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Finite Energy Functional Spaces on Unbounded Domains with a Cut

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<P>Abstract We study in this thesis functional spaces involved in crack problems in unbounded domains. These spaces are defined by closing spaces of Sobolev <B><I>H<SUP>1</SUP></I></B> regularity functions (or vector fields) of bounded support, by the <B><I>L<SUP>2</SUP></I></B> norm of the gradient. In the case of linear elasticity, the closure is done under the <B><I>L<SUP>2</SUP></I></B> norm of the symmetric gradient. Our main result states that smooth functions are in this closure if and only if their gradient, (respectively symmetric gradient for the elasticity case), is in <B><I>L<SUP>2</SUP></I></B>. We provide examples of functions in these newly defined spaces that are not in <B><I>L<SUP>2</SUP></I></B>. We show however that some limited growth in dimension 2, or some decay in dimension 3 must hold for functions in those spaces: this is due to Hardy's inequalities.</P>

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  • English
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  • etd-052409-201502
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  • 2009
Date created
  • 2009-05-24
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