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Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks

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Abstract

We prove a regularity result for the anisotropic linear elasticity equation\({P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}\) , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain \({\Omega \subset \mathbb{R}^3}\) in weighted Sobolev spaces \({\mathcal {K}^{m+1}_{a+1}(\Omega)}\) , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of \({\mathcal {K}^{m}_{a}}\) -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.

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References

  • Agmon, S.: Lectures on elliptic boundary value problems. Prepared for publication by Frank Jones, B. Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2. D. Van Nostrand Co., Inc., Princeton, London, 1965

  • Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math. 17, 35–92 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  • Ammann, B., Ionescu, A., Nistor, V. Sobolev spaces on Lie manifolds and regularity for polyhedral domains. Doc. Math., 11, 161–206 (electronic) 2006

    MathSciNet  MATH  Google Scholar 

  • Ammann B., Nistor V.: Weighted sobolev spaces and regularity for polyhedral domains. Comput. Methods Appl. Mech. Eng. 196, 3650–3659 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Apel T., Nicaise S.: The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Methods Appl. Sci. 21(6), 519–549 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Apel, T., Sändig, A.-M., Solov′ev, S.: Computation of 3D vertex singularities for linear elasticity: error estimates for a finite element method on graded meshes. M2AN Math. Model. Numer. Anal., 36(6), 1043–1070 (2003), 2002

  • Arnold D., Falk R., Winther R.: Multigrid preconditioning in H(div) on non-convex polygons. Comput. Appl. Math. 17(3), 303–315 (1998)

    MathSciNet  MATH  Google Scholar 

  • Aubin, T.: Espaces de Sobolev sur les variétés riemanniennes. Bull. Sci. Math. (2), 100(2), 149–173 (1976)

    MathSciNet  MATH  Google Scholar 

  • Aubin, T., Li, Y.Y.: On the best Sobolev inequality. J. Math. Pures Appl. (9), 78(4), 353–387 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Babuška I.: The rate of convergence for the finite element method. SIAM J. Numer. Anal. 8, 304–315 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  • Babuška, I., Aziz, A.K.: Survey lectures on the mathematical foundations of the finite element method. In: The mathematical foundations of the finite element method with applications to partial differential equations. Proceedings of Symposium, University of Maryland, Baltimore, pp. 1–359. Academic Press, New York, 1972 (With the collaboration of G. Fix and R. Kellogg)

  • Babuška I., Sobolev S.L.: Optimization of numerical processes. Appl. Math. 10, 96–129 (1965)

    MathSciNet  Google Scholar 

  • Babuška, I., Nistor, V.: Boundary value problems in spaces of distributions and their numerical investigation. IMA (Preprint)

  • Bacuta, C., Mazzucato, A.L., Nistor, V., Zikatanov, L.: Interface and mixed boundary value problems on n-dimensional polyhedral domains. Preliminary version as IMA, #1984, August 2004 (preprint)

  • Bacuta C., Nistor V., Zikatanov L.T.: Improving the rate of convergence of high-order finite elements on polyhedra. I. A priori estimates. Numer. Funct. Anal. Optim. 26(6), 613–639 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Bramble, J.: Multigrid Methods, vol. 294, Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow, 1993

  • Bramble, J.H.: A proof of the inf-sup condition for the Stokes equations on Lipschitz domains. Math. Models Methods Appl. Sci., 13(3), 361–371, 2003. Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday

  • Brenner, S., Scott, R.: The mathematical theory of finite element methods, vol. 15, Texts in Applied Mathematics. Springer, New York, 2nd edn, 2002

  • Băcuţă C., Nistor V., Zikatanov L.: Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps. Numer. Math. 100, 165–184 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Cheeger M., Gromov J., Taylor M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17(1), 15–53 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen W., Jost J.: A Riemannian version of Korn’s inequality. Calc. Var. Partial Differ. Equ. 14(4), 517–530 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Chkadua O., Duduchava R.: Pseudodifferential equations on manifolds with boundary: Fredholm property and asymptotic. Math. Nachr. 222, 79–139 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Costabel M.: Boundary integral operators on curved polygons. Ann. Math. Pura Appl. 4(133), 305–326 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Costabel M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19(3), 613–626 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • Costabel, M., Dauge, M.: Computation of corner singularities in linear elasticity. In: Boundary value problems and integral equations in nonsmooth domains (Luminy, 1993), vol. 167, Lecture Notes in Pure and Appl. Math., pp. 59–68. Dekker, New York, 1995

