The limitations of the Poincaré inequality for Grušin type operators
Tóm tắt
We examine the validity of the Poincaré inequality for degenerate, second-order, elliptic operators H in divergence form on
$${L_2(\mathbf{R}^{n}\times \mathbf{R}^{m})}$$
. 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
δ is a generalized Grušin operator,
$$H_\delta=-\nabla_{x_1}\,|x_1|^{\left(2\delta_1,2\delta_1'\right)} \,\nabla_{x_1}-|x_1|^{\left(2\delta_2,2\delta_2'\right)} \,\nabla_{x_2}^2.$$
Here
$${x_1 \in \mathbf{R}^n,\; x_2 \in \mathbf{R}^m,\;\delta_1,\delta_1'\in[0,1\rangle,\;\delta_2,\delta_2'\geq0}$$
and
$${|x_1|^{\left(2\delta,2\delta'\right)}=|x_1|^{2\delta}}$$
if
$${|x_1|\leq 1}$$
and
$${|x_1|^{\left(2\delta,2\delta'\right)}=|x_1|^{2\delta'}}$$
if
$${|x_1|\geq 1}$$
. We prove that the Poincaré inequality, formulated in terms of the geometry corresponding to the control distance of H, is valid if n ≥ 2, or if n = 1 and
$${\delta_1\vee\delta_1'\in[0,1/2\rangle}$$
but it fails if n = 1 and
$${\delta_1\vee\delta_1'\in[1/2,1\rangle}$$
. The failure is caused by the leading term. If
$${\delta_1\in[1/2, 1\rangle}$$
, it is an effect of the local degeneracy
$${|x_1|^{2\delta_1}}$$
, but if
$${\delta_1\in[0, 1/2\rangle}$$
and
$${\delta_1'\in [1/2,1\rangle}$$
, it is an effect of the growth at infinity of
$${|x_1|^{2\delta_1'}}$$
. If n = 1 and
$${\delta_1\in[1/2, 1\rangle}$$
, then the semigroup S generated by the Friedrichs’ extension of H is not ergodic. The subspaces
$${x_1\geq 0}$$
and
$${x_1\leq 0}$$
are S-invariant, and the Poincaré inequality is valid on each of these subspaces. If, however,
$${n=1,\; \delta_1\in[0, 1/2\rangle}$$
and
$${\delta_1'\in [1/2,1\rangle}$$
, then the semigroup S is ergodic, but the Poincaré inequality is only valid locally. Finally, we discuss the implication of these results for the Gaussian and non-Gaussian behaviour of the semigroup S.
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