The Neumann problem for Sub-Laplacians on Carnot groups and the extension theorem for Sobolev spaces
The main aim of the paper is to prove the existence of a unique variational solution to the Neumann boundary value problem for Sub-Laplacians on Carnot groups. As in the classical approach, it is based on the use of suitable extension and trace theorems for appropriate Sobolev spaces. The central result concerns the construction of a linear bounded extension operator \(\mathcal E:\mathcal L^{k,p}(\Omega)\to\mathcal L^{k,p}(G)\), where \(\mathcal L^{k,p}\) is the Folland-Stein non-isotropic Sobolev space, \(1\leq p\leq\infty\), \(k\geq1\), \(G\) is a Carnot group of step \(r<\infty\), and \(\Omega\subset G\) is an \((\varepsilon,\delta)\) domain with \(\text{rad}(\Omega)>0\). The main outline of the proof follows \textit{P. W. Jones} [Acta Math. 147, 71--88 (1981; Zbl 0489.30017)]. An important role in the proof is played by a theorem on the appropriate approximation of functions from \(\mathcal L^{k,p}(\Omega)\) by functions from \(C^\infty(G)\). The extension theorem is then applied to prove a trace theorem for functions from \(\mathcal L^{1,p}(\Omega)\), a compact embedding of \(\mathcal L^{k,p}(\Omega)\) in \(\mathcal L^{k-1,q}(\Omega)\), an existence theorem for the \(L^2\)-variational Neumann problem on \(G\), and several further corollaries.
scientific article; zbMATH DE number 2099209