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

Transcription

1 88 th Annual Meeting of GAMM, 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 and elasticity Dirk Pauly Fakultät für Mathematik, UDE, Germany 88 th Annual Meeting of GAMM, 217 ection 23: Applied Operator Theory We gratefully thank the Organizers: W Rainer Picard & ascha Trostorff March 8, 217, Weimar, Germany

2 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 ome Well-Known and ome Not so Well-Known De-Rham complex (grad =, rot = curl =, div = ) D( grad) L 2 (Ω) grad D( rot) div D(div) rot D( div) rot D(rot) div grad D(grad) typical fos electro statics (Ω R 3 bounded weak Lipschitz domain) rote = F, div E = grad E = G, typical Helmholtz type decompositions =N(div) L 2 = R( grad) H D R(rot) =N( rot) typical Friedrichs/Poincaré type estimates - Maxwell estimates u D( grad) E D( rot) R(rot) = D( rot) N(div) H D E D( rot) D(div) H D π R R ι R R E H D = N( rot) N(div) u L 2 c f gradul 2, E L 2 c m rote L 2, E L 2 ĉ rote L 2 + div E L 2

3 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 ome Well-Known and ome Not so Well-Known De-Rham complex (d exterior derivative, δ = ± d co-derivative) d q 1 D( d q) δ q D(δ q) d q D( d q+1 ) δ q+1 D(δ q+1 ) d q+1 δ q+2 typical fos gen. el./mag. stat. (Ω R N bd, weak-lip or Ω Riemannian manifold) dqω = f, δ q ω = d q 1ω = g, ω H q D = N( d q) N(δ q) typical Helmholtz type decompositions =N(δ q) L 2,q = R( d q 1 ) H q R(δ D q+1 ) =N( d q) typical Friedrichs/Poincaré type estimates - Maxwell estimates ω D( d q) R(δ q+1 ) = D( d q) N(δ q) H q D ζ D(δ q+1 ) R( d q) = D(δ q+1 ) N( d q+1 ) H q+1 D ω D( d q) D(δ q) H q D ω L 2,q c q d qω L 2,q+1, ζ L 2,q+1 c q δ q+1 ζ L 2,q, ω L 2,q ĉ d qω L 2,q+1 + δ q ω L 2,q 1

4 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 ome Well-Known and ome Not so Well-Known bi-harmonic Grad grad - div Div cmplx: L 2 (Ω, R) L 2 (Ω, R 3 3 sym ) L2 (Ω, R 3 3 dev ) L2 (Ω, R 3 ) RT D( Grad grad) L 2 (Ω) Grad grad D( Rot ) div Div D(div Div ) Rot D( Div T ) sym Rot T D(sym Rot T ) Div T dev Grad D(dev Grad) typical fos bi-harmonic eq. (Ω R 3 bounded weak Lipschitz domain) or Rot = F, div Div = Grad grad = G, Div T T = F, sym Rot T = Rot T = G, typical Helmholtz type decompositions =N(div Div ) L 2 = R( Grad grad) H D R(sym Rot T ) =N( Rot ) typical Friedrichs/Poincaré type estimates D( Rot ) N(div Div ) H D =R(sym Rot T ) T D( Div T ) D(sym Rot T ) H N T π RT RT H D = N( Rot ) N(div Div ) T HN T = N( Div T ) N(sym Rot T ) L 2 c R Rot L 2 T, T L 2 T ĉ Div T T L 2 + sym Rot T T L 2 ι RT RT

5 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 ome Well-Known and ome Not so Well-Known elasticity RotRot complex: L 2 (Ω, R 3 ) L 2 (Ω, R 3 3 sym ) L2 (Ω, R 3 3 sym ) L2 (Ω, R 3 ) RM sym Grad D( sym Grad) D( Rot Rot ) L 2 (Ω) Div D(Div ) Rot Rot D( Div ) Rot Rot D(Rot Rot ) typical fos elasticity (Ω R 3 bounded weak Lipschitz domain) Div sym Grad D(sym Grad) Rot Rot = F, Div = sym Grad = G, H D = N( Rot Rot ) N(Div ) typical Helmholtz type decompositions =N(Div ) L 2 = R( sym Grad) H D R(Rot Rot ) =N( Rot Rot ) π RM RM ι RM RM typical Friedrichs/Poincaré type estimates D( Rot Rot ) N(Div ) H D =R(Rot Rot ) D( Div ) D(Rot Rot ) HN L2 c RR Rot Rot L 2, L 2 ĉ Div L 2 + Rot Rot L 2

