On groups of Hölder diffeomorphisms and their regularity
|
|
|
- Kelley Eaton
- 5 years ago
- Views:
Transcription
1 On groups of Hölder diffeomorphisms and their regularity Joint work with Armin Rainer David Nenning University of Vienna Valencia, October 17, 217
2 Diffeomorphism groups generated by time-dependent vector fields
3 Diffeomorphism groups generated by time-dependent vector fields C 1 (Rd, R d )... Space of continuously differentiable functions vanishing at infinity together with their first derivative.
4 Diffeomorphism groups generated by time-dependent vector fields C 1(Rd, R d )... Space of continuously differentiable functions vanishing at infinity together with their first derivative. For u : I R d R d such that u(, x) is measurable for all x R d, u(t, ) C 1 1 u(t, ) C the ODE for all t I, 1 dt <, t Φ(t) = x + u(r, Φ(r)) dr s admits a unique solution for each fixed s I, x R d ; denoted as Φ u (t, s, x) = x + φ u (t, s, x).
5 Diffeomorphism groups generated by time-dependent vector fields C 1(Rd, R d )... Space of continuously differentiable functions vanishing at infinity together with their first derivative. For u : I R d R d such that u(, x) is measurable for all x R d, u(t, ) C 1 1 u(t, ) C the ODE for all t I, 1 dt <, t Φ(t) = x + u(r, Φ(r)) dr s admits a unique solution for each fixed s I, x R d ; denoted as Φ u (t, s, x) = x + φ u (t, s, x). Φ u (t, s, ) is a C 1 -diffeomorphism and φ u (t, s, ) C 1.
6 Diffeomorphism groups generated by time-dependent vector fields
7 Diffeomorphism groups generated by time-dependent vector fields For a locally convex space E C 1 (Rd, R d ), we set X E... Space of pointwise time-dependent E-vector fields u : I R d R d with u(, x) is measurable for all x R d, u(t, ) E for all t I, 1 p(u(t, )) dt < for all continuous seminorms p.
8 Diffeomorphism groups generated by time-dependent vector fields For a locally convex space E C 1 (Rd, R d ), we set X E... Space of pointwise time-dependent E-vector fields u : I R d R d with u(, x) is measurable for all x R d, u(t, ) E for all t I, 1 p(u(t, )) dt < for all continuous seminorms p. Definition (Trouvé group) G E := { Φ u (1,, ) : u X E }.
9 Diffeomorphism groups generated by time-dependent vector fields For a locally convex space E C 1 (Rd, R d ), we set X E... Space of pointwise time-dependent E-vector fields u : I R d R d with u(, x) is measurable for all x R d, u(t, ) E for all t I, 1 p(u(t, )) dt < for all continuous seminorms p. Definition (Trouvé group) G E := { Φ u (1,, ) : u X E }. Lemma G E is a group with respect to composition.
10 Diffeomorphism groups generated by time-dependent vector fields For a locally convex space E C 1 (Rd, R d ), we set X E... Space of pointwise time-dependent E-vector fields u : I R d R d with u(, x) is measurable for all x R d, u(t, ) E for all t I, 1 p(u(t, )) dt < for all continuous seminorms p. Definition (Trouvé group) G E := { Φ u (1,, ) : u X E }. Lemma G E is a group with respect to composition. Definition (ODE-closedness) E is called ODE-closed iff G E Id +E.
11 Hölder Diffeomorphism groups
12 Hölder Diffeomorphism groups C n,β (R d, R d )... Space of n times continuously differentiable functions vanishing at infinity together with their derivatives up to order n and β-hölder continuous n-th derivative with global Hölder constant, i.e. for f C n,β, f (k) (x) as x for k n, and f n,β := max{ f (k) L (R d,l k (R d,r d )) : k n; sup x,y R d f (n) (x) f (n) (y) Ln(Rd,Rd ) x y β } <.
13 Hölder Diffeomorphism groups C n,β (R d, R d )... Space of n times continuously differentiable functions vanishing at infinity together with their derivatives up to order n and β-hölder continuous n-th derivative with global Hölder constant, i.e. for f C n,β, f (k) (x) as x for k n, and f n,β := max{ f (k) L (R d,l k (R d,r d )) : k n; f (n) (x) f (n) (y) Ln(Rd sup,rd ) x,y R d x y β } <. The set of orientation preserving diffeomorphisms that differ from the identity by a C n,β -function is denoted as Diff C n,β := { Φ Id +C n,β : det Φ (x) > x R d}. It is a Banach manifold modelled on C n,β with global chart Φ Φ Id. (Diff C n,β ) shall denote the connected component of the identity.
14 Hölder Diffeomorphism groups
15 Hölder Diffeomorphism groups Theorem 1 Diff C n,β is a group with respect to composition. Right translations are smooth, but left translations are in general discontinuous.
