Why the trace of $u'$ is zero?
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I have question on Theorem $4$ of Evan's book page $340$ or ($6.3$ Regularity).
Evan wrote in the seventh part:
Choose $s >0$ so small that the half-ball $U' := B^0(0,s) cap {y_n >0}$ lies in $Phi(U cap B(x^0,r)).$ Set $V' := B^0(0,s/2) cap {y_n >0}.$ Finally define
$$u'(y) := u(Psi(y)) quad (y in U').$$
It is straightforward to check $u' in H^1 (U')$ and $u' =0 quad text{on }partial U' cap {y_n = 0}. $
I do not know how he get $u' =0 quad text{on }partial U' $. Can someone explain it to me? Thank you.
pde
asked Nov 18 at 23:45
Zack Ni
3,422629
add a comment |
up vote
0
down vote
favorite
I have question on Theorem $4$ of Evan's book page $340$ or ($6.3$ Regularity).
Evan wrote in the seventh part:
Choose $s >0$ so small that the half-ball $U' := B^0(0,s) cap {y_n >0}$ lies in $Phi(U cap B(x^0,r)).$ Set $V' := B^0(0,s/2) cap {y_n >0}.$ Finally define
$$u'(y) := u(Psi(y)) quad (y in U').$$
It is straightforward to check $u' in H^1 (U')$ and $u' =0 quad text{on }partial U' cap {y_n = 0}. $
I do not know how he get $u' =0 quad text{on }partial U' $. Can someone explain it to me? Thank you.
pde
asked Nov 18 at 23:45
Zack Ni
3,422629
$u'$ is derivative. can you use $tilde{u}$ instead
– mathworker21
Nov 19 at 0:09
@mathworker21 Yeah but it will cause great confusion because the book used $u'$
– Zack Ni
Nov 19 at 0:10
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have question on Theorem $4$ of Evan's book page $340$ or ($6.3$ Regularity).
Evan wrote in the seventh part:
Choose $s >0$ so small that the half-ball $U' := B^0(0,s) cap {y_n >0}$ lies in $Phi(U cap B(x^0,r)).$ Set $V' := B^0(0,s/2) cap {y_n >0}.$ Finally define
$$u'(y) := u(Psi(y)) quad (y in U').$$
It is straightforward to check $u' in H^1 (U')$ and $u' =0 quad text{on }partial U' cap {y_n = 0}. $
I do not know how he get $u' =0 quad text{on }partial U' $. Can someone explain it to me? Thank you.
pde
asked Nov 18 at 23:45
Zack Ni
3,422629
I have question on Theorem $4$ of Evan's book page $340$ or ($6.3$ Regularity).
Evan wrote in the seventh part:
Choose $s >0$ so small that the half-ball $U' := B^0(0,s) cap {y_n >0}$ lies in $Phi(U cap B(x^0,r)).$ Set $V' := B^0(0,s/2) cap {y_n >0}.$ Finally define
$$u'(y) := u(Psi(y)) quad (y in U').$$
It is straightforward to check $u' in H^1 (U')$ and $u' =0 quad text{on }partial U' cap {y_n = 0}. $
I do not know how he get $u' =0 quad text{on }partial U' $. Can someone explain it to me? Thank you.
pde
pde
asked Nov 18 at 23:45
Zack Ni
3,422629
asked Nov 18 at 23:45
Zack Ni
3,422629
asked Nov 18 at 23:45
Zack Ni
3,422629
asked Nov 18 at 23:45
Zack Ni
3,422629
asked Nov 18 at 23:45
Zack Ni
3,422629
3,422629
$u'$ is derivative. can you use $tilde{u}$ instead
– mathworker21
Nov 19 at 0:09
@mathworker21 Yeah but it will cause great confusion because the book used $u'$
– Zack Ni
Nov 19 at 0:10
add a comment |
$u'$ is derivative. can you use $tilde{u}$ instead
– mathworker21
Nov 19 at 0:09
@mathworker21 Yeah but it will cause great confusion because the book used $u'$
– Zack Ni
Nov 19 at 0:10
$u'$ is derivative. can you use $tilde{u}$ instead
– mathworker21
Nov 19 at 0:09
$u'$ is derivative. can you use $tilde{u}$ instead
– mathworker21
Nov 19 at 0:09
@mathworker21 Yeah but it will cause great confusion because the book used $u'$
– Zack Ni
Nov 19 at 0:10
@mathworker21 Yeah but it will cause great confusion because the book used $u'$
– Zack Ni
Nov 19 at 0:10
add a comment |
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$u'$ is derivative. can you use $tilde{u}$ instead
– mathworker21
Nov 19 at 0:09
@mathworker21 Yeah but it will cause great confusion because the book used $u'$
– Zack Ni
Nov 19 at 0:10