Spectral Theory in Riemannian Geometry

Lablée, Olivier

doi:10.4171/151

Cited by 36 publications

(36 citation statements)

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“…For s = 1, one can show that the definition from above coincides with the classical Sobolev space and u 2 L 2 + ∇u 2 L 2 1 2 defines an equivalent norm on H 1 (M). We refer to [25] for an explanation of the gradient as an element of the tangential bundle of M.…”

Section: Definitions and Auxiliary Resultsmentioning

confidence: 99%

Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space

Brzeźniak

Hornung

Weis

2018

Probab. Theory Relat. Fields

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Section: Definitions and Auxiliary Resultsmentioning

confidence: 99%

Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space

Brzeźniak

Hornung

Weis

2018

Probab. Theory Relat. Fields

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“…With this lemma we have proven that for every ρ ∈ H 1 (C) , ρ ≥ 0 a. e., we find a minimiser h of J ϑ with which we may then define ρ a = g ϑ • h, and further ρ i = ρ − ρ a such that (h, ρ a , ρ i ) solves (26) with vanishing diffusivities. It is easily checked that the constructed linker densities are non-negative.…”

Section: Model Without Diffusionmentioning

confidence: 89%

“…We argue by a Petrov-Galerkin-type approximation: 1) Let (ϕ i ) i∈N be an orthonormal Schauder basis of L 2 (C) with eigenvalues (λ i ) i∈N (sorted ascendingly) consisting of eigenfunctions of the Laplace-Beltrami operator ∆ C (due to the divergence theorem and ∂C = ∅, the eigenfunctions ϕ i , i ≥ 2, are mean value free; for a spectral theorem on Riemannian manifolds cf. [26], Theorem 4.…”

Section: B Existence and Uniqueness Of Solutions For The Height Equationmentioning

confidence: 99%

A PDE model for bleb formation and interaction with linker proteins

Werner

Burger

Pietschmann

2020

Transactions of Mathematics and Its Applications

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The aim of this paper is to further develop mathematical models for bleb formation in cells, including cell-membrane interactions with linker proteins. This leads to nonlinear reaction-diffusion equations on a surface coupled to fluid dynamics in the bulk. We provide a detailed mathematical analysis and investigate some singular limits of the model, connecting it to previous literature. Moreover, we provide numerical simulations in different scenarios, confirming that the model can reproduce experimental results on bleb initation. Cell blebbing, Surface PDEs, Fluid-Structure Interaction, Free Boundary Problems. T for the symmetrised gradient. The set of k-times, k ≥ 0, weak differentiable, X-valued functions, where X is a vector space, on some open set Ω ⊆ R 3 is H k (Ω, X). We will also employ the subspaces of mean-value-free functions:of functions with time-constant mean value:T ∈ [0, ∞), of solenoidals functions:, and the traces of solenoidal functions:

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“…We briefly outline its derivation on a general Riemannian Manifold, together with its weak formulation. The subject is classical, and we refer for instance to [33,31] for an introduction to Riemannian Geometry, and to [32], [37] for a detailed treatment on the properties of the Laplace-Beltrami operator (notably, its spectral properties).…”

Section: Laplace-beltrami and Related Equationsmentioning

confidence: 99%

NURBS-SEM: A hybrid spectral element method on NURBS maps for the solution of elliptic PDEs on surfaces

Pitton

Heltai

2018

Computer Methods in Applied Mechanics and Engineering

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Non Uniform Rational B-spline (NURBS) patches are a standard way to describe complex geometries in Computer Aided Design tools, and have gained a lot of popularity in recent years also for the approximation of partial differential equations, via the Isogeometric Analysis (IGA) paradigm. However, spectral accuracy in IGA is limited to relatively small NURBS patch degrees (roughly p ≤ 8), since local condition numbers grow very rapidly for higher degrees. On the other hand, traditional Spectral Element Methods (SEM) guarantee spectral accuracy but often require complex and expensive meshing techniques, like transfinite mapping, that result anyway in inexact geometries. In this work we propose a hybrid NURBS-SEM approximation method that achieves spectral accuracy and maintains exact geometry representation by combining the advantages of IGA and SEM.As a prototypical problem on non trivial geometries, we consider the Laplace-Beltrami and Allen-Cahn equations on a surface. On these problems, we present a comparison of several instances of NURBS-SEM with the standard Galerkin and Collocation Isogeometric Analysis (IGA).

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Spectral Theory in Riemannian Geometry

Cited by 36 publications

References 0 publications

Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space

Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space

A PDE model for bleb formation and interaction with linker proteins

NURBS-SEM: A hybrid spectral element method on NURBS maps for the solution of elliptic PDEs on surfaces

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