16 Hölder Diffeomorphism groups Theorem 1 Diff C n,β is a group with respect to composition. Right translations are smooth, but left translations are in general discontinuous. Theorem 2 G n,β = (Diff C n,β ). In particular C n,β is ODE-closed.
17 Proof(sketch): Diff C n,β is a group
18 Proof(sketch): Diff C n,β is a group Composition closedness: Classical proofs for composition closedness of C n,β (R d, R d ) can be modified for (Id +φ, Id +ψ) (Id +φ) (Id +ψ). Right translation (in chart representation) is affine and continuous and therefore smooth.
19 Proof(sketch): Diff C n,β is a group Composition closedness: Classical proofs for composition closedness of C n,β (R d, R d ) can be modified for (Id +φ, Id +ψ) (Id +φ) (Id +ψ). Right translation (in chart representation) is affine and continuous and therefore smooth. Inversion closedness: Let Φ = Id +φ, Φ 1 = Ψ = Id +ψ. Then ψ Φ = φ C n,β. Apply Faà di Bruno s formula, i.e. (ψ Φ) (n) (x) = sym n l=1 γ Γ(l,n) c γ ψ (l) (Φ(x)) (Φ (γ 1) (x),, Φ (γ l ) (x)), where Γ(l, n) := {γ N l > : γ = n}, c γ := n! l!γ!, and sym denotes the symmetrization of multilinear mappings.
20 Proof(sketch): Diff C n,β is a group
21 Proof(sketch): Diff C n,β is a group This yields ψ (n) (Φ(x))(Φ (x),, Φ (x)) ψ (n) (Φ(y))(Φ (y),, Φ (y)) = (φ (n) (x) φ (n) (y)) ( n 1 sym c γ ψ (l) (Φ(x)) (Φ (γ1) (x),, Φ (γ l ) (x)) n 1 l=1 γ Γ(l,n) l=1 γ Γ(l,n) ) c γ ψ (l) (Φ(y)) (Φ (γ1) (y),, Φ (γ l ) (y)), which (after some manipulations) yields ψ C n,β.
22 Proof(sketch): G n,β (Diff C n,β )
23 Proof(sketch): G n,β (Diff C n,β ) Let u be a pointwise time-dependent C n,β -vector field.
24 Proof(sketch): G n,β (Diff C n,β ) Let u be a pointwise time-dependent C n,β -vector field. Recall: u : I R d R d such that u(, x) is measurable for all x R d, u(t, ) C n,β for all t I, 1 u(t, ) n,β dt <.
25 Proof(sketch): G n,β (Diff C n,β ) Let u be a pointwise time-dependent C n,β -vector field. Recall: u : I R d R d such that u(, x) is measurable for all x R d, u(t, ) C n,β for all t I, 1 u(t, ) n,β dt <. Show: t φ u (t, ) C(I, C n,β ), where t Φ u (t, x) = x + φ u (t, x) = x + u(s, Φ u (s, x)) ds.
26 Proof(sketch): G n,β (Diff C n,β )
27 Proof(sketch): G n,β (Diff C n,β ) It is well known that φ u (t, ) C n and there are constants C 1, C 2 such that for all t I φ u (t, ) C n C 1 e C 1 2 u(s, ) C n ds.
28 Proof(sketch): G n,β (Diff C n,β ) It is well known that φ u (t, ) C n and there are constants C 1, C 2 such that for all t I φ u (t, ) C n C 1 e C 1 2 u(s, ) C n ds. And n x φ u (t, x) fulfills n x φ u (t, x) = t n x (u(s, Φ u (s, x)) ds. (1)
29 Proof(sketch): G n,β (Diff C n,β ) It is well known that φ u (t, ) C n and there are constants C 1, C 2 such that for all t I φ u (t, ) C n C 1 e C 1 2 u(s, ) C n ds. And n x φ u (t, x) fulfills Consider n x φ u (t, x) = t n x (u(s, Φ u (s, x)) ds. (1) V n x,y (t) := n x φ u (t, x) n x φ u (t, y) x y β.
30 Proof(sketch): G n,β (Diff C n,β ) It is well known that φ u (t, ) C n and there are constants C 1, C 2 such that for all t I φ u (t, ) C n C 1 e C 1 2 u(s, ) C n ds. And n x φ u (t, x) fulfills Consider n x φ u (t, x) = t n x (u(s, Φ u (s, x)) ds. (1) V n x,y (t) := n x φ u (t, x) n x φ u (t, y) x y β. Apply Faà di Bruno s formula to the integrand in (1); yields linear ODE for V n x,y.
