Mixing
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?
fa.functional-analysis real-analysis ap.analysis-of-pdes sobolev-spaces calculus-of-variations
asked 4 hours ago
user111164
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up vote
3
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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?
fa.functional-analysis real-analysis ap.analysis-of-pdes sobolev-spaces calculus-of-variations
asked 4 hours ago
user111164
11312
add a comment |Â
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?
fa.functional-analysis real-analysis ap.analysis-of-pdes sobolev-spaces calculus-of-variations
asked 4 hours ago
user111164
11312
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
fa.functional-analysis real-analysis ap.analysis-of-pdes sobolev-spaces calculus-of-variations
asked 4 hours ago
user111164
11312
asked 4 hours ago
user111164
11312
asked 4 hours ago
user111164
11312
asked 4 hours ago
user111164
11312
asked 4 hours ago
user111164
11312
11312
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add a comment |Â
2 Answers
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2
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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
answered 3 hours ago
Michael Renardy
10.4k13137
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2
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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.
answered 1 hour ago
Piotr Hajlasz
5,31632053
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
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
answered 3 hours ago
Michael Renardy
10.4k13137
add a comment |Â
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
answered 3 hours ago
Michael Renardy
10.4k13137
add a comment |Â
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
answered 3 hours ago
Michael Renardy
10.4k13137
This paper discusses counterexamples to Korn's inequality in $L^1$ spaces:
https://www.mis.mpg.de/preprints/2003/preprint2003_93.pdf
answered 3 hours ago
Michael Renardy
10.4k13137
answered 3 hours ago
Michael Renardy
10.4k13137
answered 3 hours ago
Michael Renardy
10.4k13137
answered 3 hours ago
Michael Renardy
10.4k13137
10.4k13137
add a comment |Â
add a comment |Â
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.
answered 1 hour ago
Piotr Hajlasz
5,31632053
add a comment |Â
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.
answered 1 hour ago
Piotr Hajlasz
5,31632053
add a comment |Â
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.
answered 1 hour ago
Piotr Hajlasz
5,31632053
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.
answered 1 hour ago
Piotr Hajlasz
5,31632053
answered 1 hour ago
Piotr Hajlasz
5,31632053
answered 1 hour ago
Piotr Hajlasz
5,31632053
answered 1 hour ago
Piotr Hajlasz
5,31632053
5,31632053
add a comment |Â
add a comment |Â
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