Continuity of solutions to space-varying pointwise linear elliptic equations

Journal article, 2017

We consider pointwise linear elliptic equations of the form Lxux = ?x on a smooth compact manifold where the operators Lx are in divergence form with real, bounded, measurable coefficients that vary in the space variable x. We establish L2 -continuity of the solutions at x whenever the coefficients of Lx are L?-continuous at x and the initial datum is L2-continuous at x. This is obtained by reducing the continuity of solutions to a homogeneous Kato square root problem. As an application, we consider a time evolving family of metrics gt that is tangential to the Ricci flow almost-everywhere along geodesics when starting with a smooth initial metric. Under the assumption that our initial metric is a rough metric on M with a C heat kernel on a "non-singular" nonempty open subset N, we show that x ? gt(x) is continuous whenever x ? N.