Uniqueness of Integrable Solutions ∇ζ = Gζ, ζ∣_Γ = 0 for Integrable Tensor‐Coefficients G and Applications to Elasticity

Abstract

Let Ω ⊂ R N be bounded Lipschitz and ∅ = Γ ⊂ ∂Ω relatively open. We show that the solution to the linear first order system) for some p, q > 1 with 1/p + 1/q = 1 and det P ≥ c + > 0. We give a new proof for the so called 'in-finitesimal rigid displacement lemma' in curvilinear coordinates:Then there are a ∈ R 3 and a constant skew-symmetric matrix A ∈ so(3), such that Φ = AΨ + a.

Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields

“…It is well known, that Korn's inequality does not imply Poincaré's inequality, however, due to the presence of the Curl-part we get back both estimates from our generalization (54). Indeed, for compatible P = ∇u we recover a tangential Korn inequality (Corollary 3.9) and for skew-symmetric P = A a Poincaré inequality (Corollary 3.…”

“…They can be generalized to many different settings, including the geometrically nonlinear counterpart [41,59], mixed growth conditions [25], incompatible fields (also with dislocations) [7,69,74,75,76,77] and trace-free infinitesimal strain measures [28,49]. Other generalizations are applicable to Orlicz-spaces [12,13,18,42] and SBD functions with small jump sets [17,38,39], thin domains [47,48,58] and John domains [1,29,31] as well as the case of non-constant coefficients [54,70,78,86]. Piecewise Korn-type inequalities subordinate to a FEM-mesh and involving jumps across element boundaries have also been investigated, see e.g.…”

The relaxed linear micromorphic continuum: Existence, uniqueness and continuous dependence in dynamics

“…More precisely, our model is a subset of the classical micromorphic model in which we allow the usual micromorphic tensors [2] to become positive-semidefinite only [1]. The proof of the well-posedness of this model necessitates the application of new mathematical tools [35–41]. The curvature dependence is reduced to a dependence only on the micro-dislocation tensor α : = Curl = − Curl P R 3 × 3 instead of γ = ∇ P R 27 = R 3 × 3 × 3 , and the local response is reduced to a dependence on the symmetric part of the elastic distortion (relative distortion) ε e = syme = sym ( ∇ u − P ) , while the full kinematical degrees of freedom for u and P are kept, notably rotation of the microstructure remains possible .…”

“…Then, we transform the initial boundary value problem in an abstract evolution equation in an appropriate Hilbert space and we use the results of the semigroups theory of linear operators [33, 34] in order to obtain the existence results. The main point in establishing the desired estimates is represented by the new coercive inequalities recently proved by Neff et al [35–37] and by Bauer et al [38–40] (see also [41]). The results established in our paper can be easily extended to theories which include electromagnetic and thermal interactions [42–45].…”

The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations

“…As mentioned, compared to other existence results established in the static case of the theory of micromorphic elastic materials, we allow the usual elasticity tensors to become positive-semidefinite. The main point in establishing the desired estimates is represented by the new coercive inequalities recently proved by Neff, Pauly and Witsch [87,88,89] and by Bauer, Neff, Pauly and Starke [5,6] (see also [42]).…”