The div-curl-lemma by the FA-Toolbox

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1 The div-curl-lemma by the FA-Toolbox Dirk Pauly Fakultät für Mathematik 40. Nordwestdeutsches Funktionalanalysis Kolloquium 23. Juni 2018 Gastgeberin: Birgit Jacob Arbeitsgruppe Funktionalanalysis Fakultät für Mathematik und Naturwissenschaften Bergische Universität Wuppertal

2 div-curl-lemma classical div-curl-lemma Let Ω R 3 be open. Lemma (classical div-curl-lemma) Assumptions: (i) (E n), (H n) bounded in (Ω) (ii) (rot E n) bounded in (Ω) (ii ) (div H n) bounded in (Ω) E, H and subsequences s.t. E n E, rot E n rot E and H n H, div H n div H and ϕ C (Ω) Ω ϕ(e n H n) Ω ϕ(e H) classical div-curl-lemma is local!

3 div-curl-lemma div-curl-lemma Let Ω R 3 be a bounded weak Lipschitz domain with boundary Γ and weak Lipschitz boundary parts Γ t and Γ n = Γ Γ t. Lemma (div-curl-lemma (global version)) Assumptions: (i) (E n), (H n) bounded in (Ω) (ii) (rot E n) bounded in (Ω) (ii ) (div H n) bounded in (Ω) (iii) ν E n = 0 on Γ t (iii ) ν H n = 0 on Γ n E, H and subsequences s.t. E n E, rot E n rot E and H n H, div H n div H and Ω E n H n Ω E H Proof. generalize and fa-toolbox crucial points: complex property and compact embedding

4 div-curl-lemma literature original papers (local div-curl-lemma): Murat, F.: Compacité par compensation, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 1978 Tartar, L.: Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics, Heriot-Watt symposium, 1979 recent papers (global div-curl-lemma): Gloria, A., Neukamm, S., Otto, F.: Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics, (IM) Invent. Math., 2015 Kozono, H., Yanagisawa, T.: Global compensated compactness theorem for general differential operators of first order, (ARMA) Arch. Ration. Mech. Anal., 2013 Schweizer, B.: On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma, accepted preprint, 2018 Waurick, M.: A Functional Analytic Perspective to the div-curl Lemma, (JOP) J. Operator Theory, 2018

5 div-curl-lemma fa-toolbox for linear problems/systems idea: solve problem with general and simple linear functional analysis ( fa-toolbox)... literature: probably very well known for ages, but hard to find... Friedrichs, Weyl, Hörmander, Fredholm, von Neumann, Riesz, Banach,...? Why not rediscover?

6 div-curl-lemma A 0 -A 1-lemma (generalized global div-curl-lemma) Let A 0 D(A 0 ) H 0 H 1, A 1 D(A 1 ) H 1 H 2 (possibly and generally unbounded) be two densely defined and closed linear operators on three Hilbert spaces H 0, H 1, H 2 with Hilbert space adjoints A 0 D(A 0 ) H 1 H 0, A 1 D(A 1 ) H 2 H 1. Moreover, let A 1 A 0 = 0, i.e. R(A 0 ) N(A 1 ). (complex property) Lemma (A 0 -A 1-lemma) Let D(A 1 ) D(A 0 ) H 1 be compact, and (i) (x n) bounded in D(A 1 ), (ii) (y n) bounded in D(A 0 ). x D(A 1 ), y D(A 0 ) and subsequences s.t. x n x in D(A 1 ) and y n y in D(A 0 ) as well as x n, y n H1 x, y H1. Proof.... blackboard...

7 div-curl-lemma A 0 -A 1-lemma (generalized global div-curl-lemma) compact embedding in global div-curl-lemma reads: D(A 1 ) D(A 0 ) H 1 {E (Ω) rot E (Ω), div E (Ω), ν E = 0 on Γ t, ν E = 0 on Γ n} (Ω) is compact (Weck s selection theorem, 74, also Bauer, Costabel, Kuhn, Jochmann, Osterbrink, P, Picard, Schomburg, Weber, Witsch)

8 applications: fos & sos (first and second order systems) classical de Rham complex in 3D ( -rot-div-complex) Ω R 3 bounded weak Lipschitz domain, Ω = Γ = Γ t Γ n (electro-magneto dynamics, Maxwell s equations) {0} ι {0} π {0} div rot rot div π R ιr R mixed boundary conditions and inhomogeneous and anisotropic media {0} or R ι Γ t π div Γn ε ε rot Γ t ε 1 rot Γn div Γ t Γn π ι R or {0}

9 applications: fos & sos (first and second order systems) classical de Rham complex in 3D ( -rot-div-complex) Ω R 3 bounded weak Lipschitz domain, Ω = Γ = Γ t Γ n (electro-magneto dynamics, Maxwell s equations with mixed boundary conditions) {0} or R ι Γ t π div Γn ε ε rot Γ t ε 1 rot Γn div Γ t Γn π ι R or {0} related fos Γ t u = A in Ω rot Γt E = J in Ω div H = k in Ω πv = b in Ω Γt πu = a in Ω div Γn εe = j in Ω ε 1 rot Γn H = K in Ω Γn v = B in Ω related sos div Γn ε Γ t u = j in Ω ε 1 rot Γn rot Γ t E = K in Ω Γn div Γt H = B in Ω πu = a in Ω div Γn εe = j in Ω ε 1 rot Γn H = K in Ω corresponding compact embeddings: D( Γ t ) D(π) = D( Γt ) = H1 Γt L2 (Rellich s selection theorem) D(rot Γ t ) D( div Γn ε) = R Γt ε 1 D Γn ε (Weck s selection theorem, 74) D(div Γ t ) D(ε 1 rot Γn ) = D Γ t R Γn L2 (Weck s selection theorem, 74) D( Γn ) D(π) = D( Γn ) = H 1 Γn L2 (Rellich s selection theorem) Weck s selection theorem for weak Lip. dom. and mixed bc: Bauer/P/Schomburg ( 16)

