$BV(Omega)$ is embedded compactly in $L^1 _{mathrm{loc}} (Omega)$
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
edited Nov 26 '18 at 14:08
Davide Giraudo
125k16150260
asked Jul 21 '18 at 11:53
Rosy
1045
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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
edited Nov 26 '18 at 14:08
Davide Giraudo
125k16150260
asked Jul 21 '18 at 11:53
Rosy
1045
add a comment |
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
edited Nov 26 '18 at 14:08
Davide Giraudo
125k16150260
asked Jul 21 '18 at 11:53
Rosy
1045
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
functional-analysis analysis pde bounded-variation regularity-theory-of-pdes
edited Nov 26 '18 at 14:08
Davide Giraudo
125k16150260
asked Jul 21 '18 at 11:53
Rosy
1045
edited Nov 26 '18 at 14:08
Davide Giraudo
125k16150260
asked Jul 21 '18 at 11:53
Rosy
1045
edited Nov 26 '18 at 14:08
Davide Giraudo
125k16150260
edited Nov 26 '18 at 14:08
Davide Giraudo
125k16150260
edited Nov 26 '18 at 14:08
Davide Giraudo
125k16150260
125k16150260
asked Jul 21 '18 at 11:53
Rosy
1045
asked Jul 21 '18 at 11:53
Rosy
1045
asked Jul 21 '18 at 11:53
Rosy
1045
1045
add a comment |
add a comment |
1 Answer
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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.
answered Jul 21 '18 at 13:42
Thomas
16.7k21631
add a comment |
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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.
answered Jul 21 '18 at 13:42
Thomas
16.7k21631
add a comment |
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.
answered Jul 21 '18 at 13:42
Thomas
16.7k21631
add a comment |
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.
answered Jul 21 '18 at 13:42
Thomas
16.7k21631
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.
answered Jul 21 '18 at 13:42
Thomas
16.7k21631
answered Jul 21 '18 at 13:42
Thomas
16.7k21631
answered Jul 21 '18 at 13:42
Thomas
16.7k21631
answered Jul 21 '18 at 13:42
Thomas
16.7k21631
16.7k21631
add a comment |
add a comment |
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