Gaussian bounds, strong ellipticity and uniqueness criteria
Consider a quadratic form \(h(\varphi)=\sum_{i,j=1}^d (\partial_i\varphi, c_{ij}\partial_j\varphi)\) defined on \(W^{1,2}(\mathbb{R}^d),\) where \(c_{ij}=c_{ji}\) are real-valued, locally bounded and measurable functions such that the matrix \(C=\{c_{ij}\}\) is non-negatively defined. If \(C\) is strongly elliptic, that is, \(0<\mu I\leq C\leq \lambda I\) for \(\lambda,\mu>0,\) then \(h\) is closable, and the closure determines a positive self-adjoint operator \(H\) on \(L^2(\mathbb{R}^d)\) which generates a submarkovian semigroup \(S\) with a positive distributional kernel \(K\) and this kernel satisfies Gaussian upper and lower bounds. Moreover, \(S\) is conservative, that is, \(S_t1=1\) for all \(t>0\).NEWLINENEWLINEIn the paper under review, the author examines the converse statements. First, it is established that \(C\) is strongly elliptic if and only if \(h\) is closable, the semigroup \(S\) is conservative and \(K\) satisfies Gaussian bounds. Secondly, the author proves that, if the coefficients \(c_{ij}\) 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 for \(K\). Finally, taking the coefficients in \(W^{1,\infty}_{\text{loc}}(\mathbb{R}^d),\) it is shown 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 \(K.\)
scientific article; zbMATH DE number 6355605
scientific article