10 applications: fos & sos (first and second order systems) de Rham complex in ND or on Riemannian manifolds (d-complex) Ω R N bd w. Lip. dom. or Ω Riemannian manifold with cpt cl. and Lip. boundary Γ (generalized Maxwell equations) {0} ι {0} π {0},0 d δ,1 d δ...,q d δ,q+1...,n 1 d δ,n π R ιr R

11 applications: fos & sos (first and second order systems) de Rham complex in ND or on Riemannian manifolds (d-complex) Ω R N bd w. Lip. dom. or Ω Riemannian manifold with cpt cl. and Lip. boundary Γ (generalized Maxwell equations) {0} or R related fos ι,0 d 0 Γt π δ 1 Γn,1 d 1 Γt δ 2 Γn d q Γt E = F q d...,q Γt δ q+1 Γn,q+1...,N 1 in Ω d N 1 Γt δ N Γn,N π ι R or {0} δ q Γn E = G in Ω related sos δ q+1 Γn dq Γt E = F in Ω δ q Γn E = G in Ω includes: EMS rot / div, Laplacian, rot rot, and more... corresponding compact embeddings: D(d q Γt ) D(δq Γn ) L2,q (Weck s selection theorems, 74) Weck s selection theorem for Lip. manifolds and mixed bc: Bauer/P/Schomburg ( 17)

12 applications: fos & sos (first and second order systems) elasticity complex in 3D (sym -Rot Rot S -Div S-complex) Ω R 3 bounded strong Lipschitz domain {0} ι {0} π {0} sym Div S S Rot Rot S Rot Rot S S Div S sym π RM ι RM RM

13 applications: fos & sos (first and second order systems) elasticity complex in 3D (sym -Rot Rot S -Div S-complex) Ω R 3 bounded strong Lipschitz domain {0} ι {0} π {0} sym Div S S Rot Rot S Rot Rot S S Div S sym π RM ι RM RM related fos (Rot Rot S,Γ, Rot Rot S first order operators!) sym Γ v = M in Ω Rot Rot S,Γ M = F in Ω Div S,Γ N = g in Ω πv = r in Ω πv = 0 in Ω Div S M = f in Ω Rot Rot S N = G in Ω sym v = M in Ω related sos (Rot Rot S Rot Rot S,Γ second order operator!) Div S sym Γ v = f in Ω Rot Rot S Rot Rot S,Γ M = G in Ω sym Div S,Γ N = M in Ω πv = 0 in Ω Div S M = f in Ω Rot Rot S N = G in Ω corresponding compact embeddings: D(sym Γ ) D(π) = D( Γ ) = H 1 Γ L2 D(Rot Rot S,Γ ) D(Div S) S (Rellich s selection theorem and Korn ineq.) (new selection theorem) D(Div S,Γ ) D(Rot Rot S ) L2 S (new selection theorem) D(π) D(sym ) = D( ) = H 1 (Rellich s selection theorem and Korn ineq.) two new selection theorems for strong Lip. dom.: P/Schomburg/Zulehner ( 18)

14 applications: fos & sos (first and second order systems) biharmonic / general relativity complex in 3D ( -Rot S -Div T -complex) Ω R 3 bounded strong Lipschitz domain {0} ι {0} π {0} div Div S S Rot S sym Rot T T Div T dev π RT ι RT RT

15 applications: fos & sos (first and second order systems) biharmonic / general relativity complex in 3D ( -Rot S -Div T -complex) Ω R 3 bounded strong Lipschitz domain {0} ι {0} π {0} div Div S S Rot S sym Rot T T Div T dev π RT ι RT RT related fos ( Γ, div Div S first order operators!) Γ u = M in Ω Rot S,Γ M = F in Ω Div T,Γ N = g in Ω πv = r in Ω πu = 0 in Ω div Div S M = f in Ω sym Rot T N = G in Ω dev v = T in Ω related sos (div Div S Γ = 2 Γ second order operator!) div Div S Γ u = 2 Γ u = f in Ω sym Rot T Rot S,Γ M = G in Ω dev Div T,Γ N = T in Ω πu = 0 in Ω div Div S M = f in Ω sym Rot T N = G in Ω corresponding compact embeddings: D( Γ ) D(π) = D( Γ ) = H 2 Γ L2 D(Rot S,Γ ) D(div Div S ) S (Rellich s selection theorem) (new selection theorem) D(Div T,Γ ) D(sym Rot T ) T (new selection theorem) D(π) D(dev ) = D(dev ) = D( ) = H 1 (Rellich s selection theorem and Korn type ineq.) two new selection theorems for strong Lip. dom. and Korn Type ineq.: P/Zulehner ( 16)

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16 commercials... the world is full of complexes... ;) relaxing at AANMPDE 11 11th Workshop on Analysis and Advanced Numerical Methods for Partial Differential Equations (not only) for Junior Scientists August , Särkisaari, Finland organizers: Ulrich Langer, Dirk Pauly, Sergey Repin conference site

On Grad-grad, div-div, and Rot-Rot complexes for problems related to the biharmonic equation and elasticity

88 th Annual Meeting of GAMM, Ilmenau@Weimar 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 On Grad-grad, div-div, and Rot-Rot complexes for problems related to the biharmonic equation