About pseudo-differential operators
$begingroup$
Let $Omega$ be an open and connect subset of $mathbb{R}^2$,we denote by $partial Omega$ its boundary the latter is supposed to be smooth ($mathcal{C}^infty)$, its outword normal vector is denoted by $n$. Let $f: Omega mapsto mathbb{R}$ such that $f(x)geq alpha > 0$. Now, let $A : mathbb{H}^{1/2}(partial Omega)mapsto mathbb{H}^{-1/2}(partial Omega) $.
Such that $A(varphi)=frac{partial u }{n} $, with $u$ is the unique solution in $mathbb{H}^1(Omega)$ of
$div(fnabla u)=0$ and $u_{|partialOmega}=varphi$.
Can I say that $A$ is a pseudo-differential Operator ?
pde regularity-theory-of-pdes elliptic-operators pseudo-differential-operators fractional-sobolev-spaces
asked Jan 14 at 20:39
MahranMahran
634
$endgroup$
add a comment |
$begingroup$
Let $Omega$ be an open and connect subset of $mathbb{R}^2$,we denote by $partial Omega$ its boundary the latter is supposed to be smooth ($mathcal{C}^infty)$, its outword normal vector is denoted by $n$. Let $f: Omega mapsto mathbb{R}$ such that $f(x)geq alpha > 0$. Now, let $A : mathbb{H}^{1/2}(partial Omega)mapsto mathbb{H}^{-1/2}(partial Omega) $.
Such that $A(varphi)=frac{partial u }{n} $, with $u$ is the unique solution in $mathbb{H}^1(Omega)$ of
$div(fnabla u)=0$ and $u_{|partialOmega}=varphi$.
Can I say that $A$ is a pseudo-differential Operator ?
pde regularity-theory-of-pdes elliptic-operators pseudo-differential-operators fractional-sobolev-spaces
asked Jan 14 at 20:39
MahranMahran
634
$endgroup$
add a comment |
$begingroup$
Let $Omega$ be an open and connect subset of $mathbb{R}^2$,we denote by $partial Omega$ its boundary the latter is supposed to be smooth ($mathcal{C}^infty)$, its outword normal vector is denoted by $n$. Let $f: Omega mapsto mathbb{R}$ such that $f(x)geq alpha > 0$. Now, let $A : mathbb{H}^{1/2}(partial Omega)mapsto mathbb{H}^{-1/2}(partial Omega) $.
Such that $A(varphi)=frac{partial u }{n} $, with $u$ is the unique solution in $mathbb{H}^1(Omega)$ of
$div(fnabla u)=0$ and $u_{|partialOmega}=varphi$.
Can I say that $A$ is a pseudo-differential Operator ?
pde regularity-theory-of-pdes elliptic-operators pseudo-differential-operators fractional-sobolev-spaces
asked Jan 14 at 20:39
MahranMahran
634
$endgroup$
Let $Omega$ be an open and connect subset of $mathbb{R}^2$,we denote by $partial Omega$ its boundary the latter is supposed to be smooth ($mathcal{C}^infty)$, its outword normal vector is denoted by $n$. Let $f: Omega mapsto mathbb{R}$ such that $f(x)geq alpha > 0$. Now, let $A : mathbb{H}^{1/2}(partial Omega)mapsto mathbb{H}^{-1/2}(partial Omega) $.
Such that $A(varphi)=frac{partial u }{n} $, with $u$ is the unique solution in $mathbb{H}^1(Omega)$ of
$div(fnabla u)=0$ and $u_{|partialOmega}=varphi$.
Can I say that $A$ is a pseudo-differential Operator ?
pde regularity-theory-of-pdes elliptic-operators pseudo-differential-operators fractional-sobolev-spaces
pde regularity-theory-of-pdes elliptic-operators pseudo-differential-operators fractional-sobolev-spaces
asked Jan 14 at 20:39
MahranMahran
634
asked Jan 14 at 20:39
MahranMahran
634
edited Jan 16 at 13:07
Mahran
asked Jan 14 at 20:39
MahranMahran
634
asked Jan 14 at 20:39
MahranMahran
634
asked Jan 14 at 20:39
MahranMahran
634
634
add a comment |
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
Finally, The answer is yes. In deed the result remains true if we remplace $div(f nabla .)$by any other second order elliptic opertor. Morover, $A$ is of order one. The proof can be found here : https://arxiv.org/pdf/1212.6785.pdf
answered Jan 15 at 14:48
MahranMahran
634
$endgroup$
add a comment |
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1 Answer
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1 Answer
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oldest
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$begingroup$
Finally, The answer is yes. In deed the result remains true if we remplace $div(f nabla .)$by any other second order elliptic opertor. Morover, $A$ is of order one. The proof can be found here : https://arxiv.org/pdf/1212.6785.pdf
answered Jan 15 at 14:48
MahranMahran
634
$endgroup$
add a comment |
$begingroup$
Finally, The answer is yes. In deed the result remains true if we remplace $div(f nabla .)$by any other second order elliptic opertor. Morover, $A$ is of order one. The proof can be found here : https://arxiv.org/pdf/1212.6785.pdf
answered Jan 15 at 14:48
MahranMahran
634
$endgroup$
add a comment |
$begingroup$
Finally, The answer is yes. In deed the result remains true if we remplace $div(f nabla .)$by any other second order elliptic opertor. Morover, $A$ is of order one. The proof can be found here : https://arxiv.org/pdf/1212.6785.pdf
answered Jan 15 at 14:48
MahranMahran
634
$endgroup$
Finally, The answer is yes. In deed the result remains true if we remplace $div(f nabla .)$by any other second order elliptic opertor. Morover, $A$ is of order one. The proof can be found here : https://arxiv.org/pdf/1212.6785.pdf
answered Jan 15 at 14:48
MahranMahran
634
answered Jan 15 at 14:48
MahranMahran
634
answered Jan 15 at 14:48
MahranMahran
634
answered Jan 15 at 14:48
MahranMahran
634
634
add a comment |
add a comment |
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