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Continuity of solutions to space-varying pointwise linear elliptic equations

L. Bandara
2017 Publicacions matemàtiques  
We consider pointwise linear elliptic equations of the form Lxux = ηx 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 L 2 -continuity of the solutions at x whenever the coefficients of Lx are L ∞ -continuous at x and the initial datum is L 2 -continuous at x. This is obtained by reducing the continuity of solutions to a homogeneous Kato square root problem. As an application,
more » ... s 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 1 heat kernel on a "non-singular" nonempty open subset N , we show that x → gt(x) is continuous whenever x ∈ N . 2010 Mathematics Subject Classification: 58J05, 58J60, 47J35, 58D25.
doi:10.5565/publmat_61117_09 fatcat:7dloi3yitrf27j2wpu4dkukkli