  • Costabel M., Dauge M.: Crack singularities for general elliptic systems. Math. Nachr. 235, 29–49 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Costabel, M., Dauge, M., Nicaise, S.: Corner singularities of Maxwell interface and eddy current problems. In: Operator theoretical methods and applications to mathematical physics, vol. 147, Oper. Theory Adv. Appl., pp. 241–256. Birkhäuser, Basel, 2004

  • Costanzo F., Walton J.: Steady growth of a crack with a rate and temperature sensitive cohesive zone. J. Mech. Phys. Solids 50, 1649–1679 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Dauge, M.: Elliptic boundary value problems on corner domains, vol. 1341 of Lecture Notes in Mathematics. Springer, Berlin, 1988 (Smoothness and asymptotics of solutions)

  • Duarte A., Hamzeh O., Liszka T., Tworzydlo W.: A generalized finite element method for the simulation of three-dimensional dynamic crack propagation. Comput. Methods Appl. Mech. Eng. 190, 2227–2262 (2001)

    Article  ADS  MATH  Google Scholar 

  • Duduchava R., Natroshvili D.: Mixed crack type problem in anisotropic elasticity. Math. Nachr. 191, 83–107 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Duduchava R., Speck F.-O.: Singular integral equations in special weighted spaces. Georgian. Math. J. 7(4), 633–642 (2000)

    MathSciNet  MATH  Google Scholar 

  • Duvaut, G., Lions, J.-L.: Inequalities in mechanics and physics. Translated from the French by C. W. John, Grundlehren der Mathematischen Wissenschaften, 219. Springer, Berlin, 1976

  • Evans, L.C.: Partial differential equations, vol. 19, Graduate Studies in Mathematics. American Mathematical Society, Providence, 1998

  • Falk R.: Nonconforming finite element methods for the equations of linear elasticity. Math. Comp. 57(196), 529–550 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Falk R., Osborn J.: Remarks on mixed finite element methods for problems with rough coefficients. Math. Comp. 62(205), 1–19 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Fichera G.: Linear elliptic differential systems and eigenvalue problems, vol. 8, Lecture Notes in Mathematics. Springer, Berlin (1965)

    Book  Google Scholar 

  • Folland, G.: Introduction to partial differential equations. Princeton University Press, Princeton, 2nd edition, 1995

  • Friedrichs, K.O.: On the boundary-value problems of the theory of elasticity and Korn’s inequality. Ann. Math. (2), 48, 441–471 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  • Friesecke G., James R., Müller S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55(11), 1461–1506 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Grisvard, P.: Elliptic problems in nonsmooth domains, vol. 24, Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, 1985

  • Grisvard P.: Singularités en elasticité. Arch. Ration. Mech. Anal. 107(2), 157–180 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Grisvard P.: Singularities in boundary value problems, vol. 22, Research in Applied Mathematics. Masson, Paris (1992)

    MATH  Google Scholar 

  • Hackbusch W.: Multigrid methods and applications, vol. 4, Springer Series in Computational Mathematics. Springer, Berlin (1985)

    Google Scholar 

  • Hunsicker, E., Mazzeo, R.: Harmonic forms on manifolds with edges. Int. Math. Res. Not., 2005(52), 3229–3272

  • Jerison D., Kenig C.: The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc. (N.S.) 4(2), 203–207 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  • Jerison D., Kenig C.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130(1), 161–219 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  • Jerison, D.S., Kenig, C.E.: Boundary value problems on Lipschitz domains. Studies in partial differential equations, vol. 23, MAA Stud. Math., pp. 1–68. Math. Assoc. America, Washington, DC, 1982

  • Kanninen M., Popelar C.: Advanced Fracutre Mechanics. Oxford University Press, Oxford (1985)

    MATH  Google Scholar 

  • Kapanadze D., Schulze B.-W.: Crack theory and edge singularities, vol. 561, Mathematics and its Applications. Kluwer, Dordrecht (2003)

