Continuity of solutions to space-varying pointwise linear elliptic equations

Abstract

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 $x\in \mathcal{N}$.

References in corpus (1)

• Nonlinear diffusion equations and curvature conditions in metric measure spaces

Cited by in corpus (3)

• Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics
• Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundary
• Functional calculus and harmonic analysis in geometry