Weighted Sobolev spaces and boundary behavior of solutions to degenerate hypoelliptic equations
Let \(X_1,X_2,\dots,X_k\) be \(C^\infty\)-vector fields in \(\mathbb{R}^N\) satisfying Hörmander's conditions. The author studies regularity theory and potential theory for the quasilinear degenerate hypoelliptic equation of the form \[ -\text{div} {\mathcal A}(x,\nabla_{\mathcal L} u)=0, \tag{1} \] where \(\nabla_{\mathcal L}u=(X_1u,X_2u,\dots,X_ku),\;{\mathcal A}:\Omega\times{\mathbb{R}}^k \rightarrow {\mathbb{R}}^k\) is a mapping satisfying certain structural assumptions, in particular \({\mathcal A}(x,\xi)\approx w(x)|\xi|^p\), \(\Omega\) is a domain in \({\mathbb{R}}^N, w(x)\) is a \(p\)-admissible weight. In the particular case of Euclidean spaces and standard vector fields the corresponding theory can be found in the recent monograph by \textit{J. Heinonen, T. Kilpeläinen} and \textit{O. Martio} [``Nonlinear potential theory of degenerate elliptic equations'', Oxford (1993; Zbl 0780.31001)] where the notion of \(p\)-admissibility and conditions on \(\mathcal A\) are given. The author studies also boundary behavior of the solutions to (1) introducing a new notion of an improper boundary. The generalization of the Wiener criterion of regularity of a boundary point is proved.
quasilinear degenerate hypoelliptic equation
scientific article; zbMATH DE number 923498