Necessary and sufficient conditions for rigidity of the perimeter inequality under spherical symmetrisation are given. That is, a characterisation for the uniqueness (up to orthogonal transformations) of the extremals is provided. This is obtained through a careful analysis of the equality cases, and studying fine properties of the circular symmetrisation, which was firstly introduced by Pólya in 1950.
We analyze the Γ -convergence of sequences of free-discontinuity functionals arising in the modeling of linear elastic solids with surface discontinuities, including phenomena as fracture, damage, or material voids. We prove compactness with respect to Γ -convergence and represent the Γ -limit in an integral form defined on the space of generalized special functions of bounded deformation (GSBD p ). We identify the integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions. Eventually, we investigate sequences of corresponding boundary value problems and show convergence of minimum values and minimizers. In particular, our techniques allow to characterize relaxations of functionals on GSBD p , and cover the classical case of periodic homogenization.
In this work, we provide a characterization result for lower semicontinuity of surface energies defined on piecewise rigid functions, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is a skew symmetric matrix. This characterization is achieved by means of an integral condition, called BD-ellipticity, which is in the spirit of BV -ellipticity defined by Ambrosio and Braides [5]. By specific examples we show that this novel concept is in fact stronger compared to its BV analog. We further provide a sufficient condition implying BD-ellipticity which we call symmetric joint convexity. This notion can be checked explicitly for certain classes of surface energies which are relevant for applications, e.g., for variational fracture models. Finally, we give a direct proof that surface energies with symmetric jointly convex integrands are lower semicontinuous also on the larger space of GSBD p functions.
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