Integral Representation for Functionals Defined on SBDp in Dimension Two

We prove an integral representation result for functionals with growth conditions which give coercivity on the space S B D p ( Ω ) , for Ω ⊂ R 2 , which is a bounded open Lipschitz set, and p ∈ ( 1 , ∞ ) . The space SBD p of functions whose distributional strain is the sum of an L p part and a bound...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2017-03, Vol.223 (3), p.1337-1374
Main Authors: Conti, Sergio, Focardi, Matteo, Iurlano, Flaviana
Format: Article
Language:English
Subjects:
Analysis of PDEs
Classical Mechanics
Complex Systems
Fluid- and Aerodynamics
Mathematical and Computational Physics
Mathematics
Physics
Physics and Astronomy
Theoretical
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Summary:We prove an integral representation result for functionals with growth conditions which give coercivity on the space S B D p ( Ω ) , for Ω ⊂ R 2 , which is a bounded open Lipschitz set, and p ∈ ( 1 , ∞ ) . The space SBD p of functions whose distributional strain is the sum of an L p part and a bounded measure supported on a set of finite H 1 -dimensional measure appears naturally in the study of fracture and damage models. Our result is based on the construction of a local approximation by W 1, p functions. We also obtain a generalization of Korn’s inequality in the SBD p setting.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-016-1059-y