| 31. |
- Bandara, L., et al.
(author)
-
Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundary
- 2021
-
In: 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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Journal article (peer-reviewed)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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| 32. |
- Bauer, M., et al.
(author)
-
Semi-invariant Riemannian metrics in hydrodynamics
- 2020
-
In: Calculus of Variations and Partial Differential Equations. - : Springer Science and Business Media LLC. - 0944-2669 .- 1432-0835. ; 59:2
-
Journal article (peer-reviewed)abstract
- Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa-Holm equations are well-studied examples. A beautiful approach to well-posedness is to go from the Eulerian to a Lagrangian description. Geometrically it corresponds to a geodesic initial value problem on the infinite-dimensional group of diffeomorphisms with a right invariant Riemannian metric. By establishing regularity properties of the Riemannian spray one can then obtain local, and sometimes global, existence and uniqueness results. There are, however, many hydrodynamic-type equations, notably shallow water models and compressible Euler equations, where the underlying infinite-dimensional Riemannian structure is not fully right invariant, but still semi-invariant with respect to the subgroup of volume preserving diffeomorphisms. Here we study such metrics. For semi-invariant metrics of Sobolev Hk-type we give local and some global well-posedness results for the geodesic initial value problem. We also give results in the presence of a potential functional (corresponding to the fluid's internal energy). Our study reveals many pitfalls in going from fully right invariant to semi-invariant Sobolev metrics; the regularity requirements, for example, are higher. Nevertheless the key results, such as no loss or gain in regularity along geodesics, can be adopted.
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| 33. |
- Berman, Robert, 1976
(author)
-
On the strict convexity of the K-energy
- 2019
-
In: Pure and Applied Mathematics Quarterly. - 1558-8599 .- 1558-8602. ; 15:4, s. 983-999
-
Journal article (peer-reviewed)abstract
- Let (X, L) be a polarized projective complex manifold. We show, by a simple toric one-dimensional example, that Mabuchi's K-energy functional on the geodesically complete space of bounded positive (1, 1)-forms in c(1)(L), endowed with the Mabuchi-Donaldson-Semmes metric, is not strictly convex modulo automorphisms. However, under some further assumptions the strict convexity in question does hold in the toric case. This leads to a uniqueness result saying that a finite energy minimizer of the K-energy (which exists on any toric polarized manifold (X, L) which is uniformly K-stable) is uniquely determined modulo automorphisms under the assumption that there exists some minimizer with strictly positive curvature current.
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| 34. |
- Frahm, J., et al.
(author)
-
An extension problem related to the fractional Branson-Gover operators
- 2020
-
In: Journal of Functional Analysis. - : Elsevier BV. - 0022-1236 .- 1096-0783. ; 278:5
-
Journal article (peer-reviewed)abstract
- The Branson-Gover operators are conformally invariant differential operators of even degree acting on differential forms. They can be interpolated by a holomorphic family of conformally invariant integral operators called fractional Branson-Gover operators. For Euclidean spaces we show that the fractional Branson-Gover operators can be obtained as Dirichlet-to-Neumann operators of certain conformally invariant boundary value problems, generalizing the work of Caffarelli-Silvestre for the fractional Laplacians to differential forms. The relevant boundary value problems are studied in detail and we find appropriate Sobolev type spaces in which there exist unique solutions and obtain the explicit integral kernels of the solution operators as well as some of their properties. (C) 2019 Elsevier Inc. All rights reserved.
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| 35. |
- Jensen, M., et al.
(author)
-
Finite element convergence for the time-dependent Joule heating problem with mixed boundary conditions
- 2022
-
In: Ima Journal of Numerical Analysis. - : Oxford University Press (OUP). - 0272-4979 .- 1464-3642. ; 42:1, s. 199-228
-
Journal article (peer-reviewed)abstract
- We prove strong convergence for a large class of finite element methods for the time-dependent Joule heating problem in three spatial dimensions with mixed boundary conditions on Lipschitz domains. We consider conforming subspaces for the spatial discretization and the backward Euler scheme for the temporal discretization. Furthermore, we prove uniqueness and higher regularity of the solution on creased domains and additional regularity in the interior of the domain. Due to a variational formulation with a cut-off functional, the convergence analysis does not require a discrete maximum principle, permitting approximation spaces suitable for adaptive mesh refinement, responding to the difference in regularity within the domain.