6 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 General H i Hilbert spaces densely defined, closed (unbounded), linear operators A i with adjoints A i A i D(A i ) H i H i+1, A i D(A i ) H i+1 H i, i Z Hilbert complex (sequence) with adjoint Hilbert complex: D(A i 1 ) A i 1 D(A i ) A i 1 D(A i 1 ) A i D(A i+1 ) complex: range kernel, i.e., A i A i 1 =, A i 1 A i A i+1 A i D(A i ) A i+1 D(A i+1 ) =, i.e., R(A i 1 ) N(A i ), R(A i ) N(A i 1 ) related problem: find x D(A i ) D(A i 1 ) s.t. A i x = f, A i 1 x = g, π i x = h, where f R(A i ), g R(A i 1 ) and h H i with kernel/cohomology group H i = N(A i ) N(A i 1 )

7 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 General densely defined, closed (unbounded), linear operators with adjoints A i D(A i ) H i H i+1, A i D(A i ) H i+1 H i, i Z Hilbert complex with adjoint Hilbert complex: D(A i 1 ) A i 1 D(A i ) A i 1 D(A i 1 ) A i D(A i+1 ) A i+1 A i D(A i ) A i+1 D(A i+1 ) complex is closed, iffallrangesareclosed,i.e.,r(a i ) = R(A i ) complex is exact, iffranges=kernels,i.e.,r(a i 1 ) = N(A i ) complex is compact, iff all embeddingsarecompact,i.e.,d(a i ) D(A i 1 ) H i

8 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 General Hodge/Helmholtz/Weyl decomposition (projection theorem) H i = N(A i ) Hi R(A i ), N(A i ) H i = R(A i ), H i = R(A i 1 ) Hi N(A i 1 ), N(A i 1 ) H i = R(A i 1 ) reduced operators (inj., dd,cl.(unbd.),lin.operatorswithadjoints) A i D(A i ) R(A i ) R(A i ), A i D(A i ) R(A i ) R(A i ), R(A i ) = R(A i ), D(A i ) = D(A i ) R(A i ), D(A i ) = D(A i ) R(A i ), R(A i ) = R(A i ) complex is closed R(A i ) = R(A i ) R(A i ) = R(A i ) A 1 i continuous, i.e., A 1 i R(A i ) D(A i ) continuous (A i ) 1 continuous, i.e., (A i ) 1 R(A i ) D(A i ) continuous Friedrichs/Poincaré type estimates for A i,i.e., c Ai > x D(A i ) x Hi c Ai A i x Hi+1 Friedrichs/Poincaré type estimates for A i,i.e., (c Ai = c A i ) c A i > y D(A i ) y H i+1 c A i A i y H i

9 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 General Hodge/Helmholtz/Weyl decomposition (projection theorem) H i = N(A i ) Hi R(A i ), H i = R(A i 1 ) Hi H i Hi R(A i ), H i = R(A i 1 ) Hi N(A i 1 ) reduced operators (inj., dd, cl. (unbd.), lin. operators with adjoints) A i D(A i ) R(A i ) R(A i ), A i D(A i ) R(A i ) R(A i ), R(A i ) = R(A i ), D(A i ) = D(A i ) R(A i ), D(A i ) = D(A i ) R(A i ), R(A i ) = R(A i ) complex is exact R(A i 1 ) = N(A i ) H i = N(A i ) N(A i 1 ) = {} R(A i 1) closed =R(A i 1 ) note: R(A i 1 ) = N(A i ) H i = {}

