Local compactness for linear elasticity in irregular domains

Abstract

Communicated by R. LeisFor the theory of boundary value problems in linear elasticity, it is of crucial importance that the space of vector-valued L2-functions whose symmetrized Jacobians are square-integrable should be compactly embedded in Lz. For regions with the cone property this is usually achieved by combining Korn's inequalities and Rellich's selection theorem. We shall show that in a class of less regular regions Korn's second inequality fails whereas the desired compact embedding still holds true.

Homogenization in Fractional Elasticity

“…This, in turn, can be realized if Ω is bounded and satisfies suitable geometric requirements, see e.g. [21] and the references therein. In the homogeneous Dirichlet case the closedness of the range is warranted if one assumes that Ω is bounded.…”

Weighted Poincaré and Korn inequalities for Hölder α domains

“…The following example, based on that given by Weck in [19], shows that the result of the previous theorem is optimal.…”

“…On the other hand, it is known that (1.2) is not valid for an arbitrary bounded domain. Indeed, counter-examples showing that the inequality does not hold true for domains with external cusps has been given in [9,19]. Also, in the old paper [7], Friedrichs gave a very nice counter-example for an inequality for complex analytic functions which can be derived from (1.1) in the second case.…”

“…Although, as mentioned above, our motivation for these estimates are the generalized Korn inequalities, we believe that they are of interest in themselves, in particular they improve previously known results and moreover they are optimal. In Section 3, we derive the weighted Korn inequalities, we show that they are optimal and finally, we show how the results on compactness given in [19] can be derived from our inequalities by using imbedding theorems for weighted Sobolev spaces proved in [16].…”

“…Acknowledgments: We thank P. De Napoli who gave us reference [3] which was fundamental for our work, and P. Secchi who brought to our attention reference [19] which in part motivated this research. We also thank C. D'Andrea, J. Fernández Bonder and N. Wolanski for helpful comments and references.…”