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Continuity of solutions to space-varying pointwise linear elliptic equations. (English) Zbl 1361.58011

Summary: We consider pointwise linear elliptic equations of the form \(\mathrm{L}_x u_x = \eta_x\) on a smooth compact manifold where the operators \(\mathrm{L}_x\) are in divergence form with real, bounded, measurable coefficients that vary in the space variable \(x\). We establish \(\mathrm{L}^{2}\)-continuity of the solutions at \(x\) whenever the coefficients of \(\mathrm{L}_x\) are \(\mathrm{L}^{\infty}\)-continuous at \(x\) and the initial datum is \(\mathrm{L}^{2}\)-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 \(\mathrm{g}_t\) 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 \(\mathcal{M}\) with a \(\mathrm{C}^{1}\) heat kernel on a “non-singular” nonempty open subset \(\mathcal{N}\), we show that \(x \mapsto \mathrm{g}_t(x)\) is continuous whenever \(x \in \mathcal{N}\).

MSC:

58J05 Elliptic equations on manifolds, general theory
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
47J35 Nonlinear evolution equations
58D25 Equations in function spaces; evolution equations
35R01 PDEs on manifolds
35R06 PDEs with measure
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