RENORMALIZED LAPLACIANS ON A CLASS OF HILBERT MANIFOLDS AND A BOCHNER–WEITZENBÖCK TYPE FORMULA FOR CURRENT GROUPS

2000 ◽  
Vol 03 (01) ◽  
pp. 53-98 ◽  
Author(s):  
MARC ARNAUDON ◽  
YANA BELOPOLSKAYA ◽  
SYLVIE PAYCHA
Keyword(s):  
Hilbert Space ◽  
Riemannian Manifold ◽  
Differential Operators ◽  
Sobolev Class ◽  
Hilbert Manifold ◽  
One Dimensional ◽  
Type Formula ◽  
Pseudo Differential Operators ◽  
Underlying Manifold ◽  
Class Of Functions

We define renormalized Laplacians and investigate their properties for a class of C2 functions on Hilbert manifolds modeled on a Hilbert space Hs(M, V) of sections of Sobolev class Hs of some vector bundle V based on a closed Riemannian manifold M, and such that the transition maps of the Hilbert manifold are pseudo-differential operators acting on smooth sections of the bundle V. Among these manifolds we find current groups Hs(M, G), s> dim M/2 where M is a closed manifold and G a Lie group. Weighted Laplacians are renormalized Laplacians which coincide with ordinary Laplacians when the underlying manifold M reduces to a point. We prove a Bochner–Weitzenböck type formula for weighted Laplacians and we point out how in some cases it can reduce to a relation of the type [Δ,∇]f= Ricci (∇f, ·) for a class of functions on certain current groups. Another type of renormalized Laplacian we define are Lévy-type Laplacians which coincide with Lévy Laplacians when the underlying manifold M is one-dimensional. We describe them as limit generators for a one-parameter family of regularized Brownian motions.

GENERATORS OF MEHLER-TYPE SEMIGROUPS AS PSEUDO-DIFFERENTIAL OPERATORS

2002 ◽  
Vol 05 (03) ◽  
pp. 297-315 ◽  
Author(s):  
PAUL LESCOT ◽  
MICHAEL RÖCKNER
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Invariant Measure ◽  
Differential Operators ◽  
Pseudo Differential Operator ◽  
Type Formula ◽  
Pseudo Differential Operators

We study semigroups (Pt)t ≥ 0 on a Hilbert space E, given by a Mehler-type formula: [Formula: see text] Under reasonable assumptions, the Lp(E,μ)-generator [Formula: see text] of (Pt)t ≥ 0 turns out to be expressible as a pseudo-differential operator, provided μ is an invariant measure for (Pt)t ≥ 0. The question of Lp-uniqueness is also answered positively.

scholarly journals WEIGHTED TRACES ON ALGEBRAS OF PSEUDO-DIFFERENTIAL OPERATORS AND GEOMETRY ON LOOP GROUPS

2002 ◽  
Vol 05 (04) ◽  
pp. 503-540 ◽  
Author(s):  
A. CARDONA ◽  
C. DUCOURTIOUX ◽  
J. P. MAGNOT ◽  
S. PAYCHA
Keyword(s):  
Elliptic Operator ◽  
Differential Operators ◽  
Loop Groups ◽  
Linear Functionals ◽  
Geometric Framework ◽  
Infinite Dimensional ◽  
Type Formula ◽  
Pseudo Differential Operators ◽  
Set Up ◽  
The Way

Using weighted traces which are linear functionals of the type [Formula: see text] defined on the whole algebra of (classical) pseudo-differential operators (P.D.Os) and where Q is some admissible invertible elliptic operator, we investigate the geometry of loop groups in the light of the cohomology of pseudo-differential operators. We set up a geometric framework to study a class of infinite dimensional manifolds in which we recover some results on the geometry of loop groups, using again weighted traces. Along the way, we investigate properties of extensions of the Radul and Schwinger cocycles defined with the help of weighted traces.

Two versions of pseudo-differential operators involving the Kontorovich–Lebedev transform in L 2(ℝ+;dx/x)

2018 ◽  
Vol 30 (1) ◽  
pp. 31-42 ◽  
Author(s):  
Akhilesh Prasad ◽  
Upain K. Mandal
Keyword(s):  
Hilbert Space ◽  
Differential Operators ◽  
Pseudo Differential Operators ◽  
Symbol Class

Abstract The Pseudo-differential operators (p.d.o.) {L(x,A_{x})} and {\mathcal{L}(x,A_{x})} involving the Kontorovich–Lebedev transform are defined. An estimate for these operators in the Hilbert space {L^{2}(\mathbb{R}_{+};\frac{dx}{x})} is obtained. A symbol class Λ is defined and it is shown that the product of any two symbols from this class is again in Λ. At the end, commutators for the p.d.o. and their boundedness results are discussed.

Homogenized model for a laminar ferromagnetic medium

2003 ◽  
Vol 133 (3) ◽  
pp. 567-598 ◽  
Author(s):  
H. Haddar ◽  
P. Joly
Keyword(s):  
Strong Convergence ◽  
Space Dimension ◽  
Differential Operators ◽  
Periodic Coefficients ◽  
Microscopic Scale ◽  
One Dimensional ◽  
Pseudo Differential Operators ◽  
Dimension 2 ◽  
Homogenized Model ◽  
Ferromagnetic Medium

We study the homogenization of ferromagnetic equations with periodic coefficients in space dimension 2. The obtained nonlinear homogenized law can only be written using a two-scale framework, which couples microscopic and macroscopic scales. It also involves corrector terms, at the microscopic scale, in the form of pseudo-differential operators. We prove the L2 two-scale strong convergence in the laminar case (one-dimensional periodicity).

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

scholarly journals On the Stability in Bonnet's Theorem of the Surface Theory

2007 ◽  
Vol 14 (3) ◽  
pp. 543-564
Author(s):  
Yuri G. Reshetnyak
Keyword(s):  
Differential Operators ◽  
Sobolev Class ◽  
Integrability Condition ◽  
Complete Integrability ◽  
Integral Representations ◽  
Completely Integrable Systems ◽  
Frobenius Theorem ◽  
Fundamental Tensor ◽  
The Stability ◽  
Codazzi Equations

Abstract In the space , 𝑛-dimensional surfaces are considered having the parametrizations which are functions of the Sobolev class with 𝑝 > 𝑛. The first and the second fundamental tensor are defined. The Peterson–Codazzi equations for such functions are understood in some generalized sense. It is proved that if the first and the second fundamental tensor of one surface are close to the first and, respectively, to the second fundamental tensor of the other surface, then these surfaces will be close up to the motion of the space . A difference between the fundamental tensors and the nearness of the surfaces are measured with the help of suitable 𝑊-norms. The proofs are based on a generalization of Frobenius' theorem about completely integrable systems of the differential equations which was proved by Yu. E. Borovskiĭ. The integral representations of functions by differential operators with complete integrability condition are used, which were elaborated by the author in his other works.

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