Fixing the rank during approximations of linear maps in Sobolev spaces

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Let $Omega subseteq mathbbR^d$ be an open bounded domain. Fix some integer $1<k<d$.



Let $f in W^1,k(Omega;mathbbR^d)$ be a continuous map with $det df > 0$ a.e. Consider the induced exterior map $Omega to textGL(bigwedge^k mathbbR^d)$ defined by $x to bigwedge^k df_x $. (Since $det df > 0$ a.e. it follows that $bigwedge^k df_x in textGL(bigwedge^k mathbbR^d)$ a.e.).




We can think of $textGL(bigwedge^k mathbbR^d)$ as a subset of $textEnd(bigwedge^k mathbbR^d)cong mathbbR^binomdk^2 $. With this identification in mind, suppose that $bigwedge^k df in Cbig(Omega,mathbbR^binomdk^2 big) cap W^1,kbig(Omega,mathbbR^binomdk^2 big).$



Question: Do there exist $u_n in C^inftybig(Omega,textGL(bigwedge^k mathbbR^d)big)$, such that $u_n to bigwedge^k df$ in $W^1,kbig(Omega,textGL(bigwedge^k mathbbR^d)big) $?




Of course, there are always $u_n in C^infty(Omega,mathbbR^binomdk^2)$ such that $u_n to bigwedge^k df$ in
$W^1,kbig(Omega,mathbbR^binomdk^2 big)$. (just approximate each component of $bigwedge^k df$ separately).



The question is whether or not we can choose the approximations of the different components in such a way that their "combined" map will stay invertible almost everywhere.



Comment: I really only care about the "local" behaviour- that is approximating in $W^1,k_loc$.



Also, note that it's important $k<d$. If we knew that $bigwedge^k df in W^1,p$ for some $p>d$, then the answer would definitely positive; indeed, convergence in $W^1,p$ for $p>d$, implies uniform convergence, so $u_n to bigwedge^k df$ uniformly. Since "being invertible" is an open condition, we are done.




real-analysis linear-algebra sobolev-spaces matrix-rank exterior-algebra




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    Let $Omega subseteq mathbbR^d$ be an open bounded domain. Fix some integer $1<k<d$.



    Let $f in W^1,k(Omega;mathbbR^d)$ be a continuous map with $det df > 0$ a.e. Consider the induced exterior map $Omega to textGL(bigwedge^k mathbbR^d)$ defined by $x to bigwedge^k df_x $. (Since $det df > 0$ a.e. it follows that $bigwedge^k df_x in textGL(bigwedge^k mathbbR^d)$ a.e.).




    We can think of $textGL(bigwedge^k mathbbR^d)$ as a subset of $textEnd(bigwedge^k mathbbR^d)cong mathbbR^binomdk^2 $. With this identification in mind, suppose that $bigwedge^k df in Cbig(Omega,mathbbR^binomdk^2 big) cap W^1,kbig(Omega,mathbbR^binomdk^2 big).$



    Question: Do there exist $u_n in C^inftybig(Omega,textGL(bigwedge^k mathbbR^d)big)$, such that $u_n to bigwedge^k df$ in $W^1,kbig(Omega,textGL(bigwedge^k mathbbR^d)big) $?




    Of course, there are always $u_n in C^infty(Omega,mathbbR^binomdk^2)$ such that $u_n to bigwedge^k df$ in
    $W^1,kbig(Omega,mathbbR^binomdk^2 big)$. (just approximate each component of $bigwedge^k df$ separately).



    The question is whether or not we can choose the approximations of the different components in such a way that their "combined" map will stay invertible almost everywhere.



    Comment: I really only care about the "local" behaviour- that is approximating in $W^1,k_loc$.



    Also, note that it's important $k<d$. If we knew that $bigwedge^k df in W^1,p$ for some $p>d$, then the answer would definitely positive; indeed, convergence in $W^1,p$ for $p>d$, implies uniform convergence, so $u_n to bigwedge^k df$ uniformly. Since "being invertible" is an open condition, we are done.




    real-analysis linear-algebra sobolev-spaces matrix-rank exterior-algebra




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      Let $Omega subseteq mathbbR^d$ be an open bounded domain. Fix some integer $1<k<d$.



