Analysis of degenerate elliptic operators of Gru\v{s}in type

TL;DR

This paper studies degenerate elliptic operators of Grušin type, establishing kernel bounds, conservativeness of the associated semigroup, and analyzing the effects of degeneracy on positivity and continuity.

Contribution

It provides new bounds for the heat kernel of Grušin-type operators and characterizes their geometric and analytic properties under degeneracy.

Findings

The semigroup is conservative and its kernel satisfies specific upper bounds.

The kernel may not be strictly positive or continuous due to degeneracy.

Explicit examples illustrate the impact of degeneracy on kernel properties.

Abstract

We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_2(\Ri^{n}\times\Ri^{m})$. We assume the coefficients are real symmetric and $a_1H_\delta\geq H\geq a_2H_\delta$ for some $a_1,a_2>0$ where \[ H_\delta=-{\nabla}_{x_1}\cdot(c_{\delta_1, \delta'_1}(x_1)\,\nabla_{x_1})-c_{\delta_2, \delta'_2}(x_1)\,\nabla_{x_2}^2 \;. \] Here $x_1\in\Ri^n$, $x_2\in\Ri^m$ and $c_{\delta_i, \delta'_i}$ are positive measurable functions such that $c_{\delta_i, \delta'_i}(x)$ behaves like $|x|^{\delta_i}$ as $x\to0$ and $|x|^{\delta_i'}$ as $x\to\infty$ with $\delta_1,\delta_1'\in[0,1\rangle$ and $\delta_2,\delta_2'\geq0$. Our principal results state that the submarkovian semigroup $S_t=e^{-tH}$ is conservative and its kernel $K_t$ satisfies bounds \[ 0\leq K_t(x\,;y)\leq a\,(|B(x\,;t^{1/2})|\,|B(y\,;t^{1/2})|)^{-1/2} \] where $|B(x\,;r)|$ denotes the volume of the ball $B(x\,;r)$ centred at $x$ with radius $r$ measured with respect to the Riemannian distance associated with $H$. The proofs depend on detailed subelliptic estimations on $H$, a precise characterization of the Riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation. We discuss further implications of these bounds and give explicit examples that show the kernel is not necessarily strictly positive, nor continuous.