Abstract
We consider the interior Dirichlet, Neumann and Robin problems associated to the differential system of linear elastostatics with singular data. We prove that if the assigned displacement field a on the \(C^2\) boundary S of the reference configuration of the elastic body belongs to \(W^{-1/2,2}(S) \), then there exists a unique solution to the equilibrium problem which takes the boundary datum a in a well–defined sense; similar results hold if we assign the traction or a linear combination of displacement and traction on the boundary. Moreover, natural estimates controlling the norms of the solutions with the norms of the data hold and analogous results are obtained for the exterior problems requiring the displacement vanishes at infinity.
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Notes
- 1.
We will use standard notation as in [4]. Moreover, \(W^{k,q}(\varOmega )\) is the Sobolev space of all \(\varphi \in L^1_\mathrm{loc}(\varOmega )\) such that \(\Vert \varphi \Vert _{W^{k,q}(\varOmega )}=\Vert \varphi \Vert _{L^q(\varOmega )}+\Vert \nabla _k\varphi \Vert _{L^q(\varOmega )}<+\infty \); \(W^{k,q}_0(\varOmega )\) is the completion of \(C^\infty _0(\varOmega )\) with respect to \(\Vert \varphi \Vert _{W^{k,q}(\varOmega )}\) and \(W^{-k,q'}(\varOmega )\), with \(1/q+1/{q'}=1\), is its dual space. \(W^{k-1/q,q}(S)\) is the trace space of \(W^{k,q}(\varOmega )\) and \(W^{1-k-1/q',q'}(S)\) is its dual space.
- 2.
Observe that for solutions to the traction problem we mean “normalized” displacements (see [4]).
- 3.
For isotropic bodies,
. \(\lambda \) and \(\mu \) are called the Lamé moduli. - 4.
The boundary datum a for the interior Dirichlet problem is not required to satisfy any compatibility conditions since the kernel of
is reduced to the null vector field. - 5.
This means that it has closed range and zero index.
- 6.
In (34) \(+\) is for \(\varOmega =\varOmega _i\) and − for \(\varOmega =\varOmega _e\).
- 7.
In (45) − is for \(\varOmega =\varOmega _i\) and \(+\) for \(\varOmega =\varOmega _e\).
- 8.
In (45) − is for \(\varOmega =\varOmega _i\) and \(+\) for \(\varOmega =\varOmega _e\).
. 
is reduced to the null vector field.References
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Starita, G., Tartaglione, A. (2020). Equilibrium of a Linearly Elastic Body Under Generalized Boundary Data. In: Pinelas, S., Graef, J.R., Hilger, S., Kloeden, P., Schinas, C. (eds) Differential and Difference Equations with Applications. ICDDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 333. Springer, Cham. https://doi.org/10.1007/978-3-030-56323-3_6
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