Bounded deformation vs bounded variation

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Let $BV(mathbb R^n; mathbb R^n)$ be the space of (vector-valued) functions of bounded variation and let $BD(mathbb R^n;mathbb R^n)$ the space of functions with bounded deformation. They are made up respectively of functions $u$ for which the full distributional derivative
$$
Du in mathcal M(mathbb R^n)
$$

is represented by a measure with finite total variation and of the functions for which the symmetric part of the distributional derivative
$$
Eu := fracDu+(Du)^t2 in mathcal M(mathbb R^n)
$$

is represented by a measure with finite total variation.



If $n=1$ of course the two definitions coincide. For $nge 2$ they are different, but I do not find an explicit example.




Q. Let $nge 2$. Find an element in $BD setminus BV$.




Is a characterization of such functions available somewhere in the literature?










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    Let $BV(mathbb R^n; mathbb R^n)$ be the space of (vector-valued) functions of bounded variation and let $BD(mathbb R^n;mathbb R^n)$ the space of functions with bounded deformation. They are made up respectively of functions $u$ for which the full distributional derivative
    $$
    Du in mathcal M(mathbb R^n)
    $$

    is represented by a measure with finite total variation and of the functions for which the symmetric part of the distributional derivative
    $$
    Eu := fracDu+(Du)^t2 in mathcal M(mathbb R^n)
    $$

    is represented by a measure with finite total variation.



    If $n=1$ of course the two definitions coincide. For $nge 2$ they are different, but I do not find an explicit example.




    Q. Let $nge 2$. Find an element in $BD setminus BV$.




    Is a characterization of such functions available somewhere in the literature?










    share|cite|improve this question























      up vote
      3
      down vote

      favorite









      up vote
      3
      down vote

      favorite











      Let $BV(mathbb R^n; mathbb R^n)$ be the space of (vector-valued) functions of bounded variation and let $BD(mathbb R^n;mathbb R^n)$ the space of functions with bounded deformation. They are made up respectively of functions $u$ for which the full distributional derivative
      $$
      Du in mathcal M(mathbb R^n)
      $$

      is represented by a measure with finite total variation and of the functions for which the symmetric part of the distributional derivative
      $$
      Eu := fracDu+(Du)^t2 in mathcal M(mathbb R^n)
      $$

      is represented by a measure with finite total variation.



      If $n=1$ of course the two definitions coincide. For $nge 2$ they are different, but I do not find an explicit example.




      Q. Let $nge 2$. Find an element in $BD setminus BV$.




      Is a characterization of such functions available somewhere in the literature?










      share|cite|improve this question













      Let $BV(mathbb R^n; mathbb R^n)$ be the space of (vector-valued) functions of bounded variation and let $BD(mathbb R^n;mathbb R^n)$ the space of functions with bounded deformation. They are made up respectively of functions $u$ for which the full distributional derivative
      $$
      Du in mathcal M(mathbb R^n)
      $$

      is represented by a measure with finite total variation and of the functions for which the symmetric part of the distributional derivative
      $$
      Eu := fracDu+(Du)^t2 in mathcal M(mathbb R^n)
      $$

      is represented by a measure with finite total variation.



      If $n=1$ of course the two definitions coincide. For $nge 2$ they are different, but I do not find an explicit example.




      Q. Let $nge 2$. Find an element in $BD setminus BV$.




      Is a characterization of such functions available somewhere in the literature?







      fa.functional-analysis real-analysis ap.analysis-of-pdes sobolev-spaces calculus-of-variations






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      asked 4 hours ago









      user111164

      11312




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          2 Answers
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          This paper discusses counterexamples to Korn's inequality in $L^1$ spaces:
          https://www.mis.mpg.de/preprints/2003/preprint2003_93.pdf






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            Example 7.7 in



            L. Ambrosio, A. Coscia, Alessandra, G. Dal Maso,
            Fine properties of functions with bounded deformation.
            Arch. Rational Mech. Anal. 139 (1997), no. 3, 201–238.






            share|cite|improve this answer




















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              2 Answers
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              2 Answers
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              active

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              up vote
              2
              down vote













              This paper discusses counterexamples to Korn's inequality in $L^1$ spaces:
              https://www.mis.mpg.de/preprints/2003/preprint2003_93.pdf






              share|cite|improve this answer
























                up vote
                2
                down vote













                This paper discusses counterexamples to Korn's inequality in $L^1$ spaces:
                https://www.mis.mpg.de/preprints/2003/preprint2003_93.pdf






                share|cite|improve this answer






















                  up vote
                  2
                  down vote










                  up vote
                  2
                  down vote









                  This paper discusses counterexamples to Korn's inequality in $L^1$ spaces:
                  https://www.mis.mpg.de/preprints/2003/preprint2003_93.pdf






                  share|cite|improve this answer












                  This paper discusses counterexamples to Korn's inequality in $L^1$ spaces:
                  https://www.mis.mpg.de/preprints/2003/preprint2003_93.pdf







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 3 hours ago









                  Michael Renardy

                  10.4k13137




                  10.4k13137




















                      up vote
                      2
                      down vote













                      Example 7.7 in



                      L. Ambrosio, A. Coscia, Alessandra, G. Dal Maso,
                      Fine properties of functions with bounded deformation.
                      Arch. Rational Mech. Anal. 139 (1997), no. 3, 201–238.






                      share|cite|improve this answer
























                        up vote
                        2
                        down vote













                        Example 7.7 in



                        L. Ambrosio, A. Coscia, Alessandra, G. Dal Maso,
                        Fine properties of functions with bounded deformation.
                        Arch. Rational Mech. Anal. 139 (1997), no. 3, 201–238.






                        share|cite|improve this answer






















                          up vote
                          2
                          down vote










                          up vote
                          2
                          down vote









                          Example 7.7 in



                          L. Ambrosio, A. Coscia, Alessandra, G. Dal Maso,
                          Fine properties of functions with bounded deformation.
                          Arch. Rational Mech. Anal. 139 (1997), no. 3, 201–238.






                          share|cite|improve this answer












                          Example 7.7 in



                          L. Ambrosio, A. Coscia, Alessandra, G. Dal Maso,
                          Fine properties of functions with bounded deformation.
                          Arch. Rational Mech. Anal. 139 (1997), no. 3, 201–238.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 1 hour ago









                          Piotr Hajlasz

                          5,31632053




                          5,31632053



























                               

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