Mixing
Integral Convergence Estimate with cut-off function
$begingroup$
Let $u in L^{infty}(Omega)cap H_{0}^{1}(Omega)$ and define a cut-off function $eta_{R} in C_{0}^{infty}(mathbb{R})$ for $Omega subset mathbb{R}$ an unbounded (interval) domain as follows
$$eta_{R}(x):=begin{cases}
1 &, text{if }|x|leq R\
0 &, text{if } xgeq R+1text{ or }xleq-R-1\
0<eta_{R}(x)<1 &, text{ other }x
end{cases}
$$
Now define a function $u_R = ueta_{R}$. Define a functional $I[u] = ||u||_{H_{0}^{1}(Omega)}^{2} - ||u||_{L^{p}(Omega)}^{p}$ for $2leq p <infty$.
I would like to show that $I[u_{R}] = I[u]+o(1)$ as $Rtoinfty$. This is my attempt so far:
begin{align*}
|I[u]-I[u_{R}]| &leq |, ||u||_{H_{0}^{1}(Omega)}^{2}-||u_{R}||_{H_{0}^{1}(Omega)}^{2},| + |, ||u||_{L^{p}(Omega)}^{p} - ||u_{R}||_{L^{p}(Omega)}^{p},| \
&leq |, ||u||_{L^{2}(Omega)}^{2} - ||u||_{L^{2}(Omega)}^{2},| + |, ||nabla u||_{L^{2}(Omega)}^{2}- ||nabla u_{R}||_{L^{2}(Omega)}^{2},|+ |, ||u||_{L^{p}(Omega)}^{p} - ||u_{R}||_{L^{p}(Omega)}^{p},|\
&= A + B + C
end{align*}
So, I would like to see the estimate one by one. First, I will start from $A$. Observe that
begin{align*}
A &leq int_{Omega}|u^{2}eta_{R}^{2}-u^{2}|dx \
&leq suplimits_{Omega}u^{2}int_{Omega}|1-eta_{R}^{2}|dx\
&leq 2suplimits_{Omega}u^{2}int_{Omega}|1-eta_{R}|dx
end{align*}
Similarly for $C$, I will obtain the following estimate
$$C leq psuplimits_{Omega}|u|^{p}int_{Omega}|1-eta_{R}|dx$$
Finally, for $B$, I would obtain
$$B leq int_{Omega}|partial_{x}(ueta_{R})|^{2} - (partial_{x}u)^{2}|dx$$
So, I have two main problems here.
1. How to ensure that $int_{Omega}|1-eta_{R}|dx to 0$ as $Rtoinfty$ rigorously?
2. What should I do to estimate $B$ since I am not sure what to do with the term $partial_{x}eta_{R}$ here.
Any help will much be appreciated!
functional-analysis convergence
asked Jan 11 at 3:51
Evan William ChandraEvan William Chandra
577313
$endgroup$
add a comment |
$begingroup$
Let $u in L^{infty}(Omega)cap H_{0}^{1}(Omega)$ and define a cut-off function $eta_{R} in C_{0}^{infty}(mathbb{R})$ for $Omega subset mathbb{R}$ an unbounded (interval) domain as follows
$$eta_{R}(x):=begin{cases}
1 &, text{if }|x|leq R\
0 &, text{if } xgeq R+1text{ or }xleq-R-1\
0<eta_{R}(x)<1 &, text{ other }x
end{cases}
$$
Now define a function $u_R = ueta_{R}$. Define a functional $I[u] = ||u||_{H_{0}^{1}(Omega)}^{2} - ||u||_{L^{p}(Omega)}^{p}$ for $2leq p <infty$.