10 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 General Hodge/Helmholtz/Weyl decomposition (projection theorem) D(A i ) = N(A i ) Hi D(A i ), D(A i ) D(A i 1 ) = D(A i 1 ) H i H i Hi D(A i ), D(A i 1 ) = D(A i 1 ) H i N(A i 1 ) reduced operators (inj., dd, cl. (unbd.), lin. operators with adjoints) A i D(A i ) R(A i ) R(A i ), A i D(A i ) R(A i ) R(A i ), R(A i ) = R(A i ), D(A i ) = D(A i ) R(A i ), D(A i ) = D(A i ) R(A i ), R(A i ) = R(A i ) complex is compact D(A i ) D(A i 1 ) H i D(A i ) H i D(A i 1 ) H i H i H i D(A i ) H i D(A i 1 ) H i dim H i < complex is closed and dim H i < note: D(A i ) H i A 1 i cpt σ(a i A i ) disc. (and exp. theo.) D(A i ) H i+1 (A i ) 1 cpt σ(a i A i ) disc. (and exp. theo.)

11 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 De-Rham : grad-rot-div complex (grad =, rot = curl =, div = ) D( grad) L 2 (Ω) grad D( rot) div D(div) rot D( div) rot D(rot) complex property: range kernel (rot grad =, div rot = ) R() = {} = N( grad), R( grad) N( rot), div grad D(grad) R( rot) N( div), π R R ι R R R( div) = N(π R ) = R, R(ι R ) = R = N(grad), R( grad) N(rot), R(rot) N( div), R( div) = N() = L 2 (Ω) complex closed all ranges are closed complex exact complex closed and all cohomology groups trivial, i.e., H D = N( rot) R( grad) = N(div) R(rot) = N( rot) N(div) = {} H N = N( div) R( rot) = N(rot) R(grad) = N( div) N(rot) = {} (Dirichlet fields), (Neumann fields) dimension depends only on topology, Betti numbers

12 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 De-Rham : grad-rot-div complex: D( grad) L 2 (Ω) grad D( rot) div D(div) rot D( div) rot D(rot) div grad D(grad) crucial / best compact embeddings (for, e.g., Ω bounded weak Lipschitz) D( grad) D(grad) L 2 (Rellich s selection theorem), π R R ι R R D( rot) D(div) L 2 (Weck s selection theorem: Weck 74, Weber 8, Picard 84, D(rot) D( div) L 2 Jochmann 97, Picard-Weck-Witsch 1, Bauer-P.-chomburg 16, 17), (Weck s selection theorem) closed complexes and finite cohomology groups Helmholtz decompositions Friedrichs/Poincaré type estimates continuous and compact inverses of reduced operators all from general fa-toolbox

13 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 De-Rham : grad-rot-div complex: D( grad) L 2 (Ω) grad D( rot) div D(div) e.g.: Helmholtz decompositions rot D( div) rot D(rot) div grad D(grad) L 2 = R( grad) H D R(rot) = R(grad) H N R( rot) e.g.: Friedrichs/Poincaré type estimates π R R ι R R u D( grad) u L 2 c f grad u L 2, E D(div) R( grad) = D(div) N( rot) H D E L 2 c f div E L 2, E D( rot) R(rot) = D( rot) N(div) H D E L 2 c m rot E L 2, E D(rot) R( rot) = D(rot) N( div) H N E L 2 c m rot E L 2, E D( div) R(grad) = D( div) N(rot) H N E L 2 c p div E L 2, u D(grad) R( div) = D(grad) R u L 2 c p grad u L 2

14 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 De-Rham : d complex (d exterior derivative, δ = ± d co-derivative) dq 1 D( d q) δ q D(δ q) d q D( d q+1 ) δ q+1 D(δ q+1 ) complex property: range kernel (d q d q 1 =, δ q δ q+1 = ) R( d q 1 ) N( d q), R(δ q+1 ) N(δ q) dq+1 δ q+2 complex closed all ranges are closed complex exact complex closed and all cohomology groups trivial, i.e., H q D = N( d q) N(δ q) = {} H q N = N(dq) N( δ q) = {} (Dirichlet forms), (Neumann forms) dimension depends only on topology, Betti numbers

