Harnack's inequality for parabolic operators with singular low order terms
We prove a uniform Harnack inequality for parabolic operators \(L= L_ 0+ b\cdot\nabla +c\) with time-dependent, singular low order terms \(b\) and \(c\). The unperturbed operator \(L_ 0\) may be a general time-dependent second order parabolic operator (either of divergence form \(L_ 0= {1\over 2}\nabla a\nabla+ \alpha\nabla\) or of non-divergence form \(L_ 0= {1\over 2}a \nabla^ 2+ \alpha\nabla\)) which satisfies a uniform Harnack inequality. The Harnack constant for the operator \(L\) depends only on the Harnack constant for the unperturbed operator \(L_ 0\) and the Kato norms of \(\langle b,a^{-1} b\rangle\) and \(c\) with respect to \(L_ 0\). Moreover, we prove that solutions of \(Lu=0\) are continuous (if \(c=0\) even Hölder continuous) and that boundary points are regular for \(L\) if and only if they are regular for \(L_ 0\). In particular, the sheaf of solutions defines a harmonic space.
time-dependent, singular low order terms
scientific article; zbMATH DE number 703848