Continuity of solutions to space-varying pointwise linear elliptic equations
- Autores: Lashi Bandara, Lashi Bandara
- Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 61, Nº 1, 2017, págs. 239-258
- Idioma: inglés
- DOI: 10.5565/10.5565_PUBLMAT 61117 09
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Resumen
We consider pointwise linear elliptic equations of the form Lα uα = ŋα on a smooth compact manifold where the operators Lα are in divergence form with real, bounded, measurable coefficients that vary in the space variableα. We establish L2-continuity of the solutions at α whenever the coefficients of Lα are L∞ -continuous at α and the initial datum is L2 -continuous at α. 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 ʍ with a C1 heat kernel on a “non-singular" nonempty open subset Ɲ, we show that α à gt (α) is continuous whenever α € Ɲ.