15 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 De-Rham : d complex (d exterior derivative, δ = ± d co-derivative) d q 1 D( d q) δ q D(δ q) dq D( d q+1 ) δ q+1 D(δ q+1 ) d q+1 δ q+2 crucial / best compact embeddings (for, e.g., Ω bounded weak Lipschitz domain or manifold) D( d q) D(δ q) L 2,q (Weck s selection theorem: Weck 74, Picard 84, D(d q) D( δ q) L 2,q Bauer-P.-chomburg 17), (Weck s selection theorem) closed complexes and finite cohomology groups Helmholtz decompositions Friedrichs/Poincaré type estimates continuous and compact inverses of reduced operators all from general fa-toolbox

16 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 De-Rham : d complex (d exterior derivative, δ = ± d co-derivative) d q 1 D( d q) δ q D(δ q) e.g.: Helmholtz decompositions dq D( d q+1 ) δ q+1 D(δ q+1 ) d q+1 δ q+2 L 2,q = R( d q 1 ) H q D R(δ q+1) = R(d q 1 ) H q N R( δ q+1 ) e.g.: Friedrichs/Poincaré type estimates ω D( d q) R(δ q+1 ) = D( d q) N(δ q) H q D ζ D(δ q+1 ) R( d q) = D(δ q+1 ) N( d q+1 ) H q+1 D ω L 2,q c q d ω L 2,q+1, ζ L 2,q+1 c q δζ L 2,q (same for the other boundary condition on δ q+1, note: by Hodge -operator c N q = c q,sameconstantfor4estimates)

17 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 Grad grad - div Div (Bi-Harmonic Equation) complex: L 2 (Ω, R) L 2 (Ω, R 3 3 sym) L 2 (Ω, R 3 3 dev ) L2 (Ω, R 3 ) RT D( Grad grad) L 2 (Ω) Grad grad D( Rot ) div Div D(div Div ) complex property: range kernel Rot D( Div T ) sym Rot T D(sym Rot T ) Div T dev Grad D(dev Grad) Rot Grad grad =, Div TRot =, π RT Div T =, div Div sym Rot T =, sym Rot T dev Grad =, dev Grad ι RT = complex closed all ranges are closed complex exact complex closed and all cohomology groups trivial, i.e., π RT RT ι RT RT H D = N( Rot ) N(div Div ) H N T = N( Div T ) N(sym Rot T ) (symmetric Dirichlet fields), (trace-free Neumann fields) dimension depends only on topology, Betti numbers

18 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 Grad grad - div Div (Bi-Harmonic Equation) complex: L 2 (Ω, R) L 2 (Ω, R 3 3 sym) L 2 (Ω, R 3 3 dev ) L2 (Ω, R 3 ) RT D( Grad grad) L 2 (Ω) Grad grad D( Rot ) div Div D(div Div ) Rot D( Div T ) sym Rot T D(sym Rot T ) Div T dev Grad D(dev Grad) crucial / best compact embeddings (for, e.g., Ω bounded strong Lipschitz) D( Grad grad) = H 2 L 2 (Rellich s selection theorem), D( Rot ) D(div Div ) L 2 (P.-Zulehner 16), D( Div T ) D(sym Rot T ) L 2 T (P.-Zulehner 16), D(dev Grad) = H 1 L 2 closed complexes and finite cohomology groups Helmholtz decompositions Friedrichs/Poincaré type estimates continuous and compact inverses of reduced operators all from general fa-toolbox (P.-Zulehner 16, Rellich s selection theorem) π RT RT ι RT RT

19 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 Grad grad - div Div (Bi-Harmonic Equation) complex: L 2 (Ω, R) L 2 (Ω, R 3 3 sym) L 2 (Ω, R 3 3 dev ) L2 (Ω, R 3 ) RT D( Grad grad) L 2 (Ω) Grad grad D( Rot ) div Div D(div Div ) e.g.: Helmholtz decompositions: Rot D( Div T ) sym Rot T D(sym Rot T ) L 2 = R( Grad grad) H D R(sym Rot T), e.g.: Friedrichs/Poincaré type estimates u D( Grad grad) L 2 (Ω) = R( Div T ) RT, Div T dev Grad D(dev Grad) π RT RT L 2 T = R( Rot ) H N T R(dev Grad) ι RT RT u L 2 c Gg Grad grad u L D(div Div ) R( Grad grad) = D(div Div ) N( Rot ) H D L 2 c Gg div Div L 2, D( Rot ) R(sym Rot T ) = D( Rot ) N(div Div ) H D L 2 c R Rot L 2 T, T D(sym Rot T ) R( Rot ) = D(sym Rot T ) N( Div T ) H N T T L 2 T c R sym Rot T L 2, T D( Div T ) R(dev Grad) = D( Div T ) N(sym Rot T ) H N T T L 2 T c D Div T L 2, E D(dev Grad) R( Div T ) = D(dev Grad) RT E L 2 c D dev Grad E L 2 T

