Méthodes d énergie pour le potentiel de double couche: Une inégalité de Poincaré oubliée.
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1 Méthodes d énergie pour le potentiel de double couche: Une inégalité de Poincaré oubliée. Martin Costabel IRMAR, Université de Rennes 1 2e Journée de l équipe d analyse numérique Rennes 25 oct 2007 Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 1 / 26
2 L inégalité de Poincaré Théorème Soit R 3 un domaine lipschitzien borné et + = R 3 \. Alors il existe une constante µ 1 telle que pour tout u harmonique dans +, Hloc 1 jusqu au bord et zéro à l infini et - soit u = u + sur Γ = ( potentiel de simple couche ) et u ds = 0 Γ - soit n u = n u + sur Γ ( potentiel de double couche ) on ait l estimation 1 u 2 µ u 2 µ u 2 + Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 2 / 26
3 W. Stekloff 1900 [Ann. Fac. Sci Toulouse 2 (1900) ] Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 3 / 26
4 W. Stekloff 1900 [Ann. Fac. Sci Toulouse 2 (1900) ] Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 3 / 26
5 W. Stekloff 1900 [Ann. Fac. Sci Toulouse 2 (1900) ] Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 3 / 26
6 W. Stekloff 1900 [Ann. Fac. Sci Toulouse 2 (1900) ] Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 3 / 26
7 L inégalité de Poincaré Théorème Soit R 3 un domaine lipschitzien borné et + = R 3 \. Alors il existe une constante µ 1 telle que pour tout u harmonique dans +, Hloc 1 jusqu au bord et zéro à l infini et - soit u = u + sur Γ = ( potentiel de simple couche ) et u ds = 0 Γ - soit n u = n u + sur Γ ( potentiel de double couche ) on ait l estimation 1 u 2 µ u 2 µ u 2 + Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 4 / 26
8 Références, de Poincaré à nos jours H. POINCARÉ. La méthode de Neumann et le problème de Dirichlet. Acta Math. 20 (1896) W. STEKLOFF. Sur la méthode de Neumann et le problème de Dirichlet. C. R. 130 (1900) A. KORN. Sur la méthode de Neumann et le problème de Dirichlet. C. R. 130 (1900) 557. Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 5 / 26
9 Références, de Poincaré à nos jours H. POINCARÉ. La méthode de Neumann et le problème de Dirichlet. Acta Math. 20 (1896) W. STEKLOFF. Sur la méthode de Neumann et le problème de Dirichlet. C. R. 130 (1900) A. KORN. Sur la méthode de Neumann et le problème de Dirichlet. C. R. 130 (1900) 557. Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 5 / 26
10 A. Korn 1900 Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 6 / 26
11 Références, de Poincaré à nos jours H. POINCARÉ. La méthode de Neumann et le problème de Dirichlet. Acta Math. 20 (1896) W. STEKLOFF. Sur la méthode de Neumann et le problème de Dirichlet. C. R. 130 (1900) A. KORN. Sur la méthode de Neumann et le problème de Dirichlet.. C. R. 130 (1900) 557. W. STEKLOFF. Remarque à une note de M. A. Korn: Sur la méthode de Neumann et le problème de Dirichlet. C. R. 130 (1900) W. STEKLOFF. Les méthodes générales pour résoudre les problèmes fondamentaux de la physique mathématique. Ann. Fac. Sci. Toulouse (2) 2 (1900) O. STEINBACH, W. L. WENDLAND. On C. Neumann s method for second-order elliptic systems in domains with non-smooth boundaries. J. Math. Anal. Appl. 262 (2001) M. COSTABEL. Some historical remarks on the positivity of boundary integral operators. Ch. 1 of Boundary Element Analysis - Mathematical Aspects and Applications (M. Schanz, O. Steinbach, Eds.) Lecture Notes in Applied and Computational Mechanics Vol 29, Springer, Berlin Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 7 / 26
12 Références, de Poincaré à nos jours H. POINCARÉ. La méthode de Neumann et le problème de Dirichlet. Acta Math. 20 (1896) W. STEKLOFF. Sur la méthode de Neumann et le problème de Dirichlet. C. R. 130 (1900) A. KORN. Sur la méthode de Neumann et le problème de Dirichlet.. C. R. 130 (1900) 557. W. STEKLOFF. Remarque à une note de M. A. Korn: Sur la méthode de Neumann et le problème de Dirichlet. C. R. 130 (1900) W. STEKLOFF. Les méthodes générales pour résoudre les problèmes fondamentaux de la physique mathématique. Ann. Fac. Sci. Toulouse (2) 2 (1900) O. STEINBACH, W. L. WENDLAND. On C. Neumann s method for second-order elliptic systems in domains with non-smooth boundaries. J. Math. Anal. Appl. 262 (2001) M. COSTABEL. Some historical remarks on the positivity of boundary integral operators. Ch. 1 of Boundary Element Analysis - Mathematical Aspects and Applications (M. Schanz, O. Steinbach, Eds.) Lecture Notes in Applied and Computational Mechanics Vol 29, Springer, Berlin Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 7 / 26
