Gaussian bounds, strong ellipticity and uniqueness criteria

Let h be a quadratic form with domain W1,2(Rd) given by h(φ)=∑i,j=1d(∂iφ,cij∂jφ), where cij=cji are real‐valued, locally bounded, measurable functions and C=(cij)⩾0. If C is strongly elliptic, that is, if there exist λ,μ>0 such that λI⩾C⩾μI>0, then h is closable, the closure determines a positive self‐adjoint operator H on L2(Rd) which generates a submarkovian semigroup S with a positive distributional kernel K and the kernel satisfies Gaussian upper and lower bounds. Moreover, S is conservative, that is, St1=1 for all t>0. Our aim is to examine converse statements. First, we establish that C is strongly elliptic if and only if h is closable, the semigroup S is conservative and K satisfies Gaussian bounds. Secondly, we prove that if the coefficients are such that a Tikhonov growth condition is satisfied, then S is conservative. Thus, in this case, strong ellipticity of C is equivalent to closability of h together with Gaussian bounds on K. Finally, we consider coefficients cij∈Wloc1,∞(Rd). It follows that h is closable and a growth condition of the Täcklind type is sufficient to establish the equivalence of strong ellipticity of C and Gaussian bounds on K.