31 Proof(sketch): G n,β (Diff C n,β ) It is well known that φ u (t, ) C n and there are constants C 1, C 2 such that for all t I φ u (t, ) C n C 1 e C 1 2 u(s, ) C n ds. And n x φ u (t, x) fulfills Consider n x φ u (t, x) = t n x (u(s, Φ u (s, x)) ds. (1) V n x,y (t) := n x φ u (t, x) n x φ u (t, y) x y β. Apply Faà di Bruno s formula to the integrand in (1); yields linear ODE for V n x,y. Gronwall s inequality together with integrability of t u(t, ) n,β yields uniform boundedness w.r.t. x, y, t of V n x,y (t).
32 Proof(sketch): G n,β (Diff C n,β ) It is well known that φ u (t, ) C n and there are constants C 1, C 2 such that for all t I φ u (t, ) C n C 1 e C 1 2 u(s, ) C n ds. And n x φ u (t, x) fulfills Consider n x φ u (t, x) = t n x (u(s, Φ u (s, x)) ds. (1) V n x,y (t) := n x φ u (t, x) n x φ u (t, y) x y β. Apply Faà di Bruno s formula to the integrand in (1); yields linear ODE for V n x,y. Gronwall s inequality together with integrability of t u(t, ) n,β yields uniform boundedness w.r.t. x, y, t of V n x,y (t). Integrability also gives continuity into C n,β.
33 Proof(sketch): G n,β (Diff C n,β )
34 Proof(sketch): G n,β (Diff C n,β ) For Φ = Id +φ (Diff C n,β ) there exists a polygon in (Diff C n,β with vertices Id = Φ, Φ 1 = Id +φ 1,, Φ k = Id +φ k = Φ. )
35 Proof(sketch): G n,β (Diff C n,β ) For Φ = Id +φ (Diff C n,β ) there exists a polygon in (Diff C n,β with vertices Id = Φ, Φ 1 = Id +φ 1,, Φ k = Id +φ k = Φ. ) Φ 1 = Φ u1 (1, ) with u 1 (t, x) = φ 1 ((1 t) Id +tφ 1 ) 1 (x); and u 1 is pointwise time-dependent Hölder vector field Φ 1 G n,β.
36 Proof(sketch): G n,β (Diff C n,β ) For Φ = Id +φ (Diff C n,β ) there exists a polygon in (Diff C n,β with vertices Id = Φ, Φ 1 = Id +φ 1,, Φ k = Id +φ k = Φ. ) Φ 1 = Φ u1 (1, ) with u 1 (t, x) = φ 1 ((1 t) Id +tφ 1 ) 1 (x); and u 1 is pointwise time-dependent Hölder vector field Φ 1 G n,β. Same argument gives Φ 2 Φ 1 1 G n,β. Since G n,β is a group, Φ 2 G n,β. Iterate argument another k 2 times; yields Φ k = Φ G n,β.
37 Further results
38 Further results Definition (Intermediate Hölder spaces) C n,β (R d, R d ) := α (,β) C n,α (R d, R d ), C n,β+ (R d, R d ) := α (β,1) C n,α (R d, R d ), endowed with natural projective and inductive locally convex topologies.
39 Further results Definition (Intermediate Hölder spaces) C n,β (R d, R d ) := α (,β) C n,α (R d, R d ), C n,β+ (R d, R d ) := α (β,1) C n,α (R d, R d ), endowed with natural projective and inductive locally convex topologies. Theorem Diff C n,β is a topological group, Diff C n,β+, group operations map smooth curves to continuous curves, G n,β± = (Diff C n,β± ).
40 Thanks for your attention!
Euler Equations: local existence
Euler Equations: local existence Mat 529, Lesson 2. 1 Active scalars formulation We start with a lemma. Lemma 1. Assume that w is a magnetization variable, i.e. t w + u w + ( u) w = 0. If u = Pw then u
i=1 β i,i.e. = β 1 x β x β 1 1 xβ d
66 2. Every family of seminorms on a vector space containing a norm induces ahausdorff locally convex topology. 3. Given an open subset Ω of R d with the euclidean topology, the space C(Ω) of real valued
1.4 The Jacobian of a map
1.4 The Jacobian of a map Derivative of a differentiable map Let F : M n N m be a differentiable map between two C 1 manifolds. Given a point p M we define the derivative of F at p by df p df (p) : T p
4 Hilbert spaces. The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan
The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan Wir müssen wissen, wir werden wissen. David Hilbert We now continue to study a special class of Banach spaces,
Sub-Riemannian geometry in groups of diffeomorphisms and shape spaces
Sub-Riemannian geometry in groups of diffeomorphisms and shape spaces Sylvain Arguillère, Emmanuel Trélat (Paris 6), Alain Trouvé (ENS Cachan), Laurent May 2013 Plan 1 Sub-Riemannian geometry 2 Right-invariant
Notes for Functional Analysis
Notes for Functional Analysis Wang Zuoqin (typed by Xiyu Zhai) September 29, 2015 1 Lecture 09 1.1 Equicontinuity First let s recall the conception of equicontinuity for family of functions that we learned
Functional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
Notes for Functional Analysis
Notes for Functional Analysis Wang Zuoqin (typed by Xiyu Zhai) October 16, 2015 1 Lecture 11 1.1 The closed graph theorem Definition 1.1. Let f : X Y be any map between topological spaces. We define its
Functional Analysis Exercise Class
Functional Analysis Exercise Class Week 2 November 6 November Deadline to hand in the homeworks: your exercise class on week 9 November 13 November Exercises (1) Let X be the following space of piecewise
THEODORE VORONOV DIFFERENTIABLE MANIFOLDS. Fall Last updated: November 26, (Under construction.)
