Gaussian bounds, strong ellipticity and uniqueness criteria. (7th August 2014) - Robinson, Derek W.
Abstract : Leth be a quadratic form with domainW 1, 2 ( R d ) given byh ( φ ) = ∑ i, j = 1 d ( ∂ i φ, c i j ∂ j φ ), wherec i j = c j i are real‐valued, locally bounded, measurable functions andC = ( c i j ) ⩾ 0 . IfC is strongly elliptic, that is, if there existλ, μ > 0 such thatλ I ⩾ C ⩾ μ I > 0, thenh is closable, the closure determines a positive self‐adjoint operatorH onL 2 ( R d ) which generates a submarkovian semigroupS with a positive distributional kernelK and the kernel satisfies Gaussian upper and lower bounds. Moreover, S is conservative, that is, S t 1 = 1 for allt > 0 . Our aim is to examine converse statements. First, we establish thatC is strongly elliptic if and only ifh is closable, the semigroupS is conservative andK satisfies Gaussian bounds. Secondly, we prove that if the coefficients are such that a Tikhonov growth condition is satisfied, thenS is conservative. Thus, in this case, strong ellipticity ofC is equivalent to closability ofh together with Gaussian bounds onK . Finally, we consider coefficientsc i j ∈ W loc 1, ∞ ( R d ) . It follows thath is closable and a growth condition of the Täcklind type is sufficient to establish the equivalence of strong ellipticity ofC and Gaussian bounds onK .