13 Références, de Poincaré à nos jours H. POINCARÉ. La méthode de Neumann et le problème de Dirichlet. Acta Math. 20 (1896) W. STEKLOFF. Sur la méthode de Neumann et le problème de Dirichlet. C. R. 130 (1900) A. KORN. Sur la méthode de Neumann et le problème de Dirichlet.. C. R. 130 (1900) 557. W. STEKLOFF. Remarque à une note de M. A. Korn: Sur la méthode de Neumann et le problème de Dirichlet. C. R. 130 (1900) W. STEKLOFF. Les méthodes générales pour résoudre les problèmes fondamentaux de la physique mathématique. Ann. Fac. Sci. Toulouse (2) 2 (1900) O. STEINBACH, W. L. WENDLAND. On C. Neumann s method for second-order elliptic systems in domains with non-smooth boundaries. J. Math. Anal. Appl. 262 (2001) M. COSTABEL. Some historical remarks on the positivity of boundary integral operators. Ch. 1 of Boundary Element Analysis - Mathematical Aspects and Applications (M. Schanz, O. Steinbach, Eds.) Lecture Notes in Applied and Computational Mechanics Vol 29, Springer, Berlin Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 7 / 26
14 Équation intégrale de Neumann pour le problème de Dirichlet Potentiel de double couche dans R 3 \ Γ = + : u(x) = Dg(x) = 1 n(y) (y x) 4π Γ x y 3 g(y) ds(y) Relations de saut sur Γ: n u + = n u ; u + u = g ; Déf: Kg := 1 2 (u+ + u ) Traces sur Γ: u + = ( I + K )g Problème de Dirichlet intérieur: u = f sur Γ Équation intégrale du potentiel de double couche: u = Dg avec ( 1 2 I K )g = f = 1 2 (I 2K )g = ( I ( 1 2 I + K )) g Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 8 / 26
15 Équation intégrale de Neumann pour le problème de Dirichlet Potentiel de double couche dans R 3 \ Γ = + : u(x) = Dg(x) = 1 n(y) (y x) 4π Γ x y 3 g(y) ds(y) Relations de saut sur Γ: n u + = n u ; u + u = g ; Déf: Kg := 1 2 (u+ + u ) Traces sur Γ: u + = ( I + K )g Problème de Dirichlet intérieur: u = f sur Γ Équation intégrale du potentiel de double couche: u = Dg avec ( 1 2 I K )g = f = 1 2 (I 2K )g = ( I ( 1 2 I + K )) g Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 8 / 26
16 Équation intégrale de Neumann pour le problème de Dirichlet Potentiel de double couche dans R 3 \ Γ = + : u(x) = Dg(x) = 1 n(y) (y x) 4π Γ x y 3 g(y) ds(y) Relations de saut sur Γ: n u + = n u ; u + u = g ; Déf: Kg := 1 2 (u+ + u ) Traces sur Γ: u + = ( I + K )g Problème de Dirichlet intérieur: u = f sur Γ Équation intégrale du potentiel de double couche: u = Dg avec ( 1 2 I K )g = f = 1 2 (I 2K )g = ( I ( 1 2 I + K )) g Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 8 / 26
17 La série de Neumann C. Neumann 1877: g = 2 (2K ) l f l=0 Convergence dans C 0 (Γ)/R si Γ est convexe: dθ x(y) = 1 n(y) (y x) ds(y) 4π x y 3 est une mesure de masse 1, positive si Γ convexe Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 9 / 26
18 Beer 1856, dans C. Neumann: Untersuchungen über das Logarithmische und Newtonsche Potential, Teubner Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 10 / 26
19 Beer 1856, dans C. Neumann: Untersuchungen über das Logarithmische und Newtonsche Potential, Teubner Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 10 / 26
20 La série de Neumann C. Neumann 1877: g = 2 (2K ) l f l=0 Convergence dans C 0 (Γ)/R si Γ est convexe: dθ x(y) = 1 n(y) (y x) ds(y) 4π x y 3 est une mesure de masse 1, positive si Γ convexe Poincaré 1896: u = 2 D(2K ) l f l=0 Convergence dans L (R 3 ) si Γ C 1+α Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 11 / 26
21 La série de Neumann C. Neumann 1877: g = 2 (2K ) l f l=0 Convergence dans C 0 (Γ)/R si Γ est convexe: dθ x(y) = 1 n(y) (y x) ds(y) 4π x y 3 est une mesure de masse 1, positive si Γ convexe Poincaré 1896: u = 2 D(2K ) l f l=0 Convergence dans L (R 3 ) si Γ C 1+α Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 11 / 26
22 Poincaré 1896: Équations intégrales sans equations intégrales Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 12 / 26
23 La série de Neumann C. Neumann 1877: g = 2 (2K ) l f l=0 Convergence dans C 0 (Γ)/R si Γ est convexe: dθ x(y) = 1 n(y) (y x) ds(y) 4π x y 3 est une mesure de masse 1, positive si Γ convexe Poincaré 1896: u = 2 D(2K ) l f l=0 Convergence dans L (R 3 ) si Γ C 1+α Steinbach-Wendland 2001: g = ( 1 2 I + K )l f l=0 Convergence dans H 1/2 (Γ) si Γ lipschitzien; syst. ell. d ordre 2 ds R n Ouvert: Convergence dans L 2 (Γ); ordre > 2; t ; Maxwell... Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 13 / 26
