as special cases much of the previous results by other authors in the Euclidean, Riemannian and Carnot group settings. Unlike other recent contributions in the metric measure spaces context, our work does not rely on the assumption that $(Z,d,ν)$ supports a Poincaré inequality.
