Liouville theorems in halfspaces for parabolic hypoelliptic equations
Let \(L\) be a second order, parabolic partial differential operator of the form \(L=\sum_{j=1}^{m}X_j^2 +X_0- \partial_t\) in \(\mathbb R^{N+1}\), where the \(X_j\) are first order, linear partial differential operators on a homogeneous Lie group in \(\mathbb R^N\), with infinitely differentiable coefficients. The authors extend known results about solutions of classical parabolic equations, to solutions of \(Lu=0\). The first of the two main theorems states that, if \(u\) is a lower bounded solution to \(Lu=0\) in the half-space \(\mathbb R^N\times\left]-\infty,T\right[\) for some \(T\in \mathbb R\), then \(\lim_{s\to\infty}u(\gamma(s),T-s)=\inf u\) whenever \(\gamma:\left[0,\infty\right[\to \mathbb R^N\) is a continuous curve such that \(\limsup_{s\to\infty}s^{-1}|\gamma(s)|^2<\infty\). The second states that, if \(u\) is a nonnegative solution of \(Lu=0\) in \(\mathbb R^N\times\left]-\infty,0\right[\), and continuous onto \(\mathbb R^N\times\{0\}\) with \(u(x,0)=O(|x|^n)\) as \(|x|\to\infty\) for some real number \(n\), then \(u\) is constant. Generally the references to results on classical equations are adequate, but there is one unfortunate exception. The second of the main theorems depends crucially on Corollary 4 of the paper, and the authors fail to mention that a more general result for the heat equation (than Corollary 4) was proved over thirty years ago, with a similar proof, by the reviewer in [Proc. Lond. Math. Soc. (3) 33, 251--298 (1976; Zbl 0336.35046)].
scientific article; zbMATH DE number 5173039
scientific article