Analysis of degenerate elliptic operators of Grušin type
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چکیده
We analyze degenerate, second-order, elliptic operators H in divergence form on L2(R ×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)∇ 2 x2 . Here x1 ∈ R , x2 ∈ R m and cδi,δ′ i are positive measurable functions such that cδi,δ′ i(x) behaves like |x| δi as x→ 0 and |x| ′ i as x→ ∞ with δ1, δ ′ 1 ∈ [0, 1〉 and δ2, δ ′ 2 ≥ 0. Our principal results state that the submarkovian semigroup St = e is conservative and its kernel Kt satisfies bounds 0 ≤ Kt(x ; y) ≤ a (|B(x ; t )| |B(y ; t)|) 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. AMS Subject Classification: 35J70, 35H20, 35L05, 58J35. 1. Mathematical Sciences Institute 2. Department of Mathematical Sciences Australian National University New Mexico State University Canberra, ACT 0200 P.O. Box 30001 Australia. Las Cruces NM 88003-8001, USA.
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تاریخ انتشار 2006