24 La série de Neumann C. Neumann 1877: g = 2 (2K ) l f l=0 Convergence dans C 0 (Γ)/R si Γ est convexe: dθ x(y) = 1 n(y) (y x) ds(y) 4π x y 3 est une mesure de masse 1, positive si Γ convexe Poincaré 1896: u = 2 D(2K ) l f l=0 Convergence dans L (R 3 ) si Γ C 1+α Steinbach-Wendland 2001: g = ( 1 2 I + K )l f l=0 Convergence dans H 1/2 (Γ) si Γ lipschitzien; syst. ell. d ordre 2 ds R n Ouvert: Convergence dans L 2 (Γ); ordre > 2; t ; Maxwell... Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 13 / 26
25 La série de Neumann C. Neumann 1877: g = 2 (2K ) l f l=0 Convergence dans C 0 (Γ)/R si Γ est convexe: dθ x(y) = 1 n(y) (y x) ds(y) 4π x y 3 est une mesure de masse 1, positive si Γ convexe Poincaré 1896: u = 2 D(2K ) l f l=0 Convergence dans L (R 3 ) si Γ C 1+α Steinbach-Wendland 2001: g = ( 1 2 I + K )l f l=0 Convergence dans H 1/2 (Γ) si Γ lipschitzien; syst. ell. d ordre 2 ds R n Ouvert: Convergence dans L 2 (Γ); ordre > 2; t ; Maxwell... Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 13 / 26
26 Le Grand Bouleversement D. HILBERT. Über das Dirichletsche Prinzip. Jb. DMV 8 (1900) D. HILBERT. Über das Dirichletsche Prinzip. Math. Ann. 59 (1904) D. HILBERT. Über das Dirichletsche Prinzip. J. R. A. Math. 129 (1905) I. FREDHOLM. Sur une classe d équations fonctionnelles. Acta Math. 27 (1903), no. 1, A. KORN. Über freie und erzwungene Schwingungen. Eine Einführung in die Theorie der linearen Integralgleichungen. B. G. Teubner, Leipzig A. KNESER. Die Integralgleichungen und ihre Anwendungen in der mathematischen Physik. Vieweg, Braunschweig H. B. HEYWOOD, M. R. FRÉCHET, J. HADAMARD. L équation de Fredholm et ses applications à la physique mathématique. Hermann, Paris T. LALESCO. Introduction à la théorie des équations intégrales. Avec une préface de É. Picard. Hermann, Paris V. VOLTERRA. Leçons sur les équations intégrales et les équations intégro-différentielles. Gauthier-Villars, Paris M. BÔCHER. An introduction to the study of integral equations. Cambridge University Press H. POINCARÉ. Remarques diverses sur l équation de Fredholm. Acta Math. 33 (1909) D. HILBERT. Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Teubner, Leipzig Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 14 / 26
27 Le Grand Bouleversement D. HILBERT. Über das Dirichletsche Prinzip. Jb. DMV 8 (1900) D. HILBERT. Über das Dirichletsche Prinzip. Math. Ann. 59 (1904) D. HILBERT. Über das Dirichletsche Prinzip. J. R. A. Math. 129 (1905) I. FREDHOLM. Sur une classe d équations fonctionnelles. Acta Math. 27 (1903), no. 1, A. KORN. Über freie und erzwungene Schwingungen. Eine Einführung in die Theorie der linearen Integralgleichungen. B. G. Teubner, Leipzig A. KNESER. Die Integralgleichungen und ihre Anwendungen in der mathematischen Physik. Vieweg, Braunschweig H. B. HEYWOOD, M. R. FRÉCHET, J. HADAMARD. L équation de Fredholm et ses applications à la physique mathématique. Hermann, Paris T. LALESCO. Introduction à la théorie des équations intégrales. Avec une préface de É. Picard. Hermann, Paris V. VOLTERRA. Leçons sur les équations intégrales et les équations intégro-différentielles. Gauthier-Villars, Paris M. BÔCHER. An introduction to the study of integral equations. Cambridge University Press H. POINCARÉ. Remarques diverses sur l équation de Fredholm. Acta Math. 33 (1909) D. HILBERT. Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Teubner, Leipzig Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 14 / 26
28 Le Grand Bouleversement D. HILBERT. Über das Dirichletsche Prinzip. Jb. DMV 8 (1900) D. HILBERT. Über das Dirichletsche Prinzip. Math. Ann. 59 (1904) D. HILBERT. Über das Dirichletsche Prinzip. J. R. A. Math. 129 (1905) I. FREDHOLM. Sur une classe d équations fonctionnelles. Acta Math. 27 (1903), no. 1, A. KORN. Über freie und erzwungene Schwingungen. Eine Einführung in die Theorie der linearen Integralgleichungen. B. G. Teubner, Leipzig A. KNESER. Die Integralgleichungen und ihre Anwendungen in der mathematischen Physik. Vieweg, Braunschweig H. B. HEYWOOD, M. R. FRÉCHET, J. HADAMARD. L équation de Fredholm et ses applications à la physique mathématique. Hermann, Paris T. LALESCO. Introduction à la théorie des équations intégrales. Avec une préface de É. Picard. Hermann, Paris V. VOLTERRA. Leçons sur les équations intégrales et les équations intégro-différentielles. Gauthier-Villars, Paris M. BÔCHER. An introduction to the study of integral equations. Cambridge University Press H. POINCARÉ. Remarques diverses sur l équation de Fredholm. Acta Math. 33 (1909) D. HILBERT. Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Teubner, Leipzig Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 14 / 26