I would like to show that $I[u_{R}] = I[u]+o(1)$ as $Rtoinfty$. This is my attempt so far:
begin{align*}
|I[u]-I[u_{R}]| &leq |, ||u||_{H_{0}^{1}(Omega)}^{2}-||u_{R}||_{H_{0}^{1}(Omega)}^{2},| + |, ||u||_{L^{p}(Omega)}^{p} - ||u_{R}||_{L^{p}(Omega)}^{p},| \
&leq |, ||u||_{L^{2}(Omega)}^{2} - ||u||_{L^{2}(Omega)}^{2},| + |, ||nabla u||_{L^{2}(Omega)}^{2}- ||nabla u_{R}||_{L^{2}(Omega)}^{2},|+ |, ||u||_{L^{p}(Omega)}^{p} - ||u_{R}||_{L^{p}(Omega)}^{p},|\
&= A + B + C
end{align*}
So, I would like to see the estimate one by one. First, I will start from $A$. Observe that
begin{align*}
A &leq int_{Omega}|u^{2}eta_{R}^{2}-u^{2}|dx \
&leq suplimits_{Omega}u^{2}int_{Omega}|1-eta_{R}^{2}|dx\
&leq 2suplimits_{Omega}u^{2}int_{Omega}|1-eta_{R}|dx
end{align*}
Similarly for $C$, I will obtain the following estimate
$$C leq psuplimits_{Omega}|u|^{p}int_{Omega}|1-eta_{R}|dx$$
Finally, for $B$, I would obtain
$$B leq int_{Omega}|partial_{x}(ueta_{R})|^{2} - (partial_{x}u)^{2}|dx$$
So, I have two main problems here.
1. How to ensure that $int_{Omega}|1-eta_{R}|dx to 0$ as $Rtoinfty$ rigorously?
2. What should I do to estimate $B$ since I am not sure what to do with the term $partial_{x}eta_{R}$ here.
Any help will much be appreciated!
functional-analysis convergence
asked Jan 11 at 3:51
Evan William ChandraEvan William Chandra
577313
$endgroup$
add a comment |
$begingroup$
Let $u in L^{infty}(Omega)cap H_{0}^{1}(Omega)$ and define a cut-off function $eta_{R} in C_{0}^{infty}(mathbb{R})$ for $Omega subset mathbb{R}$ an unbounded (interval) domain as follows
$$eta_{R}(x):=begin{cases}
1 &, text{if }|x|leq R\
0 &, text{if } xgeq R+1text{ or }xleq-R-1\
0<eta_{R}(x)<1 &, text{ other }x
end{cases}
$$
Now define a function $u_R = ueta_{R}$. Define a functional $I[u] = ||u||_{H_{0}^{1}(Omega)}^{2} - ||u||_{L^{p}(Omega)}^{p}$ for $2leq p <infty$.
I would like to show that $I[u_{R}] = I[u]+o(1)$ as $Rtoinfty$. This is my attempt so far:
begin{align*}
|I[u]-I[u_{R}]| &leq |, ||u||_{H_{0}^{1}(Omega)}^{2}-||u_{R}||_{H_{0}^{1}(Omega)}^{2},| + |, ||u||_{L^{p}(Omega)}^{p} - ||u_{R}||_{L^{p}(Omega)}^{p},| \
&leq |, ||u||_{L^{2}(Omega)}^{2} - ||u||_{L^{2}(Omega)}^{2},| + |, ||nabla u||_{L^{2}(Omega)}^{2}- ||nabla u_{R}||_{L^{2}(Omega)}^{2},|+ |, ||u||_{L^{p}(Omega)}^{p} - ||u_{R}||_{L^{p}(Omega)}^{p},|\
&= A + B + C
end{align*}
So, I would like to see the estimate one by one. First, I will start from $A$. Observe that
begin{align*}
A &leq int_{Omega}|u^{2}eta_{R}^{2}-u^{2}|dx \
&leq suplimits_{Omega}u^{2}int_{Omega}|1-eta_{R}^{2}|dx\
&leq 2suplimits_{Omega}u^{2}int_{Omega}|1-eta_{R}|dx
end{align*}
Similarly for $C$, I will obtain the following estimate
$$C leq psuplimits_{Omega}|u|^{p}int_{Omega}|1-eta_{R}|dx$$
Finally, for $B$, I would obtain
$$B leq int_{Omega}|partial_{x}(ueta_{R})|^{2} - (partial_{x}u)^{2}|dx$$
So, I have two main problems here.
