Compactness of the trace operator
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Is it true that for a set $Omega$ with Lipschitz boundary the trace operator $T : H^1(Omega) to L^2(partial Omega)$ is compact? Can you please give a reference?
I found a theorem in Necas' Direct methods in the Theory of Elliptic Equations, which says that if $1<p<N, 1 geq 1/q > 1/p-[1/(N-1)](p-1)/p$ then $W^{1,p}(Omega)$ injects compactly into $L^q(partial Omega)$.
Unfortunately, a case which interests me is $p=q=2$ and $N=2$, which does not seem to fit in the above result. Can you please provide a reference for this case?
sobolev-spaces weak-convergence compact-operators
edited Sep 23 '15 at 17:05
Servaes
21k33792
asked Sep 23 '15 at 17:02
anonymus2345
111
add a comment |
up vote
2
down vote
favorite
Is it true that for a set $Omega$ with Lipschitz boundary the trace operator $T : H^1(Omega) to L^2(partial Omega)$ is compact? Can you please give a reference?
I found a theorem in Necas' Direct methods in the Theory of Elliptic Equations, which says that if $1<p<N, 1 geq 1/q > 1/p-[1/(N-1)](p-1)/p$ then $W^{1,p}(Omega)$ injects compactly into $L^q(partial Omega)$.
Unfortunately, a case which interests me is $p=q=2$ and $N=2$, which does not seem to fit in the above result. Can you please provide a reference for this case?
sobolev-spaces weak-convergence compact-operators
edited Sep 23 '15 at 17:05
Servaes
21k33792
asked Sep 23 '15 at 17:02
anonymus2345
111
Duplicate of Compact embedding into boundary
– user147263
Sep 25 '15 at 6:29
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Is it true that for a set $Omega$ with Lipschitz boundary the trace operator $T : H^1(Omega) to L^2(partial Omega)$ is compact? Can you please give a reference?
I found a theorem in Necas' Direct methods in the Theory of Elliptic Equations, which says that if $1<p<N, 1 geq 1/q > 1/p-[1/(N-1)](p-1)/p$ then $W^{1,p}(Omega)$ injects compactly into $L^q(partial Omega)$.
Unfortunately, a case which interests me is $p=q=2$ and $N=2$, which does not seem to fit in the above result. Can you please provide a reference for this case?
sobolev-spaces weak-convergence compact-operators
edited Sep 23 '15 at 17:05
Servaes
21k33792
asked Sep 23 '15 at 17:02
anonymus2345
111
Is it true that for a set $Omega$ with Lipschitz boundary the trace operator $T : H^1(Omega) to L^2(partial Omega)$ is compact? Can you please give a reference?
I found a theorem in Necas' Direct methods in the Theory of Elliptic Equations, which says that if $1<p<N, 1 geq 1/q > 1/p-[1/(N-1)](p-1)/p$ then $W^{1,p}(Omega)$ injects compactly into $L^q(partial Omega)$.
Unfortunately, a case which interests me is $p=q=2$ and $N=2$, which does not seem to fit in the above result. Can you please provide a reference for this case?
sobolev-spaces weak-convergence compact-operators
sobolev-spaces weak-convergence compact-operators
edited Sep 23 '15 at 17:05
Servaes
21k33792
asked Sep 23 '15 at 17:02
anonymus2345
111
edited Sep 23 '15 at 17:05
Servaes
21k33792
asked Sep 23 '15 at 17:02
anonymus2345
111
edited Sep 23 '15 at 17:05
Servaes
21k33792
edited Sep 23 '15 at 17:05
Servaes
21k33792
edited Sep 23 '15 at 17:05
Servaes
21k33792
21k33792
asked Sep 23 '15 at 17:02
anonymus2345
111
asked Sep 23 '15 at 17:02
anonymus2345
111
asked Sep 23 '15 at 17:02
anonymus2345
111
111
Duplicate of Compact embedding into boundary
– user147263
Sep 25 '15 at 6:29
add a comment |
Duplicate of Compact embedding into boundary
– user147263
Sep 25 '15 at 6:29
Duplicate of Compact embedding into boundary
– user147263
Sep 25 '15 at 6:29
Duplicate of Compact embedding into boundary
– user147263
Sep 25 '15 at 6:29
add a comment |
1 Answer
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It does seem to fit in the above result, since $H^1 subset W^{1,2-varepsilon}$. It is better since if $p>2$, it is Holder continuous.
answered Nov 19 at 16:09
user1776247
12
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
It does seem to fit in the above result, since $H^1 subset W^{1,2-varepsilon}$. It is better since if $p>2$, it is Holder continuous.
answered Nov 19 at 16:09
user1776247
12
add a comment |
up vote
0
down vote
It does seem to fit in the above result, since $H^1 subset W^{1,2-varepsilon}$. It is better since if $p>2$, it is Holder continuous.
answered Nov 19 at 16:09
user1776247
12
add a comment |
up vote
0
down vote
up vote
0
down vote
It does seem to fit in the above result, since $H^1 subset W^{1,2-varepsilon}$. It is better since if $p>2$, it is Holder continuous.
answered Nov 19 at 16:09
user1776247
12
It does seem to fit in the above result, since $H^1 subset W^{1,2-varepsilon}$. It is better since if $p>2$, it is Holder continuous.
answered Nov 19 at 16:09
user1776247
12
answered Nov 19 at 16:09
user1776247
12
answered Nov 19 at 16:09
user1776247
12
answered Nov 19 at 16:09
user1776247
12
12
add a comment |
add a comment |
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Duplicate of Compact embedding into boundary
– user147263
Sep 25 '15 at 6:29