29 Le Grand Bouleversement D. HILBERT. Über das Dirichletsche Prinzip. Jb. DMV 8 (1900) D. HILBERT. Über das Dirichletsche Prinzip. Math. Ann. 59 (1904) D. HILBERT. Über das Dirichletsche Prinzip. J. R. A. Math. 129 (1905) I. FREDHOLM. Sur une classe d équations fonctionnelles. Acta Math. 27 (1903), no. 1, A. KORN. Über freie und erzwungene Schwingungen. Eine Einführung in die Theorie der linearen Integralgleichungen. B. G. Teubner, Leipzig A. KNESER. Die Integralgleichungen und ihre Anwendungen in der mathematischen Physik. Vieweg, Braunschweig H. B. HEYWOOD, M. R. FRÉCHET, J. HADAMARD. L équation de Fredholm et ses applications à la physique mathématique. Hermann, Paris T. LALESCO. Introduction à la théorie des équations intégrales. Avec une préface de É. Picard. Hermann, Paris V. VOLTERRA. Leçons sur les équations intégrales et les équations intégro-différentielles. Gauthier-Villars, Paris M. BÔCHER. An introduction to the study of integral equations. Cambridge University Press H. POINCARÉ. Remarques diverses sur l équation de Fredholm. Acta Math. 33 (1909) D. HILBERT. Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Teubner, Leipzig Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 14 / 26
30 Et maintenant des maths pour The Bilaplacian Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 15 / 26
31 Et maintenant des maths pour The Bilaplacian Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 15 / 26
32 Green formulas for 2 = : Smooth bounded domain in R 2 ; boundary Γ; exterior domain + 2 ( u v = u v + n u v u n v ) ds Γ 2 u v = α u α ( v + n u v τ n u τ v nu 2 n v ) ds α =2 Γ = α u α ( v + n u v + s τ n u v nu 2 n v ) ds α =2 Γ 0 σ 1: 2 ( u v = a σ (u, v) + Nσ u v M σ u n v ) ds Γ Z Z X u v + (1 σ) α u α v a σ(u, v) = σ α =2 M σ = σ u + (1 σ) 2 nu : bending moment N σ = n u + (1 σ) s τ nu : twisting moment Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 16 / 26
33 Green formulas for 2 = : Smooth bounded domain in R 2 ; boundary Γ; exterior domain + 2 ( u v = u v + n u v u n v ) ds Γ 2 u v = α u α ( v + n u v τ n u τ v nu 2 n v ) ds α =2 Γ = α u α ( v + n u v + s τ n u v nu 2 n v ) ds α =2 Γ 0 σ 1: 2 ( u v = a σ (u, v) + Nσ u v M σ u n v ) ds Γ Z Z X u v + (1 σ) α u α v a σ(u, v) = σ α =2 M σ = σ u + (1 σ) 2 nu : bending moment N σ = n u + (1 σ) s τ nu : twisting moment Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 16 / 26
34 Green formulas for 2 = : Smooth bounded domain in R 2 ; boundary Γ; exterior domain + 2 ( u v = u v + n u v u n v ) ds Γ 2 u v = α u α ( v + n u v τ n u τ v nu 2 n v ) ds α =2 Γ = α u α ( v + n u v + s τ n u v nu 2 n v ) ds α =2 Γ 0 σ 1: 2 ( u v = a σ (u, v) + Nσ u v M σ u n v ) ds Γ Z Z X u v + (1 σ) α u α v a σ(u, v) = σ α =2 M σ = σ u + (1 σ) 2 nu : bending moment N σ = n u + (1 σ) s τ nu : twisting moment Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 16 / 26
35 Green formulas for 2 = : Smooth bounded domain in R 2 ; boundary Γ; exterior domain + 2 ( u v = u v + n u v u n v ) ds Γ 2 u v = α u α ( v + n u v τ n u τ v nu 2 n v ) ds α =2 Γ = α u α ( v + n u v + s τ n u v nu 2 n v ) ds α =2 Γ 0 σ 1: 2 ( u v = a σ (u, v) + Nσ u v M σ u n v ) ds Γ Z Z X u v + (1 σ) α u α v a σ(u, v) = σ α =2 M σ = σ u + (1 σ) 2 nu : bending moment N σ = n u + (1 σ) s τ nu : twisting moment Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 16 / 26
36 Traces (smooth domain) Z Z 2 u v = a(u, v) + `Nu v Mu nv ds Γ Cauchy data: (γ 0 u, γ 1 u) := (u, nu, Nu, Mu) on Γ Traces: (γ 0, γ 1 ) : H s () H s 1 2 (Γ) H s 3 2 (Γ) H s 7 2 (Γ) H s 5 2 (Γ) Energy norm (s = 2): X := H 3 2 (Γ) H 1 2 (Γ) = γ0 H 2 () (γ 0, γ 1 ) : H 2 ( 2 ; ) X X First Green formula a(u, v) = 2 u v + γ 1 u, γ 0 v Second Green formula ( 2 u v u 2 v ) = γ 1 u, γ 0 v + γ 0 u, γ 1 v Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 17 / 26
37 Traces (smooth domain) Z Z 2 u v = a(u, v) + `Nu v Mu nv ds Γ Cauchy data: (γ 0 u, γ 1 u) := (u, nu, Nu, Mu) on Γ Traces: (γ 0, γ 1 ) : H s () H s 1 2 (Γ) H s 3 2 (Γ) H s 7 2 (Γ) H s 5 2 (Γ) Energy norm (s = 2): X := H 3 2 (Γ) H 1 2 (Γ) = γ0 H 2 () (γ 0, γ 1 ) : H 2 ( 2 ; ) X X First Green formula a(u, v) = 2 u v + γ 1 u, γ 0 v Second Green formula ( 2 u v u 2 v ) = γ 1 u, γ 0 v + γ 0 u, γ 1 v Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 17 / 26
