Resumen de Gaussian bounds, strong ellipticity and uniqueness criteria

Derek W. Robinson

• Let h be a quadratic form with domain W 1,2 (R d ) given by h(f)=? d i,j=1 (? i f,c ij ? j f), where c ij =c ji are real-valued, locally bounded, measurable functions and C=(c ij )=0 . If C is strongly elliptic, that is, if there exist ?,µ>0 such that ?I=C=µI>0 , then h is closable, the closure determines a positive self-adjoint operator H on L 2 (R d ) 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, S t 1=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 c ij ?W 1,8 loc (R d ) . 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