    Book  Google Scholar 

  • Kenig, C.: Recent progress on boundary value problems on Lipschitz domains. Pseudodifferential operators and applications (Notre Dame, Ind., 1984), vol. 43, Proceedings of Symposium in Pure Mathematics, pp. 175–205. American Mathematical Society, Providence, 1985

  • Kondrat′ev V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Transl. Moscow Math. Soc. 16, 227–313 (1967)

    MathSciNet  Google Scholar 

  • Kozlov, V., Maz′ya, V., Rossmann, J.: Spectral problems associated with corner singularities of solutions to elliptic equations, vol. 85, Mathematical Surveys and Monographs. American Mathematical Society, Providence, 2001

  • Kozlov V.A., Maz′ya V.G.: On stress singularities near the boundary of a polygonal crack. Proc. R. Soc. Edinburgh Sect. A 117(1-2), 31–37 (1991)

    Article  MathSciNet  Google Scholar 

  • Laasonen, P.: On the degree of convergence of discrete approximations for the solutions of the Dirichlet problem. Ann. Acad. Sci. Fenn. Ser. A. I. 1957(246), 19 (1957)

  • Lewis J., Parenti C.: Pseudodifferential operators of Mellin type. Comm. Partial Differ. Equ. 8(5), 477–544 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Lewis J.E.: Layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains. Trans. Am. Math. Soc. 320(1), 53–76 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Li, H., Mazzucato, A.L., Nistor, V.: Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains (Submitted)

  • Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications, vol. 1. Travaux et Recherches Mathématiques, No. 17. Dunod, Paris, 1968

  • Marsden, J.E., Hughes, T.: Mathematical Foundations of Elasticity. Dover Publications Inc., New York, 1994 (Corrected reprint of the 1983 original)

  • Maz’eja, V.G., Plamenevskiĭ, B.A.: Elliptic boundary value problems on manifolds with singularities. In: Problems in mathematical analysis, No. 6: Spectral theory, boundary value problems (Russian), pp. 85–142, 203. Izdat. Leningrad. Univ., Leningrad, 1977

  • Mazja, W.G., Nazarow, S.A., Plamenewski, B.A.: Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten. I, vol. 82, Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs]. Akademie-Verlag, Berlin, 1991. Störungen isolierter Randsingularitäten. [Perturbations of isolated boundary singularities]

  • Maz’ya V., Roßmann J.: Weighted L p estimates of solutions to boundary value problems for second order elliptic systems in polyhedral domains. ZAMM Z. Angew. Math. Mech. 83(7), 435–467 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Mazzeo R.: Elliptic theory of differential edge operators. I. Commun. Partial Differ. Equ. 16(10), 1615–1664 (1991)

    Article  MATH  Google Scholar 

  • McLean W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  • Melrose, R.: Analysis on Manifolds with Corners (in preparation)

  • Melrose R.: Geometric Scattering Theory. Stanford Lectures. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  • Mitrea, D., Mitrea, M., Taylor, M.: Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds. Mem. Am. Math. Soc. 150(713), x+120 (2001)

  • Mitrea I.: On the traction problem for the Lamé system on curvilinear polygons.J. Integral Equ. Appl. 16(2), 175–219 (2004)

    MathSciNet  MATH  Google Scholar 

  • Mitrea, I., Mitrea, M.: The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains. Trans. Am. Math. Soc. 359(9), 4143–4182 (electronic), 2007

    Article  MathSciNet  MATH  Google Scholar 

  • Mitrea M., Taylor M.: Boundary layer methods for Lipschitz domains in Riemannian manifolds. J. Funct. Anal. 163(2), 181–251 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Mitrea M., Taylor M.: Potential theory on Lipschitz domains in Riemannian manifolds: L p Hardy, and Hölder space results. Comm. Anal. Geom. 9(2), 369–421 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Morrey, C.B., Jr: Second-order elliptic systems of differential equations. In: Contributions to the theory of partial differential equations, Annals of Mathematics Studies, No. 33, pp. 101–159. Princeton University Press, Princeton, 1954

  • Nazarov S., Plamenevsky B.: Elliptic problems in domains with piecewise smooth boundaries, vol. 13, de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin (1994)

    Book  Google Scholar 

  • Nicaise S.: Regularity of the solutions of elliptic systems in polyhedral domains. Bull. Belg. Math. Soc. Simon Stevin 4(3), 411–429 (1997)

