Abstract
In this article, we are interested in the existence, uniqueness and regularity of the solution of the linear elasticity system. More precisely, the quasi-static elasticity system. In the first part, we study the existence of a weak solution and the regularity in the space \(W^{1, p}_0(\Omega ),\ \forall p \in ]1, +\infty [\) for a p-integrable source function. In the second part, the very weak solution is introduced which can be considered when the second member is a function with a very weak solution, for example, a locally integrable function. Such source functions lead to a lack of regularity for the solution in the fact that existence in classical spaces is no longer assured. So, to overcome this difficulty, the strategy consists in approaching it by another more regular problem “converging” towards the initial problem “in a direction to be specified”.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Acquistapace, P.: On BMO regularity for linear elliptic systems. Ann. Math. Pura Appl. 161, 231–270 (1992)
Bennet, C., Sharpley, R.: Interpolation of Operators. Academic Press, London (1983)
Campanato, S.: Equazioni ellittiche del secondo ordine e spazi \({L}^{2,\lambda }(\Omega )\). Annali di Matematica (IV) LXIX, 321–381 (1965)
Ciarlet, P.G.: Élasticité tridimensionnelle. Masson, Paris (1985)
Ciarlet, P.G.: Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity, Series “Studies in Mathematics and its Application”. North-Holland, Amsterdam (1988)
Díaz, J.I., Rakotoson, J.M.: Elliptic problems on the space of weighted with the distance to the boundary integrable functions revisited. Electron. J. Differ. Equ. Conf. 21, 45–59 (2014)
Díaz, J.I., Rakotoson, J.M.: On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary. J. Funct. Anal. 257, 807–831 (2009). https://doi.org/10.1016/j.jfa.2009.03.002
Díaz, J.I.: On the very weak solvability of the beam equation. Rev. R. Acad. Cien. Serie A. Mat. (RACSAM) 105, 167–172 (2011)
Díaz, J.I.: Non Hookean beams and plates: very weak solutions and their numerical analysis. Int. J. Numer. Anal. Model. 11(2), 315–331 (2014)
Dolzman, G., Mûller, S.: Estimates for Green’s matrices of elliptic systems by \(L^p\) theory, Mathematisches institut der Universitat Freiburg, Albertstr. 23b, 79104 Freiburg. Manuscripta Math. 88, 261–273 (1995)
Gilbarg, D., Trudinger, S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)
Grisvard, P.: Singularities in Boundary Value Problems. Masson, Paris (1992)
Kufner, A.: Weighted Sobolev Spaces. Teuber Verlagsgesellshaft Prague (1980)
Morrey, C.B.: Multiple integral problems in the calculus of variations and related topics. Univ. Calif. Publ. Math. (N.S.) 1, 1–130 (1943)
Rakotoson, J.M.: Réarrangement Relatif: un instrument d’estimation dans les problèmes aux limites. Springer, Berlin (2008)
Rakotoson, J.M.: New Hardy inequalities and behaviour of linear elliptic equations. J. Funct. Anal. 263, 2893–2920 (2012)
Rakotoson, J.M.: A few natural extension of the regularity of a very weak solutions. Differ. Integr. Equ. 24(11–12), 1125–1140 (2011)
Rakotoson, J.M.: Linear equation with data in non standard spaces. Rend. Lincei Mat. Appl. 26, 241–262 (2015)
Simader, C.: On Dirichlet’s Boundary Value Problem. Springer, Berlin (1972)
Shi, P., Wright, S.: \(W^{2, p}\) regularity of the displacement problem for the lame system on \(W^{2, s}\) domains. J. Math. Anal. Appl. 293, 291–305 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
El Berdan, N.K. Study of the Regularity of Solutions for a Elasticity System with Integrable Data with Respect to the Distance Function to the Boundary. Mediterr. J. Math. 15, 43 (2018). https://doi.org/10.1007/s00009-018-1088-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-018-1088-x
Keywords
- Linear elasticity system
- very weak solution
- regularity for very weak solution
- weighted spaces
- duality method