Boundary behavior of non-negative solutions to degenerate sub-elliptic equations
The authors study the nonnegative solutions to the equation \[ Lu=\sum^m_{i,j=1}X^{\ast}_i(a_{ij}X_ju)=\sum^m_{i,j=1}X^{\ast}_i(x)(a_{ij}(x)X_j(x)u(x))=0 \] in \(\Omega\) where \(X=\{X_1,\dots ,X_m\}\) is a system of \(C^{\infty}\) vector fields satisfying Hörmander's finite rank condition, \(X^{\ast}_i\) is the formal adjoint of \(X_i\) and \(\Omega\) is a non-tangentially accessible domain with respect to the Carnot--Carathéodory distance \(d\) induced by \(X\). Moreover \(\{a_{ij}(x)\}\) is assumed to be real, symmetric and such that \[ \beta^{-1}\lambda (x)|\xi |^2\leq \sum^m_{i,j=1}a_{ij}(x)\xi_i\xi_j\leq \beta \lambda (x)|\xi |^2 \] where \(\beta \geq 1\) and \(\lambda\) defines an \(A_2\)-weight with respect to the metric introduced by the system of vector fields \(X=\{X_1,\dots ,X_m\}\). More precisely the authors study the doubling property of the associated elliptic measure and the boundary behaviour of nonnegative solutions. More precisely, they prove the Hölder continuity up to the boundary of quotients of non-negative solutions which vanish continuously on a portion of the boundary. This result generalizes results proved in the eighties by Fabes, Kenig, Jerison and Serapioni and more recently by Capogna and Garofalo. The aim of the authors is to extend to the most possible general case of very recent results proved by Lewis, Lundström and Nyström concerning the boundary behavior of non-negative solutions to equations of \(p\)-Laplace type.
scientific article; zbMATH DE number 6154866