Uniqueness properties of degenerate elliptic operators
Résumé
Let be an open subset of Rd and K = − d i, j=1 ∂i ci j ∂ j + d i=1 ci ∂i + c0 a second-order partial differential operator with real-valued coefficients ci j = c ji ∈ W1,∞ loc ( ), ci , c0 ∈ L∞,loc( ) satisfying the strict ellipticity condition C = (ci j) > 0. Further let H = − d i, j=1 ∂i ci j ∂ j denote the principal part of K. Assuming an accretivity condition C ≥ κ(c⊗c T ) withκ > 0, an invariance condition (1 , Kϕ) = 0 and a growth condition which allows C(x) ∼ |x|2 log |x| as |x| → ∞we prove that K is L1-unique if and only if H is L1-unique or Markov unique.