$L^2$ norm of a PDE
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Iam working on the following problem:
Let $Omega subseteq mathbb{R}^n$ and $v$ be the solution of $(-Delta+q-lambda)v=F$ on $Omega$ such that $q$ is bounded and $v=exp(-ax)u$ where $uin H_{0}^{1}(Omega)cap H^{2}(Omega)$
For a large $Xin mathbb{R}$, we define a cut-off function $chi_{X}$ on $Omega_{X}$:
$chi_{X}=begin{cases} 1 & text{on }[0,X-1]\ 1leqchi_{X}leq0 & text{on} (X-1,X)\
0 & text{on} [X,infty)end{cases}$
Iam trying to find $L^2(Omega_{X})$-norm of $(chi_{X}-1)(-Delta v)+(chi_{X}-1)(q-lambda)v$ ..
I startred by the following:
$|(chi_{X}-1)(-Delta v)+(chi_{X}-1)(q-lambda)v|^2_{L^2(Omega_{X})}=\ |(chi_{X}-1)(-Delta+q-lambda)v|^2_{L^2(Omega_{X})}=\ int_{Omega_{X}}(chi_{X}-1)^2((-Delta+q-lambda)v)^2$
Now Iam not sure how I can finish the calculation, I know $(chi_{X}-1)$ is bounded on $Omega_{X}$ hence
$ int_{Omega_{X}}(chi_{X}-1)^2((-Delta+q-lambda)v)^2= int_{Omega_{X}}|chi_{X}-1|^2((-Delta+q-lambda)v)^2leq C int_{Omega_{X}}((-Delta+q-lambda)v)^2$
Are these steps correct?
and how I can find the $L^2$ norm on $Omega_{X}$ for the laplacian term?
functional-analysis operator-theory norm
asked Nov 22 at 16:28
S.N.A
887
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up vote
1
down vote
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Iam working on the following problem:
Let $Omega subseteq mathbb{R}^n$ and $v$ be the solution of $(-Delta+q-lambda)v=F$ on $Omega$ such that $q$ is bounded and $v=exp(-ax)u$ where $uin H_{0}^{1}(Omega)cap H^{2}(Omega)$
For a large $Xin mathbb{R}$, we define a cut-off function $chi_{X}$ on $Omega_{X}$:
$chi_{X}=begin{cases} 1 & text{on }[0,X-1]\ 1leqchi_{X}leq0 & text{on} (X-1,X)\
0 & text{on} [X,infty)end{cases}$
Iam trying to find $L^2(Omega_{X})$-norm of $(chi_{X}-1)(-Delta v)+(chi_{X}-1)(q-lambda)v$ ..
I startred by the following:
$|(chi_{X}-1)(-Delta v)+(chi_{X}-1)(q-lambda)v|^2_{L^2(Omega_{X})}=\ |(chi_{X}-1)(-Delta+q-lambda)v|^2_{L^2(Omega_{X})}=\ int_{Omega_{X}}(chi_{X}-1)^2((-Delta+q-lambda)v)^2$
Now Iam not sure how I can finish the calculation, I know $(chi_{X}-1)$ is bounded on $Omega_{X}$ hence
$ int_{Omega_{X}}(chi_{X}-1)^2((-Delta+q-lambda)v)^2= int_{Omega_{X}}|chi_{X}-1|^2((-Delta+q-lambda)v)^2leq C int_{Omega_{X}}((-Delta+q-lambda)v)^2$
Are these steps correct?
and how I can find the $L^2$ norm on $Omega_{X}$ for the laplacian term?
functional-analysis operator-theory norm
asked Nov 22 at 16:28
S.N.A
887
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Iam working on the following problem:
Let $Omega subseteq mathbb{R}^n$ and $v$ be the solution of $(-Delta+q-lambda)v=F$ on $Omega$ such that $q$ is bounded and $v=exp(-ax)u$ where $uin H_{0}^{1}(Omega)cap H^{2}(Omega)$
For a large $Xin mathbb{R}$, we define a cut-off function $chi_{X}$ on $Omega_{X}$:
$chi_{X}=begin{cases} 1 & text{on }[0,X-1]\ 1leqchi_{X}leq0 & text{on} (X-1,X)\
0 & text{on} [X,infty)end{cases}$
Iam trying to find $L^2(Omega_{X})$-norm of $(chi_{X}-1)(-Delta v)+(chi_{X}-1)(q-lambda)v$ ..
I startred by the following:
$|(chi_{X}-1)(-Delta v)+(chi_{X}-1)(q-lambda)v|^2_{L^2(Omega_{X})}=\ |(chi_{X}-1)(-Delta+q-lambda)v|^2_{L^2(Omega_{X})}=\ int_{Omega_{X}}(chi_{X}-1)^2((-Delta+q-lambda)v)^2$
Now Iam not sure how I can finish the calculation, I know $(chi_{X}-1)$ is bounded on $Omega_{X}$ hence
$ int_{Omega_{X}}(chi_{X}-1)^2((-Delta+q-lambda)v)^2= int_{Omega_{X}}|chi_{X}-1|^2((-Delta+q-lambda)v)^2leq C int_{Omega_{X}}((-Delta+q-lambda)v)^2$
Are these steps correct?
and how I can find the $L^2$ norm on $Omega_{X}$ for the laplacian term?
functional-analysis operator-theory norm
asked Nov 22 at 16:28
S.N.A
887
Iam working on the following problem:
Let $Omega subseteq mathbb{R}^n$ and $v$ be the solution of $(-Delta+q-lambda)v=F$ on $Omega$ such that $q$ is bounded and $v=exp(-ax)u$ where $uin H_{0}^{1}(Omega)cap H^{2}(Omega)$
For a large $Xin mathbb{R}$, we define a cut-off function $chi_{X}$ on $Omega_{X}$:
$chi_{X}=begin{cases} 1 & text{on }[0,X-1]\ 1leqchi_{X}leq0 & text{on} (X-1,X)\
0 & text{on} [X,infty)end{cases}$
Iam trying to find $L^2(Omega_{X})$-norm of $(chi_{X}-1)(-Delta v)+(chi_{X}-1)(q-lambda)v$ ..
I startred by the following:
$|(chi_{X}-1)(-Delta v)+(chi_{X}-1)(q-lambda)v|^2_{L^2(Omega_{X})}=\ |(chi_{X}-1)(-Delta+q-lambda)v|^2_{L^2(Omega_{X})}=\ int_{Omega_{X}}(chi_{X}-1)^2((-Delta+q-lambda)v)^2$
Now Iam not sure how I can finish the calculation, I know $(chi_{X}-1)$ is bounded on $Omega_{X}$ hence
$ int_{Omega_{X}}(chi_{X}-1)^2((-Delta+q-lambda)v)^2= int_{Omega_{X}}|chi_{X}-1|^2((-Delta+q-lambda)v)^2leq C int_{Omega_{X}}((-Delta+q-lambda)v)^2$
Are these steps correct?
and how I can find the $L^2$ norm on $Omega_{X}$ for the laplacian term?
functional-analysis operator-theory norm
functional-analysis operator-theory norm
asked Nov 22 at 16:28
S.N.A
887
asked Nov 22 at 16:28
S.N.A
887
asked Nov 22 at 16:28
S.N.A
887
asked Nov 22 at 16:28
S.N.A
887
asked Nov 22 at 16:28
S.N.A
887
887
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