4 Vector fields Last updated: November 26, 2009. (Under construction.) 4.1 Tangent vectors as derivations After we have introduced topological notions, we can come back to analysis on manifolds. Let M
ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM
ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM OLEG ZUBELEVICH DEPARTMENT OF MATHEMATICS THE BUDGET AND TREASURY ACADEMY OF THE MINISTRY OF FINANCE OF THE RUSSIAN FEDERATION 7, ZLATOUSTINSKY MALIY PER.,
Lecture 11 Hyperbolicity.
Lecture 11 Hyperbolicity. 1 C 0 linearization near a hyperbolic point 2 invariant manifolds Hyperbolic linear maps. Let E be a Banach space. A linear map A : E E is called hyperbolic if we can find closed
Progress in Several Complex Variables KIAS 2018
for Progress in Several Complex Variables KIAS 2018 Outline 1 for 2 3 super-potentials for 4 real for Let X be a real manifold of dimension n. Let 0 p n and k R +. D c := { C k (differential) (n p)-forms
1 Differentiable manifolds and smooth maps
1 Differentiable manifolds and smooth maps Last updated: April 14, 2011. 1.1 Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set
Riemann integral and volume are generalized to unbounded functions and sets. is an admissible set, and its volume is a Riemann integral, 1l E,
Tel Aviv University, 26 Analysis-III 9 9 Improper integral 9a Introduction....................... 9 9b Positive integrands................... 9c Special functions gamma and beta......... 4 9d Change of
h-principles for the curvature of (semi-)riemannian metrics
Marc Nardmann (marc.nardmann@math.uni-hamburg.de) London, 2013 March 14 h-principles for the curvature of (semi-)riemannian metrics M E + = Sym 2 + T M smooth manifold fibre bundle over M sections in E
LECTURE 22: INTEGRATION ON MANIFOLDS. 1. Orientations
LECTURE 22: INTEGRATION ON ANIFOLDS 1. Orientations Let be a smooth manifold of dimension n, and let ω Ω n () be a smooth n-form. We want to define the integral ω. First assume = R n. In calculus we learned
INTEGRATION ON MANIFOLDS and GAUSS-GREEN THEOREM
INTEGRATION ON MANIFOLS and GAUSS-GREEN THEOREM 1. Schwarz s paradox. Recall that for curves one defines length via polygonal approximation by line segments: a continuous curve γ : [a, b] R n is rectifiable
The incompressible Navier-Stokes equations in vacuum
The incompressible, Université Paris-Est Créteil Piotr Bogus law Mucha, Warsaw University Journées Jeunes EDPistes 218, Institut Elie Cartan, Université de Lorraine March 23th, 218 Incompressible Navier-Stokes
Entropy Methods for Reaction-Diffusion Equations with Degenerate Diffusion Arising in Reversible Chemistry
1 Entropy Methods for Reaction-Diffusion Equations with Degenerate Diffusion Arising in Reversible Chemistry L. Desvillettes CMLA, ENS Cachan, IUF & CNRS, PRES UniverSud, 61, avenue du Président Wilson,
DIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES. September 25, 2015
DIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES MAGGIE MILLER September 25, 2015 1. 09/16/2015 1.1. Textbooks. Textbooks relevant to this class are Riemannian Geometry by do Carmo Riemannian Geometry
LECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
A Class of Shock Wave Solutions of the Periodic Degasperis-Procesi Equation
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.9(2010) No.3,pp.367-373 A Class of Shock Wave Solutions of the Periodic Degasperis-Procesi Equation Caixia Shen
Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces
Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces YUAN-HENG WANG Zhejiang Normal University Department of Mathematics Yingbing Road 688, 321004 Jinhua
SYNGE-WEINSTEIN THEOREMS IN RIEMANNIAN GEOMETRY
SYNGE-WEINSTEIN THEOREMS IN RIEMANNIAN GEOMETRY AKHIL MATHEW Abstract. We give an exposition of the proof of a few results in global Riemannian geometry due to Synge and Weinstein using variations of the
EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS
EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS Cezar Avramescu Abstract The problem of existence of the solutions for ordinary differential equations vanishing at ± is considered. AMS
The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:
Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply
ON PERIODIC SOLUTIONS TO SOME LAGRANGIAN SYSTEM WITH TWO DEGREES OF FREEDOM
ON PERIODIC SOLUTIONS TO SOME LAGRANGIAN SYSTEM WITH TWO DEGREES OF FREEDOM OLEG ZUBELEVICH DEPT. OF THEORETICAL MECHANICS, MECHANICS AND MATHEMATICS FACULTY, M. V. LOMONOSOV MOSCOW STATE UNIVERSITY RUSSIA,
Applied Analysis (APPM 5440): Final exam 1:30pm 4:00pm, Dec. 14, Closed books.
