| 1. |
- Bandara, Lashi, 1981, et al.
(författare)
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Continuity of solutions to space-varying pointwise linear elliptic equations
- 2017
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Ingår i: Publicacions Matematiques. - 0214-1493 .- 2014-4350. ; 61, s. 239-258
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Tidskriftsartikel (refereegranskat)abstract
- We consider pointwise linear elliptic equations of the form Lxux = ηx on a smooth compact manifold where the operators Lx are in divergence form with real, bounded, measurable coefficients that vary in the space variable x. We establish L2 -continuity of the solutions at x whenever the coefficients of Lx are L∞-continuous at x and the initial datum is L2-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 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 M with a C heat kernel on a "non-singular" nonempty open subset N, we show that x → gt(x) is continuous whenever x ∈ N.
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| 2. |
- Bandara, Lashi, 1981, et al.
(författare)
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Essential self-adjointness of powers of first-order differential operators on non-compact manifolds with low-regularity metrics
- 2017
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Ingår i: Journal of Functional Analysis. - : Elsevier BV. - 0022-1236 .- 1096-0783. ; 273:12, s. 3719-3758
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Tidskriftsartikel (refereegranskat)abstract
- We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator possesses higher regularity coefficients, we show that higher powers are essentially self-adjoint if and only if this condition is satisfied. In the case that the low-regularity Riemannian metric induces a complete length space, we demonstrate essential self-adjointness of the operator and its higher powers up to the regularity of its coefficients. We also present applications to Dirac operators on Dirac bundles when the metric is non-smooth. (C) 2017 Elsevier Inc. All rights reserved.
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| 3. |
- Bandara, Lashi, 1981, et al.
(författare)
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Geometric singularities and a flow tangent to the Ricci flow
- 2017
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Ingår i: Annali della Scuola Normale Superiore di Pisa - Classe di Scienze. - 0391-173X .- 2036-2145. ; 17:2, s. 763-804
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Tidskriftsartikel (refereegranskat)abstract
- We consider a geometric flow introduced by Gigli and Mahtegazza which, in the case of a smooth compact manifold with a smooth metric, is tangential to the Ricci flow almost-everywhere along geodesies. To study spaces with geometric singularities, we consider this flow in the context of a smooth manifold with a rough metric possessing a sufficiently regular heat kernel. On an appropriate non-singular open region, we provide a family of metric tensors evolving in time and provide a regularity theory for this flow in terms of the regularity of the heat kernel. When the rough metric induces a metric measure space satisfying a Rieman-nian curvature dimension condition, we demonstrate that the distance induced by the flow is identical to the evolving distance metric defined by Gigli and Man-tegazza on appropriate admissible points. Consequently, we demonstrate that a smooth compact manifold with a finite number of geometric conical singularities remains a smooth manifold with a smooth metric away from the cone points for all future times. Moreover, we show that the distance induced by the evolving metric tensor agrees with the flow of RCD(K, N) spaces defined by Gigli and Mantegazza.
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| 4. |
- Bandara, Lashi, 1981, et al.
(författare)
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Rough metrics on manifolds and quadratic estimates
- 2016
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Ingår i: Mathematische Zeitschrift. - : Springer Science and Business Media LLC. - 0025-5874 .- 1432-1823. ; 283:3, s. 1245-1281
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Tidskriftsartikel (refereegranskat)abstract
- We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of “rough” Riemannian-like metrics that are permitted to be of low regularity and degenerate on sets of measure zero. We also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi-Lipschitz. As a consequence, we demonstrate the invariance of the Kato square root problem under Lipschitz transformations and obtain solutions to this problem on functions and forms on compact manifolds with a rough metric. Furthermore, we show that a lower bound on the injectivity radius is not a necessary condition to solve the Kato square root problem.
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| 5. |
- Bandara, Lashi, 1981, et al.
(författare)
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Self-adjointness of the Gaffney Laplacian on Vector Bundles
- 2015
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Ingår i: Mathematical Physics Analysis and Geometry. - : Springer Science and Business Media LLC. - 1385-0172 .- 1572-9656. ; 18:1
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Tidskriftsartikel (refereegranskat)abstract
- We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boundary is a necessary and sufficient condition for the self-adjointness of this operator.
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