    MathSciNet  MATH  Google Scholar 

  • Nicaise S., Sändig A.-M.: Transmission problems for the Laplace and elasticity operators: regularity and boundary integral formulation. Math. Models Methods Appl. Sci. 9(6), 855–898 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Nirenberg L.: Estimates and existence of solutions of elliptic equations. Comm. Pure Appl. Math. 9, 509–529 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  • Nitsche J.A.: On Korn’s second inequality. RAIRO Anal. Numér. 15(3), 237–248 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  • Sändig A.-M., Richter U., Sändig R.: The regularity of boundary value problems for the Lamé equations in a polygonal domain. Rostock. Math. Kolloq. 36, 21–50 (1989)

    MATH  Google Scholar 

  • Savaré G.: Regularity results for elliptic equations in Lipschitz domains. J. Funct. Anal. 152(1), 176–201 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Schwab C.: P- And Hp- Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Oxford University Press, Oxford (1999)

    Google Scholar 

  • Shubin, M.: Spectral theory of elliptic operators on noncompact manifolds. Astérisque, 207:5, 35–108, 1992 [Méthodes semi-classiques, vol.1 (Nantes, 1991)]

  • Skrzypczak, L.: Mapping properties of pseudodifferential operators on manifolds with bounded geometry. J. Lond. Math. Soc. (2) 57(3), 721–738 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Sternberg E., Koiter W.: The wedge under a concentrated couple: a paradox in the two-dimensional theory of elasticity. J. Appl. Mech. 25, 575–581 (1958)

    MathSciNet  MATH  Google Scholar 

  • Taylor M.: Pseudodifferential Operators, vol. 34, Princeton Mathematical Series, xi+452. Princeton University Press, Princeton (1981)

    Google Scholar 

  • Taylor M.: Partial differential equations I, Basic theory, vol. 115, Applied Mathematical Sciences. Springer, New York (1995)

    Google Scholar 

  • Taylor M.: Partial differential equations II, Qualitative studies of linear equations, vol. 116, Applied Mathematical Sciences. Springer, New York (1996)

    Google Scholar 

  • Taylor, M.: Tools for PDE, vol. 81, Mathematical Surveys and Monographs. American Mathematical Society, Providence, 2000 (Pseudodifferential operators, paradifferential operators, and layer potentials)

  • Triebel H.: Characterizations of function spaces on a complete Riemannian manifold with bounded geometry. Math. Nachr. 130, 321–346 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  • Verchota G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59(3), 572–611 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  • Verchota, G., Vogel, A.: The multidirectional Neumann problem in \({\mathbb {R}^4}\) (2003,Preprint)

  • Vishik, M.I.: Les opérateurs différentiels et pseudo-différentiels à une infinité de variables, les problèmes elliptiques et paraboliques. In: Colloque International CNRS sur les équations aux Dérivées Partielles Linéaires (Univ. Paris-Sud, Orsay, 1972), pp. 342–362. Astérisque, 2 et 3. Soc. Math. France, Paris, 1973

  • Wahlbin L.: On the sharpness of certain local estimates for H̊1 projections into finite element spaces: influence of a re-entrant corner. Math. Comp. 42(165), 1–8 (1984)

    MathSciNet  MATH  Google Scholar 

  • Wahlbin, L.: Local behavior in finite element methods. In: Handbook of numerical analysis, vol. II, Handb. Numer. Anal., II, pp. 353–522. North-Holland, Amsterdam, 1991

  • Wendland W.L.: Elliptic Systems in the Plane. Pitman, London (1979)

    MATH  Google Scholar 

  • Whitney, H.: Tangents to an analytic variety. Ann. Math. (2) 81, 496–549 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  • Ziemer, W.P.: Weakly Differentiable Functions, vol. 120, Graduate Texts in Mathematics. Springer, New York, 1989 (Sobolev spaces and functions of bounded variation)

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  1. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Anna L. Mazzucato & Victor Nistor

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  1. Anna L. Mazzucato
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Correspondence to Anna L. Mazzucato.

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Dedicated to Michael E. Taylor on occasion of his sixtieth birthday

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Mazzucato, A.L., Nistor, V. Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks. Arch Rational Mech Anal 195, 25–73 (2010). https://doi.org/10.1007/s00205-008-0180-y

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