Mixing
$BV(Omega)$ is embedded compactly in $L^1 _loc (Omega)$
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Definition:
We say that a function $u: Omega rightarrow mathbbR$ is a function of bounded variation iff $uin L^1(Omega)$ and $supintlimits_Omegau divphi : phi in C_c(Omega, mathbbR^d), ||phi||_infty leq 1} < +infty$.
By definition it is clear that $BV(Omega)subset L^1(Omega)$
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 regularity-theory-of-pdes
asked Jul 21 at 11:53
Rosy
643
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up vote
1
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Definition:
We say that a function $u: Omega rightarrow mathbbR$ is a function of bounded variation iff $uin L^1(Omega)$ and $supintlimits_Omegau divphi : phi in C_c(Omega, mathbbR^d), ||phi||_infty leq 1} < +infty$.
By definition it is clear that $BV(Omega)subset L^1(Omega)$
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 regularity-theory-of-pdes
asked Jul 21 at 11:53
Rosy
643
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Definition:
We say that a function $u: Omega rightarrow mathbbR$ is a function of bounded variation iff $uin L^1(Omega)$ and $supintlimits_Omegau divphi : phi in C_c(Omega, mathbbR^d), ||phi||_infty leq 1} < +infty$.
By definition it is clear that $BV(Omega)subset L^1(Omega)$
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 regularity-theory-of-pdes
asked Jul 21 at 11:53
Rosy
643
Definition:
We say that a function $u: Omega rightarrow mathbbR$ is a function of bounded variation iff $uin L^1(Omega)$ and $supintlimits_Omegau divphi : phi in C_c(Omega, mathbbR^d), ||phi||_infty leq 1} < +infty$.
By definition it is clear that $BV(Omega)subset L^1(Omega)$
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 regularity-theory-of-pdes
asked Jul 21 at 11:53
Rosy
643
asked Jul 21 at 11:53
Rosy
643
asked Jul 21 at 11:53
Rosy
643
asked Jul 21 at 11:53
Rosy
643
643
add a comment |Â
add a comment |Â
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 at 13:42
Thomas
15.7k21429
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
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 at 13:42
Thomas
15.7k21429
add a comment |Â
up vote
0
down vote
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 at 13:42
Thomas
15.7k21429
add a comment |Â
up vote
0
down vote
up vote
0
down vote
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 at 13:42
Thomas
15.7k21429
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 at 13:42
Thomas
15.7k21429
answered Jul 21 at 13:42
Thomas
15.7k21429
answered Jul 21 at 13:42
Thomas
15.7k21429
answered Jul 21 at 13:42
Thomas
15.7k21429
15.7k21429
add a comment |Â
add a comment |Â
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