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2017 Continuity of solutions to space-varying pointwise linear elliptic equations
Lashi Bandara
Publ. Mat. 61(1): 239-258 (2017). DOI: 10.5565/PUBLMAT_61117_09

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 $x \in \mathcal{N}$.

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Lashi Bandara. "Continuity of solutions to space-varying pointwise linear elliptic equations." Publ. Mat. 61 (1) 239 - 258, 2017. https://doi.org/10.5565/PUBLMAT_61117_09

Information

Received: 23 June 2015; Revised: 10 February 2016; Published: 2017
First available in Project Euclid: 22 December 2016

zbMATH: 1361.58011
MathSciNet: MR3590121
Digital Object Identifier: 10.5565/PUBLMAT_61117_09

Subjects:
Primary: 47J35 , 58D25 , 58J05 , 58J60

Keywords: continuity equation , homogeneous Kato square root problem , rough metrics

Rights: Copyright © 2017 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.61 • No. 1 • 2017
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