Uniqueness of integrable solutions to \({\nabla \zeta=G \zeta, \zeta|_\Gamma = 0}\) for integrable tensor coefficients G and applications to elasticity

Abstract

Let \({\Omega \subset \mathbb{R}^{N}}\) be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary \({\partial\Omega}\). We show that the solution to the linear first-order system

$$\nabla \zeta = G\zeta, \, \, \zeta|_\Gamma = 0 \quad \quad \quad (1)$$

is unique if \({G \in \textsf{L}^{1}(\Omega; \mathbb{R}^{(N \times N) \times N})}\) and \({\zeta \in \textsf{W}^{1,1}(\Omega; \mathbb{R}^{N})}\). As a consequence, we prove

$$||| \cdot ||| : \textsf{C}_{o}^{\infty}(\Omega, \Gamma; \mathbb{R}^{3}) \rightarrow [0, \infty), \, \, u \mapsto \parallel {\rm sym}(\nabla uP^{-1})\parallel_{\textsf{L}^{2}(\Omega)}$$

to be a norm for \({P \in \textsf{L}^{\infty}(\Omega; \mathbb{R}^{3 \times 3})}\) with Curl \({P \in \textsf{L}^{p}(\Omega; \mathbb{R}^{3 \times 3})}\), Curl \({P^{-1} \in \textsf{L}^{q}(\Omega; \mathbb{R}^{3 \times 3})}\) for some p, q > 1 with 1/p + 1/q = 1 as well as det \({P \geq c^+ > 0}\). We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let \({\Phi \in \textsf{H}^{1}(\Omega; \mathbb{R}^{3})}\) satisfy sym \({(\nabla\Phi^\top\nabla\Psi) = 0}\) for some \({\Psi \in \textsf{W}^{1,\infty}(\Omega; \mathbb{R}^{3}) \cap \textsf{H}^{2}(\Omega; \mathbb{R}^{3})}\) with det \({\nabla\Psi \geq c^+ > 0}\). Then, there exist a constant translation vector \({a \in \mathbb{R}^{3}}\) and a constant skew-symmetric matrix \({A \in \mathfrak{so}(3)}\), such that \({\Phi = A\Psi + a}\).

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References

  1. 1

    Anicic S., Le Dret H., Raoult A.: The infinitesimal rigid displacement lemma in Lipschitz co-ordinates and application to shells with minimal regularity. Math. Meth. Appl. Sci. 27(11), 1283–1299 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2

    Ciarlet P.G.: Mathematical Elasticity, vol. III: Theory of Shells, 1st edn. Amsterdam, North-Holland (1999)

    Google Scholar 

  3. 3

    Ciarlet P.G.: On Korn’s inequality. Chin. Ann. Math. 31B((5), 607–618 (2010)

    MathSciNet  Article  Google Scholar 

  4. 4

    Ciarlet, P.G., Mardare, C.: On rigid and infinitesimal rigid displacements in three-dimensional elasticity. Math. Model. Methods Appl. Sci. 13(11): 1589–1598. MR 2024464 (2004j:74014) (2003)

    Google Scholar 

  5. 5

    Klawonn A., Neff P., Rheinbach O., Vanis S.: FETI-DP domain decomposition methods for elasticity with structural changes: P-elasticity. ESAIM Math. Mod. Num. Anal. 45, 563–602 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6

    Korn, A.: Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen. Bulletin International de l’Académie des Sciences de Cracovie Classe des Sciences Mathématiques et Naturels, no. 9, Novembre, 705–724 (1909)

  7. 7

    Neff P.: On Korn’s first inequality with nonconstant coefficients. Proc. Roy. Soc. Edinb. A 132, 221–243 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8

    Neff, P.: Local existence and uniqueness for a geometrically exact membrane-plate with viscoelastic transverse shear resistance. Math. Methods Appl. Sci. (MMAS) 28, 1031–1060 (2005)

    Google Scholar 

  9. 9

    Neff P.: Local existence and uniqueness for quasistatic finite plasticity with grain boundary relaxation. Q. Appl. Math. 63, 88–116 (2005)

    MathSciNet  MATH  Google Scholar 

  10. 10

    Neff P.: Existence of minimizers for a finite-strain micromorphic elastic solid. Proc. Roy. Soc. Edinb. A 136, 997–1012 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11

    Neff P.: A geometrically exact planar Cosserat shell-model with microstructure. Existence of minimizers for zero Cosserat couple modulus. Math. Models Methods Appl. Sci. M 3(S 17(3), 363–392 (2007)

    MathSciNet  Article  Google Scholar 

  12. 12

    Neff, P., Münch, I.: Curl bounds Grad on SO (3). ESAIM Control Optim. Calc. Var. 14(1), 148–159 (2008). doi:10.1051/cocv:2007050

  13. 13

    Neff P., Pauly D., Witsch K.-J.: Maxwell meets Korn: a new coercive inequality for tensor fields in \({\mathbb{R}^{N\times N} }\) with square-integrable exterior derivative. Math. Methods Appl. Sci. 35, 65–71 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14

    Neff P., Pauly D., Witsch K.J.: A canonical extension of Korn’s first inequality to H(Curl) motivated by gradient plasticity with plastic spin. C. R. Acad. Sci. Paris Ser. I 349, 1251–1254 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15

    Neff, P., Pauly, D., Witsch, K.J.: Poincaré meets Korn via Maxwell: Extending Korn’s first inequality to incompatible tensor fields. submitted; http://arxiv.org/abs/1203.2744; Preprint SM-E-753, Universität Duisburg-Essen, Schriftenreihe der Fakultät für Mathematik, http://www.uni-due.de/ hn213me/preprints/bd753.pdf (2012)

  16. 16

    Neff, P., Pompe, W.: Counterexamples in the theory of coerciveness for linear elliptic systems related to generalizations of Korn’s second inequality (submitted) (2011)

  17. 17

    Pompe W.: Korn’s first inequality with variable coefficients and its generalizations. Comment. Math. Univ. Carolinae 44(1), 57–70 (2003)

    MathSciNet  MATH  Google Scholar 

  18. 18

    Pompe, W.: Counterexamples to Korn’s inequality with non-constant rotation coefficients. Math. Mech. Solids 16, 172–176 (2011). doi:10.1177/1081286510367554

  19. 19

    Ziemer W.P.: Weakly Differentiable Functions Graduate. Texts in Mathematics, vol. 120. Springer, Berlin (1989)

    Google Scholar 

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Lankeit, J., Neff, P. & Pauly, D. Uniqueness of integrable solutions to \({\nabla \zeta=G \zeta, \zeta|_\Gamma = 0}\) for integrable tensor coefficients G and applications to elasticity. Z. Angew. Math. Phys. 64, 1679–1688 (2013). https://doi.org/10.1007/s00033-013-0314-4

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Mathematics Subject Classification

  • 74B99
  • 35Q74
  • 35A02

Keywords

  • Korn’s inequality
  • Generalized Korn’s first inequality
  • First-order system of partial differential equations
  • Uniqueness
  • Infinitesimal rigid displacement lemma
  • Korn’s inequality in curvilinear coordinates
  • Unique continuation