Analysis of degenerate elliptic operators of Grušin type

By Derek W. Robinson, Adam S. Sikora. 2006

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AbstractWe analyse degenerate, second-order, elliptic operators H in divergence form on L2(Rn × Rm). We assume the coefficients are real symmetric and a1Hδ ≥ H ≥ a2Hδ for some a1, a2 > 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 x1 ∈ Rn, x2 ∈ Rm 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 → 0 and $${|x|^{\delta'_{i}}}$$ as $$x \to \infty$$ with $$\delta_1, \delta'_1 \in [0, 1 \rangle$$ and $$\delta_2,\delta'_2 \geq 0$$ . Our principal results state that the submarkovian semigroup $$S_t = e^{-tH}$$ is conservative and its kernel Kt 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7677b449d816260cae8ddeb3a6944ad3f5f4f79b