      Let $f in W^1,k(Omega;mathbbR^d)$ be a continuous map with $det df > 0$ a.e. Consider the induced exterior map $Omega to textGL(bigwedge^k mathbbR^d)$ defined by $x to bigwedge^k df_x $. (Since $det df > 0$ a.e. it follows that $bigwedge^k df_x in textGL(bigwedge^k mathbbR^d)$ a.e.).




      We can think of $textGL(bigwedge^k mathbbR^d)$ as a subset of $textEnd(bigwedge^k mathbbR^d)cong mathbbR^binomdk^2 $. With this identification in mind, suppose that $bigwedge^k df in Cbig(Omega,mathbbR^binomdk^2 big) cap W^1,kbig(Omega,mathbbR^binomdk^2 big).$



      Question: Do there exist $u_n in C^inftybig(Omega,textGL(bigwedge^k mathbbR^d)big)$, such that $u_n to bigwedge^k df$ in $W^1,kbig(Omega,textGL(bigwedge^k mathbbR^d)big) $?




      Of course, there are always $u_n in C^infty(Omega,mathbbR^binomdk^2)$ such that $u_n to bigwedge^k df$ in
      $W^1,kbig(Omega,mathbbR^binomdk^2 big)$. (just approximate each component of $bigwedge^k df$ separately).



      The question is whether or not we can choose the approximations of the different components in such a way that their "combined" map will stay invertible almost everywhere.



      Comment: I really only care about the "local" behaviour- that is approximating in $W^1,k_loc$.



      Also, note that it's important $k<d$. If we knew that $bigwedge^k df in W^1,p$ for some $p>d$, then the answer would definitely positive; indeed, convergence in $W^1,p$ for $p>d$, implies uniform convergence, so $u_n to bigwedge^k df$ uniformly. Since "being invertible" is an open condition, we are done.




      real-analysis linear-algebra sobolev-spaces matrix-rank exterior-algebra




      share|cite|improve this question













      Let $Omega subseteq mathbbR^d$ be an open bounded domain. Fix some integer $1<k<d$.



      Let $f in W^1,k(Omega;mathbbR^d)$ be a continuous map with $det df > 0$ a.e. Consider the induced exterior map $Omega to textGL(bigwedge^k mathbbR^d)$ defined by $x to bigwedge^k df_x $. (Since $det df > 0$ a.e. it follows that $bigwedge^k df_x in textGL(bigwedge^k mathbbR^d)$ a.e.).




      We can think of $textGL(bigwedge^k mathbbR^d)$ as a subset of $textEnd(bigwedge^k mathbbR^d)cong mathbbR^binomdk^2 $. With this identification in mind, suppose that $bigwedge^k df in Cbig(Omega,mathbbR^binomdk^2 big) cap W^1,kbig(Omega,mathbbR^binomdk^2 big).$



      Question: Do there exist $u_n in C^inftybig(Omega,textGL(bigwedge^k mathbbR^d)big)$, such that $u_n to bigwedge^k df$ in $W^1,kbig(Omega,textGL(bigwedge^k mathbbR^d)big) $?




      Of course, there are always $u_n in C^infty(Omega,mathbbR^binomdk^2)$ such that $u_n to bigwedge^k df$ in
      $W^1,kbig(Omega,mathbbR^binomdk^2 big)$. (just approximate each component of $bigwedge^k df$ separately).



      The question is whether or not we can choose the approximations of the different components in such a way that their "combined" map will stay invertible almost everywhere.



      Comment: I really only care about the "local" behaviour- that is approximating in $W^1,k_loc$.



      Also, note that it's important $k<d$. If we knew that $bigwedge^k df in W^1,p$ for some $p>d$, then the answer would definitely positive; indeed, convergence in $W^1,p$ for $p>d$, implies uniform convergence, so $u_n to bigwedge^k df$ uniformly. Since "being invertible" is an open condition, we are done.





      real-analysis linear-algebra sobolev-spaces matrix-rank exterior-algebra





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      Asaf Shachar

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