20 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 Rot Rot (Elasticity) complex: L 2 (Ω, R 3 ) L 2 (Ω, R 3 3 sym) L 2 (Ω, R 3 3 sym) L 2 (Ω, R 3 ) RM D( sym Grad) sym Grad D( Rot Rot ) Rot Rot D( Div ) Div π RM RM L 2 (Ω) Div D(Div ) complex property: range kernel Rot Rot D(Rot Rot ) sym Grad D(sym Grad) Rot Rot sym Grad =, Div Rot Rot =, π RMDiv =, Div Rot Rot =, Rot Rot sym Grad =, sym Grad ι RM = complex closed all ranges are closed complex exact complex closed and all cohomology groups trivial, i.e., ι RM RM H D H N = N( Rot Rot ) N(Div ) = N( Div ) N(Rot Rot ) (symmetric Dirichlet fields II), (symmetric Neumann fields) dimension depends only on topology, Betti numbers

21 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 Rot Rot (Elasticity) complex: L 2 (Ω, R 3 ) L 2 (Ω, R 3 3 sym) L 2 (Ω, R 3 3 sym) L 2 (Ω, R 3 ) RM D( sym Grad) sym Grad D( Rot Rot ) Rot Rot D( Div ) Div π RM RM L 2 (Ω) Div D(Div ) Rot Rot D(Rot Rot ) sym Grad D(sym Grad) crucial / best compact embeddings (for, e.g., Ω bounded strong Lipschitz) D( sym Grad) = H 1 L 2 (Rellich s selection theorem), D( Rot Rot ) D(Div ) L 2 (P.-Zulehner 17), D( Div ) D(Rot Rot ) L2 (P.-Zulehner 17), D(sym Grad) = H 1 L 2 closed complexes and finite cohomology groups Helmholtz decompositions Friedrichs/Poincaré type estimates continuous and compact inverses of reduced operators all from general fa-toolbox (Rellich s selection theorem) ι RM RM

22 88 th Annual Meeting of GAMM, 217, ection 23: Applied Operator Theory Weimar, Germany, March 8, 217 Rot Rot (Elasticity) complex: L 2 (Ω, R 3 ) L 2 (Ω, R 3 3 sym) L 2 (Ω, R 3 3 sym) L 2 (Ω, R 3 ) RM sym Grad D( sym Grad) D( Rot Rot ) L 2 (Ω) Div D(Div ) e.g.: Helmholtz decompositions: Rot Rot D( Div ) Rot Rot D(Rot Rot ) L 2 (Ω) = R( Div ) RM, Div sym Grad D(sym Grad) L 2 = R( sym Grad) H D R(Rot Rot ), L2 = R( Rot Rot ) HN R(sym Grad) e.g.: Friedrichs/Poincaré type estimates π RM RM ι RM RM E D( sym Grad) E L 2 c sg sym Grad E L 2 D(Div ) R( sym Grad) = D(Div ) N( Rot Rot ) HD L 2 c sg Div L 2, D( Rot Rot ) R(Rot Rot ) = D( Rot Rot ) N(Div ) H D L 2 c RR Rot Rot L 2 D(Rot Rot ) R( Rot Rot ) = D(Rot Rot ) N( Div ) H N L 2 c RR Rot Rot L 2 D( Div ) R(sym Grad) = D( Div ) N(Rot Rot ) HN L 2 c D Div L 2, E D(sym Grad) R( Div ) = D(sym Grad) RM E L 2 c D sym Grad E L 2