Abstract
Let Ω be an open subset of R d and \({ K=-\sum^d_{i,j=1}\partial_i\,c_{ij}\,\partial_j+\sum^d_{i=1}c_i\partial_i+c_0}\) a second-order partial differential operator with real-valued coefficients \({c_{ij}=c_{ji}\in W^{1,\infty}_{\rm loc}(\Omega),c_i,c_0\in L_{\infty,{\rm loc}}(\Omega)}\) satisfying the strict ellipticity condition \({C=(c_{ij}) >0 }\). Further let \({H=-\sum^d_{i,j=1} \partial_i\,c_{ij}\,\partial_j}\) denote the principal part of K. Assuming an accretivity condition \({C\geq \kappa (c\otimes c^{\,T})}\) with \({\kappa >0 }\), an invariance condition \({(1\!\!1_\Omega, K\varphi)=0}\) and a growth condition which allows \({\|C(x)\|\sim |x|^2\log |x|}\) as |x| → ∞ we prove that K is L 1-unique if and only if H is L 1-unique or Markov unique.
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