Dirichlet problem for some hypoelliptic operators
Let \[ L=\sum_{i=1}^n \sum_{j=1}^n a_{ij}(x) D_{ij}+ \sum_{i=1}^n b_i(x)D_i+ c(x) \] be a degenerate elliptic operator of second order with \(C^\infty\) coefficients with \(c\leq 0\) defined on an open set \(\Omega\) of \(\mathbb{R}^n\). Let us suppose that there exist vector fields \(X_1, X_2,\dots, X_r\), \(Y\) such that \(L\) can be written in the form \[ L=\sum_{k=1}^{k=r} X_k^2+ Y+c. \] Further let us assume that the Lie algebra generated by \(X_1, X_2,\dots, X_r\), \(Y\) at any point is of rank \(n\). For these operators, with the addition assumption that \(\exists \lambda<0\) such that \(c(x)\leq \lambda\), \(\forall x\in \Omega\), \textit{J. M. Bony} [Ann. Inst. Fourier 19, No. 1, 277-304 (1969; Zbl 0176.09703)] has proved the existence and uniqueness of the Dirichlet problem \(Lu=f\) in \(\omega\), \(u=\varphi\) on \(\partial\omega\), where \(f\) and \(\varphi\) are continuous and \(\omega\) is a relatively compact open subset of \(\Omega\), \(\overline{\omega} \subset\Omega\), such that \(\partial\omega\) admits a special kind of barrier. In this article we remove the above restriction on \(c\) and assume just that \(c\leq 0\), but assume that \(L\) satisfies maximum principles and prove the existence and uniqueness of the Dirichlet problem when \(f=0\).
scientific article; zbMATH DE number 1210564