Compactness and lower semicontinuity properties in $SBD(\Omega)$
Copyright policy )Compactness and lower semicontinuity properties in $SBD(\Omega)$
The space \(BD(\Omega)\) of vector fields with bounded deformation is the set of all vector fields \(u= (u^1,u^2,\dots, u^n)\in L^1(\Omega,\mathbb{R}^n)\) whose distributional gradient \(Du\) has a symmetric part which is a bounded Radon measure valued in the space of symmetric \(n\times n\) matrices. The space \(SBD(\Omega)\) of special vector fields with bounded deformation is defined as the set of all \(u\in BD(\Omega)\) with the Cantor part \(E^cu= 0\). The main result of this paper is a compactness theorem for bounded subsets of \(SBD(\Omega)\), which is an analogue of \textit{L. Ambrosio's} result for \(SBV(\Omega)\) [Boll. Unione Mat. Ital., VII. Ser. B3, No. 4, 857-881 (1989; Zbl 0767.49001)]. The result cannot be obtained directly from Ambrosio's result, due to a difficulty in finding an analogue of Korn's inequality. The proof is based on a characterization of the space \(SBD(\Omega)\) by means of suitable one-dimensional sections. A lower semicontinuity result is obtained for functionals related to variational models in fracture mechanics for linearly elastic materials.
- University of Parma Italy
- University of Siena Italy
- University of Rome Tor Vergata Italy
- University of Udine Italy
- International School for Advanced Studies Italy
bounded Radon measure, Spaces of vector- and operator-valued functions, Methods involving semicontinuity and convergence; relaxation, Korn's inequality, distributional gradient, vector fields with bounded deformation, Compactness in topological linear spaces; angelic spaces, etc., Settore MAT/05 - ANALISI MATEMATICA, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
bounded Radon measure, Spaces of vector- and operator-valued functions, Methods involving semicontinuity and convergence; relaxation, Korn's inequality, distributional gradient, vector fields with bounded deformation, Compactness in topological linear spaces; angelic spaces, etc., Settore MAT/05 - ANALISI MATEMATICA, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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