38 Traces (smooth domain) Z Z 2 u v = a(u, v) + `Nu v Mu nv ds Γ Cauchy data: (γ 0 u, γ 1 u) := (u, nu, Nu, Mu) on Γ Traces: (γ 0, γ 1 ) : H s () H s 1 2 (Γ) H s 3 2 (Γ) H s 7 2 (Γ) H s 5 2 (Γ) Energy norm (s = 2): X := H 3 2 (Γ) H 1 2 (Γ) = γ0 H 2 () (γ 0, γ 1 ) : H 2 ( 2 ; ) X X First Green formula a(u, v) = 2 u v + γ 1 u, γ 0 v Second Green formula ( 2 u v u 2 v ) = γ 1 u, γ 0 v + γ 0 u, γ 1 v Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 17 / 26
39 Representation Formula, Single and Double Layer Fundamental solution: G(x) = 1 8π x 2 log x ( G = 1 (log x + 1)) 2π Representation in u(x) = Z 2 u(y) G(x y) dy Z + ` Nu(y)G(x y) + Mu(y) n(y) G(x y) ds(y) Γ Z ` nu(y)m(y)g(x y) u(y)n(y)g(x y) ds(y) Γ = N f (x) + S γ 1 u(x) Dγ 0 u(x) Distributional definitions N f = G f S φ = G γ 0 φ Dg = G γ 1 g Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 18 / 26
40 Representation Formula, Single and Double Layer Fundamental solution: G(x) = 1 8π x 2 log x ( G = 1 (log x + 1)) 2π Representation in u(x) = Z 2 u(y) G(x y) dy Z + ` Nu(y)G(x y) + Mu(y) n(y) G(x y) ds(y) Γ Z ` nu(y)m(y)g(x y) u(y)n(y)g(x y) ds(y) Γ = N f (x) + S γ 1 u(x) Dγ 0 u(x) Distributional definitions N f = G f S φ = G γ 0 φ Dg = G γ 1 g Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 18 / 26
41 Representation Formula, Single and Double Layer Fundamental solution: G(x) = 1 8π x 2 log x ( G = 1 (log x + 1)) 2π Representation in u(x) = Z 2 u(y) G(x y) dy Z + ` Nu(y)G(x y) + Mu(y) n(y) G(x y) ds(y) Γ Z ` nu(y)m(y)g(x y) u(y)n(y)g(x y) ds(y) Γ = N f (x) + S γ 1 u(x) Dγ 0 u(x) Distributional definitions N f = G f S φ = G γ 0 φ Dg = G γ 1 g Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 18 / 26
42 The double layer potential D ` g 0 g 1 (x) = Z Γ ` N(y)G(x y)g0 (y) + M(y)G(x y)g 1 (y) ds(y) Jump relations: [γ 0 Dg] = g ; [γ 1 Dg] = 0 Z Kg(x) = One-sided traces: γ + 0 Dg = g + Kg ; γ 2 1 Dg = Wg Γ «N(y)G(x y) M(y)G(x y) g0 n(x) N(y)G(x y) n(x) M(y)G(x y) g 1 «(y)ds(y) Integral equation for the interior Dirichlet problem ` 1 I K g = f 2 Orders: ` 0 1 : Not a classical Fredholm second kind integral equation! +1 0 Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 19 / 26
43 The double layer potential D ` g 0 g 1 (x) = Z Γ ` N(y)G(x y)g0 (y) + M(y)G(x y)g 1 (y) ds(y) Jump relations: [γ 0 Dg] = g ; [γ 1 Dg] = 0 Z Kg(x) = One-sided traces: γ + 0 Dg = g + Kg ; γ 2 1 Dg = Wg Γ «N(y)G(x y) M(y)G(x y) g0 n(x) N(y)G(x y) n(x) M(y)G(x y) g 1 «(y)ds(y) Integral equation for the interior Dirichlet problem ` 1 I K g = f 2 Orders: ` 0 1 : Not a classical Fredholm second kind integral equation! +1 0 Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 19 / 26
44 The double layer potential D ` g 0 g 1 (x) = Z Γ ` N(y)G(x y)g0 (y) + M(y)G(x y)g 1 (y) ds(y) Jump relations: [γ 0 Dg] = g ; [γ 1 Dg] = 0 Z Kg(x) = One-sided traces: γ + 0 Dg = g + Kg ; γ 2 1 Dg = Wg Γ «N(y)G(x y) M(y)G(x y) g0 n(x) N(y)G(x y) n(x) M(y)G(x y) g 1 «(y)ds(y) Integral equation for the interior Dirichlet problem ` 1 I K g = f 2 Orders: ` 0 1 : Not a classical Fredholm second kind integral equation! +1 0 Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 19 / 26
45 The double layer potential D ` g 0 g 1 (x) = Z Γ ` N(y)G(x y)g0 (y) + M(y)G(x y)g 1 (y) ds(y) Jump relations: [γ 0 Dg] = g ; [γ 1 Dg] = 0 Z Kg(x) = One-sided traces: γ + 0 Dg = g + Kg ; γ 2 1 Dg = Wg Γ «N(y)G(x y) M(y)G(x y) g0 n(x) N(y)G(x y) n(x) M(y)G(x y) g 1 «(y)ds(y) Integral equation for the interior Dirichlet problem ` 1 I K g = f 2 Orders: ` 0 1 : Not a classical Fredholm second kind integral equation! +1 0 Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 19 / 26
46 Lipschitz boundaries: Spaces, traces and potentials Dirichlet trace: X = H 2 ( )/H 2 0 ( ) = H 2 ( + )/H 2 0 ( + ) = H 2 (R 2 )/H 2 0 (R 2 \ Γ) γ 0 : H 2 () X: Canonical projection Neumann trace: γ 1 = γ 1,σ : H 2 ( 2 ; ) X H 2 Γ (R2 ) defined by the first Green formula: γ 1 u, γ 0 v := R 2 u v a(u, v) Definition Single layer potential: S φ = G γ 0 φ Double layer potential : Dg = G γ 1 g With these definitions, many things work and look the same as for the Laplace operator or other second order operators: Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 20 / 26