1. How to ensure that $int_{Omega}|1-eta_{R}|dx to 0$ as $Rtoinfty$ rigorously?
2. What should I do to estimate $B$ since I am not sure what to do with the term $partial_{x}eta_{R}$ here.
Any help will much be appreciated!
functional-analysis convergence
asked Jan 11 at 3:51
Evan William ChandraEvan William Chandra
577313
$endgroup$
Let $u in L^{infty}(Omega)cap H_{0}^{1}(Omega)$ and define a cut-off function $eta_{R} in C_{0}^{infty}(mathbb{R})$ for $Omega subset mathbb{R}$ an unbounded (interval) domain as follows
$$eta_{R}(x):=begin{cases}
1 &, text{if }|x|leq R\
0 &, text{if } xgeq R+1text{ or }xleq-R-1\
0<eta_{R}(x)<1 &, text{ other }x
end{cases}
$$
Now define a function $u_R = ueta_{R}$. Define a functional $I[u] = ||u||_{H_{0}^{1}(Omega)}^{2} - ||u||_{L^{p}(Omega)}^{p}$ for $2leq p <infty$.
I would like to show that $I[u_{R}] = I[u]+o(1)$ as $Rtoinfty$. This is my attempt so far:
begin{align*}
|I[u]-I[u_{R}]| &leq |, ||u||_{H_{0}^{1}(Omega)}^{2}-||u_{R}||_{H_{0}^{1}(Omega)}^{2},| + |, ||u||_{L^{p}(Omega)}^{p} - ||u_{R}||_{L^{p}(Omega)}^{p},| \
&leq |, ||u||_{L^{2}(Omega)}^{2} - ||u||_{L^{2}(Omega)}^{2},| + |, ||nabla u||_{L^{2}(Omega)}^{2}- ||nabla u_{R}||_{L^{2}(Omega)}^{2},|+ |, ||u||_{L^{p}(Omega)}^{p} - ||u_{R}||_{L^{p}(Omega)}^{p},|\
&= A + B + C
end{align*}
So, I would like to see the estimate one by one. First, I will start from $A$. Observe that
begin{align*}
A &leq int_{Omega}|u^{2}eta_{R}^{2}-u^{2}|dx \
&leq suplimits_{Omega}u^{2}int_{Omega}|1-eta_{R}^{2}|dx\
&leq 2suplimits_{Omega}u^{2}int_{Omega}|1-eta_{R}|dx
end{align*}
Similarly for $C$, I will obtain the following estimate
$$C leq psuplimits_{Omega}|u|^{p}int_{Omega}|1-eta_{R}|dx$$
Finally, for $B$, I would obtain
$$B leq int_{Omega}|partial_{x}(ueta_{R})|^{2} - (partial_{x}u)^{2}|dx$$
So, I have two main problems here.
1. How to ensure that $int_{Omega}|1-eta_{R}|dx to 0$ as $Rtoinfty$ rigorously?
2. What should I do to estimate $B$ since I am not sure what to do with the term $partial_{x}eta_{R}$ here.
Any help will much be appreciated!
functional-analysis convergence
functional-analysis convergence
asked Jan 11 at 3:51
Evan William ChandraEvan William Chandra
577313
asked Jan 11 at 3:51
Evan William ChandraEvan William Chandra
577313
asked Jan 11 at 3:51
Evan William ChandraEvan William Chandra
577313
asked Jan 11 at 3:51
Evan William ChandraEvan William Chandra
577313
asked Jan 11 at 3:51
Evan William ChandraEvan William Chandra
577313
577313
add a comment |
add a comment |
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