| 1. |
- Aldana, C. L., et al.
(författare)
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A Polyakov Formula for Sectors
- 2018
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Ingår i: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 28:2, s. 1773-1839
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Tidskriftsartikel (refereegranskat)abstract
- We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw-Sommerfeld's heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.
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| 2. |
- Aldana, Clara L., et al.
(författare)
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Correction to: A Polyakov Formula for Sectors
- 2020
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Ingår i: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 30:3, s. 3371-3372
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- Let ?? denote a finite circular sector of opening angle ?∈(0,?) and radius one, and let ?−?Δ? denote the heat operator associated to the Dirichlet extension of the Laplacian. Based on recent joint work [2] and [3], we discovered an extra contribution to the variational Polyakov formula in [1] coming from the curved boundary component of the sector. Theorems 3 and 4 of [1] should have an added term +14?. This calculation will appear in [2]. The corrected statements of these theorems are given below.
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| 3. |
- Aldana, Clara L., et al.
(författare)
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Correction to: A Polyakov Formula for Sectors (The Journal of Geometric Analysis, (2018), 28, 2, (1773-1839), 10.1007/s12220-017-9888-y)
- 2020
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Ingår i: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 30, s. 3371-3372
-
Tidskriftsartikel (refereegranskat)abstract
- Let ?? denote a finite circular sector of opening angle ?∈(0,?) and radius one, and let ?−?Δ? denote the heat operator associated to the Dirichlet extension of the Laplacian. Based on recent joint work [2] and [3], we discovered an extra contribution to the variational Polyakov formula in [1] coming from the curved boundary component of the sector. Theorems 3 and 4 of [1] should have an added term +14?. This calculation will appear in [2]. The corrected statements of these theorems are given below.
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| 4. |
- Bandara, L., et al.
(författare)
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Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundary
- 2021
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Ingår i: Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze. - : Scuola Normale Superiore - Edizioni della Normale. - 0391-173X .- 2036-2145. ; 22:4, s. 1843-1878
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Tidskriftsartikel (refereegranskat)abstract
- Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems.
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| 5. |
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| 6. |
- Birnir, Bjorn, et al.
(författare)
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Mathematical models for erosion and the optimal transportation of sediment
- 2013
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Ingår i: International Journal of Nonlinear Sciences and Numerical Simulation. - : Walter de Gruyter GmbH. - 2191-0294 .- 1565-1339. ; 14:6, s. 232--337-
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Tidskriftsartikel (refereegranskat)abstract
- We investigate a mathematical theory for the erosion of sediment which begins with the study of a non-linear, parabolic, weighted 4-Laplace equation on a rectangular domain corresponding to a base segment of an extended landscape. Imposing natural boundary conditions, we show that the equation admits entropy solutions and prove regularity and uniqueness of weak solutions when they exist. We then investigate a particular class of weak solutions studied in previous work of the first author and produce numerical simulations of these solutions. After introducing an optimal transportation problem for the sediment flow, we show that this class of weak solutions implements the optimal transportation of the sediment.
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| 7. |
- Charalambous, N., et al.
(författare)
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Eigenvalue Estimates on Bakry–Émery Manifolds
- 2015
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Ingår i: Springer Proceedings in Mathematics. International Workshop on Elliptic and Parabolic Equations, 2013, Hannover, Germany, 10-12 September 2013. - Cham : Springer International Publishing. - 2190-5614 .- 2190-5622 .- 2194-1017 .- 2194-1009. - 9783319125466 ; 119, s. 45-61
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Konferensbidrag (refereegranskat)abstract
- We demonstrate lower bounds for the eigenvalues of compact Bakry– Émery manifolds with and without boundary. The lower bounds for the first eigenvalue rely on a generalized maximum principle which allows gradient estimates in the Riemannian setting to be directly applied to the Bakry–Émery setting. Lower bounds for all eigenvalues are demonstrated using heat kernel estimates and a suitable Sobolev inequality.
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| 8. |
- Charalambous, N., et al.
(författare)
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THE HEAT TRACE FOR THE DRIFTING LAPLACIAN AND SCHRODINGER OPERATORS ON MANIFOLDS
- 2019
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Ingår i: Asian Journal of Mathematics. - 1093-6106 .- 1945-0036. ; 23:4, s. 539-559
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Tidskriftsartikel (refereegranskat)abstract
- We study the heat trace for both Schrodinger operators as well as the drifting Laplacian on compact Riemannian manifolds. In the case of a finite regularity (bounded and pleasurable) potential or weight function, we prove the existence of a partial asymptotic expansion of the heat trace for small times as well as a suitable remainder estimate. This expansion is sharp in the following sense: further terms in the expansion exist if and only if the potential or weight function is of higher Sobolev regularity. In the case of a smooth weight function, we determine the full asymptotic expansion of the heat trace for the drifting Laplacian for small times. We then use the heat trace to study the asymptotics of the eigenvalue counting function. In both cases the Weyl law coincides with the Weyl law for the Riemannian manifold with the standard Laplace-Beltrami operator. We conclude by demonstrating isospectrality results for the drifting Laplacian on compact manifolds.
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| 9. |
- Charalambous, Nelia, et al.
(författare)
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The Laplace spectrum on conformally compact manifolds
- 2024
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Ingår i: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - 0002-9947 .- 1088-6850.
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Tidskriftsartikel (refereegranskat)abstract
- We consider the spectrum of the Laplace operator acting on L-p over a conformally compact manifold for 1 < p < infinity. We prove that for p not equal 2 this spectrum always contains an open region of the complex plane. We further show that the spectrum is contained within a certain parabolic region of the complex plane. These regions depend on the value of p, the dimension of the manifold, and the values of the sectional curvatures approaching the boundary.
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| 10. |
- Fedosova, Ksenia, et al.
(författare)
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Fourier expansions of vector-valued automorphic functions with non-unitary twists
- 2023
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Ingår i: Communications in Number Theory and Physics. - 1931-4531 .- 1931-4523. ; 17:1, s. 173-248
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Tidskriftsartikel (refereegranskat)abstract
- We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.
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