47 Lipschitz boundaries: Spaces, traces and potentials Dirichlet trace: X = H 2 ( )/H 2 0 ( ) = H 2 ( + )/H 2 0 ( + ) = H 2 (R 2 )/H 2 0 (R 2 \ Γ) γ 0 : H 2 () X: Canonical projection Neumann trace: γ 1 = γ 1,σ : H 2 ( 2 ; ) X H 2 Γ (R2 ) defined by the first Green formula: γ 1 u, γ 0 v := R 2 u v a(u, v) Definition Single layer potential: S φ = G γ 0 φ Double layer potential : Dg = G γ 1 g With these definitions, many things work and look the same as for the Laplace operator or other second order operators: Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 20 / 26
48 Lipschitz boundaries: Spaces, traces and potentials Dirichlet trace: X = H 2 ( )/H 2 0 ( ) = H 2 ( + )/H 2 0 ( + ) = H 2 (R 2 )/H 2 0 (R 2 \ Γ) γ 0 : H 2 () X: Canonical projection Neumann trace: γ 1 = γ 1,σ : H 2 ( 2 ; ) X H 2 Γ (R2 ) defined by the first Green formula: γ 1 u, γ 0 v := R 2 u v a(u, v) Definition Single layer potential: S φ = G γ 0 φ Double layer potential : Dg = G γ 1 g With these definitions, many things work and look the same as for the Laplace operator or other second order operators: Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 20 / 26
49 Lipschitz boundaries: Spaces, traces and potentials Dirichlet trace: X = H 2 ( )/H 2 0 ( ) = H 2 ( + )/H 2 0 ( + ) = H 2 (R 2 )/H 2 0 (R 2 \ Γ) γ 0 : H 2 () X: Canonical projection Neumann trace: γ 1 = γ 1,σ : H 2 ( 2 ; ) X H 2 Γ (R2 ) defined by the first Green formula: γ 1 u, γ 0 v := R 2 u v a(u, v) Definition Single layer potential: S φ = G γ 0 φ Double layer potential : Dg = G γ 1 g With these definitions, many things work and look the same as for the Laplace operator or other second order operators: Continuity S : X H 2 loc(r 2 ) ; D : X H 2 ( 2 ; ) Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 20 / 26
50 Lipschitz boundaries: Spaces, traces and potentials Dirichlet trace: X = H 2 ( )/H 2 0 ( ) = H 2 ( + )/H 2 0 ( + ) = H 2 (R 2 )/H 2 0 (R 2 \ Γ) γ 0 : H 2 () X: Canonical projection Neumann trace: γ 1 = γ 1,σ : H 2 ( 2 ; ) X H 2 Γ (R2 ) defined by the first Green formula: γ 1 u, γ 0 v := R 2 u v a(u, v) Definition Single layer potential: S φ = G γ 0 φ Double layer potential : Dg = G γ 1 g With these definitions, many things work and look the same as for the Laplace operator or other second order operators: Representation formula in u H 2 ( 2 ; ) : u = G 2 u + S γ 1 u Dγ 0 u Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 20 / 26
51 Lipschitz boundaries: Spaces, traces and potentials Dirichlet trace: X = H 2 ( )/H 2 0 ( ) = H 2 ( + )/H 2 0 ( + ) = H 2 (R 2 )/H 2 0 (R 2 \ Γ) γ 0 : H 2 () X: Canonical projection Neumann trace: γ 1 = γ 1,σ : H 2 ( 2 ; ) X H 2 Γ (R2 ) defined by the first Green formula: γ 1 u, γ 0 v := R 2 u v a(u, v) Definition Single layer potential: S φ = G γ 0 φ Double layer potential : Dg = G γ 1 g With these definitions, many things work and look the same as for the Laplace operator or other second order operators: Jump relations [γ 0 S φ] = 0 ; [γ 1 S φ] = φ ; [γ 0 Dg] = g ; [γ 1 D] = 0 Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 20 / 26
52 Lipschitz boundaries: Spaces, traces and potentials Dirichlet trace: X = H 2 ( )/H 2 0 ( ) = H 2 ( + )/H 2 0 ( + ) = H 2 (R 2 )/H 2 0 (R 2 \ Γ) γ 0 : H 2 () X: Canonical projection Neumann trace: γ 1 = γ 1,σ : H 2 ( 2 ; ) X H 2 Γ (R2 ) defined by the first Green formula: γ 1 u, γ 0 v := R 2 u v a(u, v) Definition Single layer potential: S φ = G γ 0 φ Double layer potential : Dg = G γ 1 g With these definitions, many things work and look the same as for the Laplace operator or other second order operators: Definition of boundary integral operators V φ = {γ 0 S φ}, K φ = {γ 1 S φ} on X Kg = {γ 0 Dg}, Wg = {γ 1 Dg} on X Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 20 / 26
53 Lipschitz boundaries: Spaces, traces and potentials Dirichlet trace: X = H 2 ( )/H 2 0 ( ) = H 2 ( + )/H 2 0 ( + ) = H 2 (R 2 )/H 2 0 (R 2 \ Γ) γ 0 : H 2 () X: Canonical projection Neumann trace: γ 1 = γ 1,σ : H 2 ( 2 ; ) X H 2 Γ (R2 ) defined by the first Green formula: γ 1 u, γ 0 v := R 2 u v a(u, v) Definition Single layer potential: S φ = G γ 0 φ Double layer potential : Dg = G γ 1 g With these definitions, many things work and look the same as for the Laplace operator or other second order operators: Etc... K = K, KV = VK, K W = WK, VW = 1 4 I K 2... Calderón projector, Poincaré-Steklov operator, Boundary integral equations... Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 20 / 26