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| 36. |
- Maurelli, M., et al.
(author)
-
Incompressible Euler equations with stochastic forcing: A geometric approach
- 2023
-
In: Stochastic Processes and Their Applications. - : Elsevier BV. - 0304-4149. ; 159, s. 101-148
-
Journal article (peer-reviewed)abstract
- We consider a stochastic version of Euler equations using the infinite-dimensional geometric approach as pioneered by Ebin and Marsden (1970). For the Euler equations on a compact manifold (possibly with smooth boundary) we establish local existence and uniqueness of a strong solution in spaces of Sobolev mappings (of high enough regularity). Our approach combines techniques from stochastic analysis and infinite-dimensional geometry and provides a novel toolbox to establish local well-posedness of stochastic non-linear partial differential equations.(c) 2023 Elsevier B.V. All rights reserved.
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| 37. |
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| 38. |
- Gerken, Jan, 1991, et al.
(author)
-
Equivariance versus augmentation for spherical images
- 2022
-
In: Proceedings of Machine Learning Resaerch. ; 162, s. 7404-7421
-
Conference paper (peer-reviewed)abstract
- We analyze the role of rotational equivariance in convolutional neural networks (CNNs) applied to spherical images. We compare the performance of the group equivariant networks known as S2CNNs and standard non-equivariant CNNs trained with an increasing amount of data augmentation. The chosen architectures can be considered baseline references for the respective design paradigms. Our models are trained and evaluated on single or multiple items from the MNIST- or FashionMNIST dataset projected onto the sphere. For the task of image classification, which is inherently rotationally invariant, we find that by considerably increasing the amount of data augmentation and the size of the networks, it is possible for the standard CNNs to reach at least the same performance as the equivariant network. In contrast, for the inherently equivariant task of semantic segmentation, the non-equivariant networks are consistently outperformed by the equivariant networks with significantly fewer parameters. We also analyze and compare the inference latency and training times of the different networks, enabling detailed tradeoff considerations between equivariant architectures and data augmentation for practical problems.
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| 39. |
- Balkanova, O., et al.
(author)
-
NON-VANISHING OF MAASS FORM L-FUNCTIONS AT THE CENTRAL POINT
- 2021
-
In: Proceedings of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9939 .- 1088-6826. ; 149:2, s. 509-523
-
Journal article (peer-reviewed)abstract
- In this paper, we consider the family {L-j(s)}(j=1)(infinity) of L-functions associated to an orthonormal basis {u(j)}(j=1)(infinity) of even Hecke-Maass forms for the modular group SL(2, Z) with eigenvalues {lambda(j) = kappa(2)(j) + 1/4}(j=1)(infinity). We prove the following effective non-vanishing result: At least 50% of the central values L-j(1/2) with kappa(j) <= T do not vanish as T -> infinity. Furthermore, we establish effective non-vanishing results in short intervals.
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| 40. |
- Casarino, V., et al.
(author)
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On the maximal operator of a general Ornstein-Uhlenbeck semigroup
- 2022
-
In: Mathematische Zeitschrift. - : Springer Science and Business Media LLC. - 0025-5874 .- 1432-1823. ; 301
-
Journal article (peer-reviewed)abstract
- If Q is a real, symmetric and positive definite n x n matrix, and B a real n x n matrix whose eigenvalues have negative real parts, we consider the Ornstein-Uhlenbeck semigroup on R-n with covariance Q and drift matrix B. Our main result says that the associated maximal operator is of weak type (1, 1) with respect to the invariant measure. The proof has a geometric gist and hinges on the "forbidden zones method" previously introduced by the third author.
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