Applied Analysis APPM 44: Final exam 1:3pm 4:pm, Dec. 14, 29. Closed books. Problem 1: 2p Set I = [, 1]. Prove that there is a continuous function u on I such that 1 ux 1 x sin ut 2 dt = cosx, x I. Define
WELL-POSEDNESS FOR HYPERBOLIC PROBLEMS (0.2)
WELL-POSEDNESS FOR HYPERBOLIC PROBLEMS We will use the familiar Hilbert spaces H = L 2 (Ω) and V = H 1 (Ω). We consider the Cauchy problem (.1) c u = ( 2 t c )u = f L 2 ((, T ) Ω) on [, T ] Ω u() = u H
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are
An introduction to Mathematical Theory of Control
An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018
Automorphic Equivalence Within Gapped Phases
1 Harvard University May 18, 2011 Automorphic Equivalence Within Gapped Phases Robert Sims University of Arizona based on joint work with Sven Bachmann, Spyridon Michalakis, and Bruno Nachtergaele 2 Outline:
Rough Burgers-like equations with multiplicative noise
Rough Burgers-like equations with multiplicative noise Martin Hairer Hendrik Weber Mathematics Institute University of Warwick Bielefeld, 3.11.21 Burgers-like equation Aim: Existence/Uniqueness for du
Recent Trends in Differential Inclusions
Recent Trends in Alberto Bressan Department of Mathematics, Penn State University (Aveiro, June 2016) (Aveiro, June 2016) 1 / Two main topics ẋ F (x) differential inclusions with upper semicontinuous,
0.1 Diffeomorphisms. 0.2 The differential
Lectures 6 and 7, October 10 and 12 Easy fact: An open subset of a differentiable manifold is a differentiable manifold of the same dimension the ambient space differentiable structure induces a differentiable
Piecewise Smooth Solutions to the Burgers-Hilbert Equation
Piecewise Smooth Solutions to the Burgers-Hilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA e-mails: bressan@mathpsuedu, zhang
Solution. 1 Solution of Homework 7. Sangchul Lee. March 22, Problem 1.1
Solution Sangchul Lee March, 018 1 Solution of Homework 7 Problem 1.1 For a given k N, Consider two sequences (a n ) and (b n,k ) in R. Suppose that a n b n,k for all n,k N Show that limsup a n B k :=
2. Function spaces and approximation
2.1 2. Function spaces and approximation 2.1. The space of test functions. Notation and prerequisites are collected in Appendix A. Let Ω be an open subset of R n. The space C0 (Ω), consisting of the C
Why Do Partitions Occur in Faà di Bruno s Chain Rule For Higher Derivatives?
Why Do Partitions Occur in Faà di Bruno s Chain Rule For Higher Derivatives? arxiv:math/0602183v1 [math.gm] 9 Feb 2006 Eliahu Levy Department of Mathematics Technion Israel Institute of Technology, Haifa
Fuchsian groups. 2.1 Definitions and discreteness
2 Fuchsian groups In the previous chapter we introduced and studied the elements of Mob(H), which are the real Moebius transformations. In this chapter we focus the attention of special subgroups of this
Continuity of convex functions in normed spaces
Continuity of convex functions in normed spaces In this chapter, we consider continuity properties of real-valued convex functions defined on open convex sets in normed spaces. Recall that every infinitedimensional
CALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
Calculus of Variations. Final Examination
Université Paris-Saclay M AMS and Optimization January 18th, 018 Calculus of Variations Final Examination Duration : 3h ; all kind of paper documents (notes, books...) are authorized. The total score of
Applied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.
Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R
3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
MULTIDIMENSIONAL SINGULAR λ-lemma
Electronic Journal of Differential Equations, Vol. 23(23), No. 38, pp.1 9. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) MULTIDIMENSIONAL
2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
LECTURE 5: THE METHOD OF STATIONARY PHASE
LECTURE 5: THE METHOD OF STATIONARY PHASE Some notions.. A crash course on Fourier transform For j =,, n, j = x j. D j = i j. For any multi-index α = (α,, α n ) N n. α = α + + α n. α! = α! α n!. x α =
Solutions to Problem Set 5 for , Fall 2007
Solutions to Problem Set 5 for 18.101, Fall 2007 1 Exercise 1 Solution For the counterexample, let us consider M = (0, + ) and let us take V = on M. x Let W be the vector field on M that is identically
DIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17
DIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17 6. Geodesics A parametrized line γ : [a, b] R n in R n is straight (and the parametrization is uniform) if the vector γ (t) does not depend on t. Thus,
Distributions: Topology and Sequential Compactness
Distributions: Topology and Sequential Compactness Acknowledgement Distributions: Topology and Sequential Compactness Acknowledgement First of all I want to thank Professor Mouhot for setting this essay
TIME DISCRETE GEODESIC PATHS IN THE SPACE OF IMAGES
TIME ISCRETE GEOESIC PATHS IN THE SPACE OF IMAGES B. BERELS, A. EFFLAN, M. RUMPF Abstract. In this paper the space of images is considered as a Riemannian manifold using the metamorphosis approach [28,
Hadamard and Perron JWR. October 18, On page 23 of his famous monograph [2], D. V. Anosov writes
![Hadamard and Perron JWR. October 18, On page 23 of his famous monograph [2], D. V. Anosov writes](/thumbs/88/117485901.jpg "Hadamard and Perron JWR. October 18, On page 23 of his famous monograph [2], D. V. Anosov writes")
Hadamard and Perron JWR October 18, 1999 On page 23 of his famous monograph [2], D. V. Anosov writes Every five years or so, if not more often, someone discovers the theorem of Hadamard and Perron proving
Commutative Banach algebras 79
8. Commutative Banach algebras In this chapter, we analyze commutative Banach algebras in greater detail. So we always assume that xy = yx for all x, y A here. Definition 8.1. Let A be a (commutative)
The Calderon-Vaillancourt Theorem
The Calderon-Vaillancourt Theorem What follows is a completely self contained proof of the Calderon-Vaillancourt Theorem on the L 2 boundedness of pseudo-differential operators. 1 The result Definition
Convex analysis and profit/cost/support functions
Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m Convex analysts may give one of two
LECTURE 3 Functional spaces on manifolds
LECTURE 3 Functional spaces on manifolds The aim of this section is to introduce Sobolev spaces on manifolds (or on vector bundles over manifolds). These will be the Banach spaces of sections we were after
SMSTC (2017/18) Geometry and Topology 2.
SMSTC (2017/18) Geometry and Topology 2 Lecture 1: Differentiable Functions and Manifolds in R n Lecturer: Diletta Martinelli (Notes by Bernd Schroers) a wwwsmstcacuk 11 General remarks In this lecture
L p Spaces and Convexity
L p Spaces and Convexity These notes largely follow the treatments in Royden, Real Analysis, and Rudin, Real & Complex Analysis. 1. Convex functions Let I R be an interval. For I open, we say a function
e (x y)2 /4kt φ(y) dy, for t > 0. (4)
Math 24A October 26, 2 Viktor Grigoryan Heat equation: interpretation of the solution Last time we considered the IVP for the heat equation on the whole line { ut ku xx = ( < x
56 4 Integration against rough paths
56 4 Integration against rough paths comes to the definition of a rough integral we typically take W = LV, W ; although other choices can be useful see e.g. remark 4.11. In the context of rough differential
Introduction to Computational Manifolds and Applications
IMPA - Instituto de Matemática Pura e Aplicada, Rio de Janeiro, RJ, Brazil Introduction to Computational Manifolds and Applications Part 1 - Foundations Prof. Jean Gallier jean@cis.upenn.edu Department
Riemann integral and Jordan measure are generalized to unbounded functions. is a Jordan measurable set, and its volume is a Riemann integral, R n
Tel Aviv University, 214/15 Analysis-III,IV 161 1 Improper integral 1a What is the problem................ 161 1b Positive integrands................. 162 1c Newton potential.................. 166 1d Special
zi z i, zi+1 z i,, zn z i. z j, zj+1 z j,, zj 1 z j,, zn
The Complex Projective Space Definition. Complex projective n-space, denoted by CP n, is defined to be the set of 1-dimensional complex-linear subspaces of C n+1, with the quotient topology inherited from
FIXED POINT THEOREMS OF KRASNOSELSKII TYPE IN A SPACE OF CONTINUOUS FUNCTIONS
Fixed Point Theory, Volume 5, No. 2, 24, 181-195 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.htm FIXED POINT THEOREMS OF KRASNOSELSKII TYPE IN A SPACE OF CONTINUOUS FUNCTIONS CEZAR AVRAMESCU 1 AND CRISTIAN
Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping.
