$BV(Omega)$ is embedded compactly in $L^1 _{mathrm{loc}} (Omega)$












3














Definition:
We say that a function $u: Omega rightarrow mathbb{R}$ is a function of bounded variation iff $uin L^1(Omega)$ and $supleft{intlimits_{Omega}u operatorname{div}phi : phi in C_c(Omega, mathbb{R}^d), ||phi||_{infty} leq 1right} < +{infty}$.



By definition it is clear that $mathrm{BV}(Omega)subset L^1(Omega)$.



How to show that this embedding is compact when $Omega$ is bounded set?
Seems like this is a very standard result, but I could not find the proof of this in many of the functional analysis book.










share|cite|improve this question





























    3














    Definition:
    We say that a function $u: Omega rightarrow mathbb{R}$ is a function of bounded variation iff $uin L^1(Omega)$ and $supleft{intlimits_{Omega}u operatorname{div}phi : phi in C_c(Omega, mathbb{R}^d), ||phi||_{infty} leq 1right} < +{infty}$.



    By definition it is clear that $mathrm{BV}(Omega)subset L^1(Omega)$.



    How to show that this embedding is compact when $Omega$ is bounded set?
    Seems like this is a very standard result, but I could not find the proof of this in many of the functional analysis book.










    share|cite|improve this question



























      3












      3








      3







      Definition:
      We say that a function $u: Omega rightarrow mathbb{R}$ is a function of bounded variation iff $uin L^1(Omega)$ and $supleft{intlimits_{Omega}u operatorname{div}phi : phi in C_c(Omega, mathbb{R}^d), ||phi||_{infty} leq 1right} < +{infty}$.



      By definition it is clear that $mathrm{BV}(Omega)subset L^1(Omega)$.



      How to show that this embedding is compact when $Omega$ is bounded set?
      Seems like this is a very standard result, but I could not find the proof of this in many of the functional analysis book.










      share|cite|improve this question















      Definition:
      We say that a function $u: Omega rightarrow mathbb{R}$ is a function of bounded variation iff $uin L^1(Omega)$ and $supleft{intlimits_{Omega}u operatorname{div}phi : phi in C_c(Omega, mathbb{R}^d), ||phi||_{infty} leq 1right} < +{infty}$.



      By definition it is clear that $mathrm{BV}(Omega)subset L^1(Omega)$.



      How to show that this embedding is compact when $Omega$ is bounded set?
      Seems like this is a very standard result, but I could not find the proof of this in many of the functional analysis book.







      functional-analysis analysis pde bounded-variation regularity-theory-of-pdes






      share|cite|improve this question















      share|cite|improve this question













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      edited Nov 26 '18 at 14:08









      Davide Giraudo

      125k16150260




      125k16150260










      asked Jul 21 '18 at 11:53









      Rosy

      1045




      1045






















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          This is Theorem 5.3.4 in W.P.Ziemer: Weakly Differentiable Functions or Theorem 5.2.3 in Evans, Gariepy: Measure Theory and Fine Properties of Functions.






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            This is Theorem 5.3.4 in W.P.Ziemer: Weakly Differentiable Functions or Theorem 5.2.3 in Evans, Gariepy: Measure Theory and Fine Properties of Functions.






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              This is Theorem 5.3.4 in W.P.Ziemer: Weakly Differentiable Functions or Theorem 5.2.3 in Evans, Gariepy: Measure Theory and Fine Properties of Functions.






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                This is Theorem 5.3.4 in W.P.Ziemer: Weakly Differentiable Functions or Theorem 5.2.3 in Evans, Gariepy: Measure Theory and Fine Properties of Functions.






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                This is Theorem 5.3.4 in W.P.Ziemer: Weakly Differentiable Functions or Theorem 5.2.3 in Evans, Gariepy: Measure Theory and Fine Properties of Functions.







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                answered Jul 21 '18 at 13:42









                Thomas

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