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Continuity of solutions to space-varying pointwise linear elliptic equations

Lashi Bandara

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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 $x \in \mathcal{N}$.

Article information

Publ. Mat., Volume 61, Number 1 (2017), 239-258.

Received: 23 June 2015
Revised: 10 February 2016
First available in Project Euclid: 22 December 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J05: Elliptic equations on manifolds, general theory [See also 35-XX] 58J60: Relations with special manifold structures (Riemannian, Finsler, etc.) 47J35: Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25] 58D25: Equations in function spaces; evolution equations [See also 34Gxx, 35K90, 35L90, 35R15, 37Lxx, 47Jxx]

Continuity equation rough metrics homogeneous Kato square root problem


Bandara, Lashi. Continuity of solutions to space-varying pointwise linear elliptic equations. Publ. Mat. 61 (2017), no. 1, 239--258. doi:10.5565/PUBLMAT_61117_09.

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