Analysis of degenerate elliptic operators of Grushin type

Original · EN

We analyze 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 Hδ=-∇ₓ₁ cδ₁, δ'₁(x₁) ∇ₓ₁-cδ₂, δ'₂(x₁) ∇ₓ₂². Here x₁∈ Rⁿ, x₂∈ Rᵐ and cδᵢ, δ'ᵢ are positive measurable functions such that cδᵢ, δ'ᵢ(x) behaves like |x|δⁱ as x→0 and |x|δⁱ' as x→∞ with δ₁,δ₁'∈[0,1> and δ₂,δ₂'≥0. Our principal results state that the submarkovian semigroup Sₜ=e-tH is conservative and its kernel Kₜ satisfies bounds 0≤ Kₜ(x;y)≤ a (|B(x;t¹/²)| |B(y;t¹/²)|)⁻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.