Minimization Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping. 1 Minimization A Topological Result. Let S be a topological
Functional Analysis. Martin Brokate. 1 Normed Spaces 2. 2 Hilbert Spaces The Principle of Uniform Boundedness 32
Functional Analysis Martin Brokate Contents 1 Normed Spaces 2 2 Hilbert Spaces 2 3 The Principle of Uniform Boundedness 32 4 Extension, Reflexivity, Separation 37 5 Compact subsets of C and L p 46 6 Weak
Uniformly accurate averaging numerical schemes for oscillatory evolution equations
Uniformly accurate averaging numerical schemes for oscillatory evolution equations Philippe Chartier University of Rennes, INRIA Joint work with M. Lemou (University of Rennes-CNRS), F. Méhats (University
LOCAL EXISTENCE THEORY OF THE VACUUM EINSTEIN EQUATIONS
LOCAL EXISTENCE THEORY OF THE VACUUM EINSTEIN EQUATIONS TSZ KIU AARON CHOW Abstract. In this paper, we treat the vacuum Einstein equations as a local Cauchy problem. We show that it reduces to a system
Convergence results for solutions of a first-order differential equation
vailable online at www.tjnsa.com J. Nonlinear ci. ppl. 6 (213), 18 28 Research rticle Convergence results for solutions of a first-order differential equation Liviu C. Florescu Faculty of Mathematics,
MATH 220: MIDTERM OCTOBER 29, 2015
MATH 22: MIDTERM OCTOBER 29, 25 This is a closed book, closed notes, no electronic devices exam. There are 5 problems. Solve Problems -3 and one of Problems 4 and 5. Write your solutions to problems and
Functional Analysis Exercise Class
Functional Analysis Exercise Class Wee November 30 Dec 4: Deadline to hand in the homewor: your exercise class on wee December 7 11 Exercises with solutions Recall that every normed space X can be isometrically
Sobolev Spaces. Chapter 10
Chapter 1 Sobolev Spaces We now define spaces H 1,p (R n ), known as Sobolev spaces. For u to belong to H 1,p (R n ), we require that u L p (R n ) and that u have weak derivatives of first order in L p
Let us recall in a nutshell the definition of some important algebraic structure, increasingly more refined than that of group.
Chapter 1 SOME MATHEMATICAL TOOLS 1.1 Some definitions in algebra Let us recall in a nutshell the definition of some important algebraic structure, increasingly more refined than that of group. Ring A
Lecture Notes in Real Analysis Anant R. Shastri Department of Mathematics Indian Institute of Technology Bombay
Lecture Notes in Real Analysis 2010 Anant R. Shastri Department of Mathematics Indian Institute of Technology Bombay August 6, 2010 Lectures 1-3 (I-week) Lecture 1 Why real numbers? Example 1 Gaps in the
L. Levaggi A. Tabacco WAVELETS ON THE INTERVAL AND RELATED TOPICS
Rend. Sem. Mat. Univ. Pol. Torino Vol. 57, 1999) L. Levaggi A. Tabacco WAVELETS ON THE INTERVAL AND RELATED TOPICS Abstract. We use an abstract framework to obtain a multilevel decomposition of a variety
MATH 217A HOMEWORK. P (A i A j ). First, the basis case. We make a union disjoint as follows: P (A B) = P (A) + P (A c B)
MATH 217A HOMEWOK EIN PEASE 1. (Chap. 1, Problem 2. (a Let (, Σ, P be a probability space and {A i, 1 i n} Σ, n 2. Prove that P A i n P (A i P (A i A j + P (A i A j A k... + ( 1 n 1 P A i n P (A i P (A
Proof. We indicate by α, β (finite or not) the end-points of I and call
C.6 Continuous functions Pag. 111 Proof of Corollary 4.25 Corollary 4.25 Let f be continuous on the interval I and suppose it admits non-zero its (finite or infinite) that are different in sign for x tending
LECTURE 10: THE PARALLEL TRANSPORT
LECTURE 10: THE PARALLEL TRANSPORT 1. The parallel transport We shall start with the geometric meaning of linear connections. Suppose M is a smooth manifold with a linear connection. Let γ : [a, b] M be
Threshold behavior and non-quasiconvergent solutions with localized initial data for bistable reaction-diffusion equations
Threshold behavior and non-quasiconvergent solutions with localized initial data for bistable reaction-diffusion equations P. Poláčik School of Mathematics, University of Minnesota Minneapolis, MN 55455
Geodesics on Shape Spaces with Bounded Variation and Sobolev Metrics
Geodesics on Shape Spaces with Bounded Variation and Sobolev Metrics arxiv:142.654v6 [math.oc] 26 Dec 214 Giacomo Nardi, Gabriel Peyré, François-Xavier Vialard Ceremade, Université Paris-Dauphine Abstract
Continuous Functions on Metric Spaces