54 And finally: The Energy Removal of zero-energy fields in X 0 = {g X p P 1 : g, γ 0 p = 0}; X 0 = X/γ 0 P 1 ; X 0 = (X 0) The Neumann problem u H 2 () : 2 u = 0, γ 1 u = g is solvable g X 0 Finiteness of energy in + For u = S φ, φ X : a + (u, u) < φ X 0 and a + (u, u) = 0 φ = 0 For u = Dg, g X : a + (u, u) < and a + (u, u) = 0 g γ 0 P 1 Lemma The total energy a (u, u) + a + (u, u) defines positive quadratic forms on X 0 via single layer potentials u = S φ on X 0 via double layer potentials u = Dg Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 21 / 26
55 And finally: The Energy Removal of zero-energy fields in X 0 = {g X p P 1 : g, γ 0 p = 0}; X 0 = X/γ 0 P 1 ; X 0 = (X 0) The Neumann problem u H 2 () : 2 u = 0, γ 1 u = g is solvable g X 0 Finiteness of energy in + For u = S φ, φ X : a + (u, u) < φ X 0 and a + (u, u) = 0 φ = 0 For u = Dg, g X : a + (u, u) < and a + (u, u) = 0 g γ 0 P 1 Lemma The total energy a (u, u) + a + (u, u) defines positive quadratic forms on X 0 via single layer potentials u = S φ on X 0 via double layer potentials u = Dg Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 21 / 26
56 And finally: The Energy Removal of zero-energy fields in X 0 = {g X p P 1 : g, γ 0 p = 0}; X 0 = X/γ 0 P 1 ; X 0 = (X 0) The Neumann problem u H 2 () : 2 u = 0, γ 1 u = g is solvable g X 0 Finiteness of energy in + For u = S φ, φ X : a + (u, u) < φ X 0 and a + (u, u) = 0 φ = 0 For u = Dg, g X : a + (u, u) < and a + (u, u) = 0 g γ 0 P 1 Lemma The total energy a (u, u) + a + (u, u) defines positive quadratic forms on X 0 via single layer potentials u = S φ on X 0 via double layer potentials u = Dg Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 21 / 26
57 The Poincaré Fundamental Lemma for 2 Lemma There exists µ = µ(γ) 1 such that if u = S φ, φ X 0, or u = Dg, g X, then Proof of the Corollary: 1 µ a (u, u) a + (u, u) µa (u, u) a(u, u) (µ + 1)a (u, u) a(u, u) (1 + µ)a + (u, u) a + (u, u) = a(u, u) a (u, u) µ µ+1a(u, u) a (u, u) = a(u, u) a + (u, u) µ µ+1a(u, u) a + (u, u) a (u, u) µ 1 µ+1 a(u, u) Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 22 / 26
58 The Poincaré Fundamental Lemma for 2 Proof Martin Costabel of the (Rennes) Corollary: Inégalité de Poincaré oubliée 2e J d équipe 22 / 26 Lemma There exists µ = µ(γ) 1 such that if u = S φ, φ X 0, or u = Dg, g X, then Corollary 1 µ a (u, u) a + (u, u) µa (u, u) On the Hilbert space X 0 with the norm of the total energy g 2 a = a(u, u) = a (u, u) + a + (u, u) ; (u = Dg) the operators A + and A defined by the bilinear forms a + and a are positive definite, selfadjoint bounded operators satisfying A + + A = I. The 3 operators A +, A and A + A are contractions: A + a µ µ + 1 ; A a µ µ + 1 ; A+ A a µ 1 µ + 1
59 The Poincaré Fundamental Lemma for 2 Lemma There exists µ = µ(γ) 1 such that if u = S φ, φ X 0, or u = Dg, g X, then Proof of the Corollary: 1 µ a (u, u) a + (u, u) µa (u, u) a(u, u) (µ + 1)a (u, u) a(u, u) (1 + µ)a + (u, u) a + (u, u) = a(u, u) a (u, u) µ µ+1a(u, u) a (u, u) = a(u, u) a + (u, u) µ µ+1a(u, u) a + (u, u) a (u, u) µ 1 µ+1 a(u, u) Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 22 / 26
60 The biharmonic double layer potential operator From the first Green formulas and the jump relations, one has the expressions for the total and partial energies u = Dg, g X 0 : a + (u, u) = Wg, ( 1 2 I + K )g ; a(u, u) = Wg, g Hence we can identify: g 2 a = Wg, g ; A + = 1 2 I + K ; A + A = 2K Theorem The operators 1 2 I + K are positive definite selfadjoint operators on X 0 with the energy norm. The operators 1 2 I + K and 2K are contractions. The Dirichlet problem in : 2 u = 0, γ 0 u = f X can be solved by a double layer potential u = Dg, where g is given by the convergent Neumann series g = ( 1 2 I K ) 1 f = ( 1 2 I + K )l f Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 23 / 26 l=0
61 The biharmonic double layer potential operator Theorem The operators 1 2 I + K are positive definite selfadjoint operators on X 0 with the energy norm. The operators 1 2 I + K and 2K are contractions. The Dirichlet problem in : 2 u = 0, γ 0 u = f X can be solved by a double layer potential u = Dg, where g is given by the convergent Neumann series g = ( 1 2 I K ) 1 f = ( 1 2 I + K )l f On the quotient space X 0, the following Neumann series is also convergent: g = ( 1 2 (I 2K )) 1 f = 2 (2K ) l f l=0 l=0 Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 23 / 26
