Mixing
Applying Green's formula
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I've been having some trouble proving this, so any help or suggestions would be great. Thanks in advance!
Show using a Green's formula that, for any $u in H^2(Omega)$ satisfying
$$u = frac{partial u}{partial n} = 0$$
we have
$$int_Omega |Delta u|^2 , dx , dy = int_Omega left[(u_{xx})^2 + (u_{yy})^2 - (u_{xy})^2 + (u_{yx})^2 right] , dx , dy$$
My first instinct is to use integration by parts on the left hand side, but I'm a little confused because it says $u in H^2(Omega)$, so doesn't that mean I can't have three derivatives on $u$?
By the way, we can assume that $Omega$ has sufficiently smooth boundary. Also, $frac{partial}{partial n}$ denotes differentiation in the outward normal direction to the boundary $Gamma$.
multivariable-calculus
asked Aug 16 '14 at 21:21
EpicMochi
638413
add a comment |
up vote
1
down vote
favorite
I've been having some trouble proving this, so any help or suggestions would be great. Thanks in advance!
Show using a Green's formula that, for any $u in H^2(Omega)$ satisfying
$$u = frac{partial u}{partial n} = 0$$
we have
$$int_Omega |Delta u|^2 , dx , dy = int_Omega left[(u_{xx})^2 + (u_{yy})^2 - (u_{xy})^2 + (u_{yx})^2 right] , dx , dy$$
My first instinct is to use integration by parts on the left hand side, but I'm a little confused because it says $u in H^2(Omega)$, so doesn't that mean I can't have three derivatives on $u$?
By the way, we can assume that $Omega$ has sufficiently smooth boundary. Also, $frac{partial}{partial n}$ denotes differentiation in the outward normal direction to the boundary $Gamma$.
multivariable-calculus
asked Aug 16 '14 at 21:21
EpicMochi
638413
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I've been having some trouble proving this, so any help or suggestions would be great. Thanks in advance!
Show using a Green's formula that, for any $u in H^2(Omega)$ satisfying
$$u = frac{partial u}{partial n} = 0$$
we have
$$int_Omega |Delta u|^2 , dx , dy = int_Omega left[(u_{xx})^2 + (u_{yy})^2 - (u_{xy})^2 + (u_{yx})^2 right] , dx , dy$$
My first instinct is to use integration by parts on the left hand side, but I'm a little confused because it says $u in H^2(Omega)$, so doesn't that mean I can't have three derivatives on $u$?
By the way, we can assume that $Omega$ has sufficiently smooth boundary. Also, $frac{partial}{partial n}$ denotes differentiation in the outward normal direction to the boundary $Gamma$.
multivariable-calculus
asked Aug 16 '14 at 21:21
EpicMochi
638413
I've been having some trouble proving this, so any help or suggestions would be great. Thanks in advance!
Show using a Green's formula that, for any $u in H^2(Omega)$ satisfying
$$u = frac{partial u}{partial n} = 0$$
we have
$$int_Omega |Delta u|^2 , dx , dy = int_Omega left[(u_{xx})^2 + (u_{yy})^2 - (u_{xy})^2 + (u_{yx})^2 right] , dx , dy$$
My first instinct is to use integration by parts on the left hand side, but I'm a little confused because it says $u in H^2(Omega)$, so doesn't that mean I can't have three derivatives on $u$?
By the way, we can assume that $Omega$ has sufficiently smooth boundary. Also, $frac{partial}{partial n}$ denotes differentiation in the outward normal direction to the boundary $Gamma$.
multivariable-calculus
multivariable-calculus
asked Aug 16 '14 at 21:21
EpicMochi
638413
asked Aug 16 '14 at 21:21
EpicMochi
638413
asked Aug 16 '14 at 21:21
EpicMochi
638413
asked Aug 16 '14 at 21:21
EpicMochi
638413
asked Aug 16 '14 at 21:21
EpicMochi
638413
638413
add a comment |
add a comment |
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