Continuous Functions on Metric Spaces Math 201A, Fall 2016 1 Continuous functions Definition 1. Let (X, d X ) and (Y, d Y ) be metric spaces. A function f : X Y is continuous at a X if for every ɛ > 0
Representations of moderate growth Paul Garrett 1. Constructing norms on groups
(December 31, 2004) Representations of moderate growth Paul Garrett Representations of reductive real Lie groups on Banach spaces, and on the smooth vectors in Banach space representations,
MEASURE VALUED DIRECTIONAL SPARSITY FOR PARABOLIC OPTIMAL CONTROL PROBLEMS
MEASURE VALUED DIRECTIONAL SPARSITY FOR PARABOLIC OPTIMAL CONTROL PROBLEMS KARL KUNISCH, KONSTANTIN PIEPER, AND BORIS VEXLER Abstract. A directional sparsity framework allowing for measure valued controls
Partial Differential Equations, 2nd Edition, L.C.Evans The Calculus of Variations
Partial Differential Equations, 2nd Edition, L.C.Evans Chapter 8 The Calculus of Variations Yung-Hsiang Huang 2018.03.25 Notation: denotes a bounded smooth, open subset of R n. All given functions are
arxiv: v1 [math.ds] 29 May 2018
![arxiv: v1 [math.ds] 29 May 2018](/thumbs/94/120869406.jpg "arxiv: v1 [math.ds] 29 May 2018")
NOTES ON HYPERBOLIC DYNAMICS SEMYON DYATLOV arxiv:1805.11660v1 [math.ds] 29 May 2018 Abstract. These expository notes present a proof of the Stable/Unstable Manifold Theorem (also known as the Hadamard
Integration on Measure Spaces
Chapter 3 Integration on Measure Spaces In this chapter we introduce the general notion of a measure on a space X, define the class of measurable functions, and define the integral, first on a class of
Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem
56 Chapter 7 Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem Recall that C(X) is not a normed linear space when X is not compact. On the other hand we could use semi
at time t, in dimension d. The index i varies in a countable set I. We call configuration the family, denoted generically by Φ: U (x i (t) x j (t))
Notations In this chapter we investigate infinite systems of interacting particles subject to Newtonian dynamics Each particle is characterized by its position an velocity x i t, v i t R d R d at time
Lecture Notes Math 632, PDE Spring Semester Sigmund Selberg Visiting Assistant Professor Johns Hopkins University
Lecture Notes Math 63, PDE Spring Semester 1 Sigmund Selberg Visiting Assistant Professor Johns Hopkins University CHAPTER 1 The basics We consider the equation 1.1. The wave equation on R 1+n u =, where
4 Countability axioms
4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said
On Fréchet algebras with the dominating norm property
On Fréchet algebras with the dominating norm property Tomasz Ciaś Faculty of Mathematics and Computer Science Adam Mickiewicz University in Poznań Poland Banach Algebras and Applications Oulu, July 3 11,
Math 147, Homework 5 Solutions Due: May 15, 2012
Math 147, Homework 5 Solutions Due: May 15, 2012 1 Let f : R 3 R 6 and φ : R 3 R 3 be the smooth maps defined by: f(x, y, z) = (x 2, y 2, z 2, xy, xz, yz) and φ(x, y, z) = ( x, y, z) (a) Show that f is
Differentiability with Respect to Lag Function for Nonlinear Pantograph Equations
Differentiability with Respect to Lag Function for Nonlinear Pantograph Equations Alexandru Tămăşan Differentiability with respect to λ (, 1) is provided for the nonlinear pantograph equations y (t) =
Review of Multi-Calculus (Study Guide for Spivak s CHAPTER ONE TO THREE)
Review of Multi-Calculus (Study Guide for Spivak s CHPTER ONE TO THREE) This material is for June 9 to 16 (Monday to Monday) Chapter I: Functions on R n Dot product and norm for vectors in R n : Let X
An introduction to some aspects of functional analysis
An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms
Maxwell s equations in Carnot groups
Maxwell s equations in Carnot groups B. Franchi (U. Bologna) INDAM Meeting on Geometric Control and sub-riemannian Geometry Cortona, May 21-25, 2012 in honor of Andrey Agrachev s 60th birthday Researches
MA108 ODE: Picard s Theorem
MA18 ODE: Picard s Theorem Preeti Raman IIT Bombay MA18 Existence and Uniqueness The IVP s that we have considered usually have unique solutions. This need not always be the case. MA18 Example Example:
1 The Local-to-Global Lemma
Point-Set Topology Connectedness: Lecture 2 1 The Local-to-Global Lemma In the world of advanced mathematics, we are often interested in comparing the local properties of a space to its global properties.