62 The contraction constant (Poincaré estimate) For the norm 1 2 (I 2K W we have seen 1 I K Wg, ( 1 2 W = sup I K )g 2 g X 0 Wg, g a (u, u) = sup{ u is a double layer potential } a + (u, u) + a (u, u) a + (u, u) = 1 inf{ u is a double layer potential } a + (u, u) + a (u, u) In a similar way, we get, by representing single layer potentials by their Dirichlet data 1 I + K V 1 g, ( 1 2 V 1 = sup I + K )g 2 g X V 1 g, g a (u, u) = sup{ u is a single layer potential } a + (u, u) + a (u, u) a + (u, u) = 1 inf{ u is a single layer potential } a + (u, u) + a (u, u) Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 24 / 26
63 The contraction constant (Poincaré estimate) For the norm 1 2 (I 2K W we have seen 1 I K Wg, ( 1 2 W = sup I K )g 2 g X 0 Wg, g a (u, u) = sup{ u is a double layer potential } a + (u, u) + a (u, u) a + (u, u) = 1 inf{ u is a double layer potential } a + (u, u) + a (u, u) In a similar way, we get, by representing single layer potentials by their Dirichlet data 1 I + K V 1 g, ( 1 2 V 1 = sup I + K )g 2 g X V 1 g, g a (u, u) = sup{ u is a single layer potential } a + (u, u) + a (u, u) a + (u, u) = 1 inf{ u is a single layer potential } a + (u, u) + a (u, u) Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 24 / 26
64 The contraction constant (Steinbach-Wendland estimate) Recall: The Poincaré-Steklov operator in : S : γu γ 1 u (Lu = 0) S = ( 1 2 I + K )V 1 ( S.L.: u=s ϕ ; γu=v ϕ ; γ 1 u=( 1 2 I+K )ϕ ) = W ( 1 2 I K ) 1 ( D.L.: u=dv ; γu=( 1 2 I+K )v ; γ 1u= Wv ) = W + S( 1 2 I + K ) ( S(I ( 1 2 I+K ))=W ) = W + ( 1 2 I + K )V 1 ( 1 2I + K ) symmetric form Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 25 / 26
65 The contraction constant (Steinbach-Wendland estimate) Recall: The Poincaré-Steklov operator in : S : γu γ 1 u (Lu = 0) S = ( 1 2 I + K )V 1 ( S.L.: u=s ϕ ; γu=v ϕ ; γ 1 u=( 1 2 I+K )ϕ ) = W ( 1 2 I K ) 1 ( D.L.: u=dv ; γu=( 1 2 I+K )v ; γ 1u= Wv ) = W + S( 1 2 I + K ) ( S(I ( 1 2 I+K ))=W ) = W + ( 1 2 I + K )V 1 ( 1 2I + K ) symmetric form Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 25 / 26
66 The contraction constant (Steinbach-Wendland estimate) Recall: The Poincaré-Steklov operator in : S : γu γ 1 u (Lu = 0) S = ( 1 2 I + K )V 1 ( S.L.: u=s ϕ ; γu=v ϕ ; γ 1 u=( 1 2 I+K )ϕ ) = W ( 1 2 I K ) 1 ( D.L.: u=dv ; γu=( 1 2 I+K )v ; γ 1u= Wv ) = W + S( 1 2 I + K ) ( S(I ( 1 2 I+K ))=W ) = W + ( 1 2 I + K )V 1 ( 1 2I + K ) symmetric form Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 25 / 26
67 The contraction constant (Steinbach-Wendland estimate) Recall: The Poincaré-Steklov operator in : S : γu γ 1 u (Lu = 0) S = ( 1 2 I + K )V 1 ( S.L.: u=s ϕ ; γu=v ϕ ; γ 1 u=( 1 2 I+K )ϕ ) = W ( 1 2 I K ) 1 ( D.L.: u=dv ; γu=( 1 2 I+K )v ; γ 1u= Wv ) = W + S( 1 2 I + K ) ( S(I ( 1 2 I+K ))=W ) = W + ( 1 2 I + K )V 1 ( 1 2I + K ) symmetric form Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 25 / 26
68 The contraction constant (Steinbach-Wendland estimate) Recall: The Poincaré-Steklov operator in : S : γu γ 1 u (Lu = 0) S = ( 1 2 I + K )V 1 ( S.L.: u=s ϕ ; γu=v ϕ ; γ 1 u=( 1 2 I+K )ϕ ) = W ( 1 2 I K ) 1 ( D.L.: u=dv ; γu=( 1 2 I+K )v ; γ 1u= Wv ) = W + S( 1 2 I + K ) ( S(I ( 1 2 I+K ))=W ) = W + ( 1 2 I + K )V 1 ( 1 2I + K ) symmetric form Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 25 / 26
69 The contraction constant (Steinbach-Wendland estimate) If a, b R and b b 2 + a and a > 0, then a b a < 1 If A, B are bounded selfadjoint operators and B = B 2 + A and A ai > 0, then q B a < Let B = 1 I + K in X 2 V 1. The symmetric representation of S V 1 ( 1 2 I + K ) = S = ( 1 2 I + K )V 1 ( 1 2 I + K ) + W shows that B = B 2 + A, A c 0 I > 0 with c 0 = inf v X 0 v, Wv v, V 1 v Hence B V c 0 < 1 Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 26 / 26
70 The contraction constant (Steinbach-Wendland estimate) If a, b R and b b 2 + a and a > 0, then a b a < 1 If A, B are bounded selfadjoint operators and B = B 2 + A and A ai > 0, then q B a < Let B = 1 I + K in X 2 V 1. The symmetric representation of S V 1 ( 1 2 I + K ) = S = ( 1 2 I + K )V 1 ( 1 2 I + K ) + W shows that B = B 2 + A, A c 0 I > 0 with c 0 = inf v X 0 v, Wv v, V 1 v Hence B V c 0 < 1 Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 26 / 26
71 The contraction constant (Steinbach-Wendland estimate) If a, b R and b b 2 + a and a > 0, then a b a < 1 If A, B are bounded selfadjoint operators and B = B 2 + A and A ai > 0, then q B a < Let B = 1 I + K in X 2 V 1. The symmetric representation of S V 1 ( 1 2 I + K ) = S = ( 1 2 I + K )V 1 ( 1 2 I + K ) + W shows that B = B 2 + A, A c 0 I > 0 with c 0 = inf v X 0 v, Wv v, V 1 v Hence B V c 0 < 1 Martin Costabel (Rennes) Inégalité de Poincaré oubliée 2e J d équipe 26 / 26
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