Gaussian bounds, strong ellipticity and uniqueness criteria

Abstract

Let $h$ be a quadratic form with domain $W_0^{1,2}(\Ri^d)$ given by \[ h(φ)=\sum^d_{i,j=1}(\partial_iφ,c_{ij}\,\partial_jφ) \] where $c_{ij}=c_{ji}$ are real-valued, locally bounded, measurable functions and $C=(c_{ij})\ge 0$. If $C$ is strongly elliptic, i.e.\ if there exist $\lambda ,\mu >0$ such that $\lambda I\ge C\ge \mu I>0$, then $h$ is closable, the closure determines a positive self-adjoint operator $H$ on $L_2(\Ri^d)$ 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, i.e.\ $S_t\one=\one$ 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 $c_{ij}\in W^{1,\infty}_{\rm loc}(\Ri^d)$. 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 $K$.

15 pages