In which sense is u a solution for (1) on $ Omega=B_1(0) $?
Clash Royale CLAN TAG#URR8PPP
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Consider $$ beginequation
begincases
-Deltaomega=f on Omega \
omega=0 on partialOmega
endcases (1)endequation $$
Definition 1 : $$ Let uin W^1,2_0(Omega) such that \
int_Omega(nabla u*nablaphi-fphi)dx=0 for all phiin W^1,2_0(Omega) , $$
$$ then u is called a weak solution for (1) on Omega .$$
$$ \ $$
Definition 2 :$$ Let uin C_0(overline Omega) such that \
int_Omega(- u*Deltaphi-fphi)dx=0 for all phiin C^infty_0(Omega) ,$$
$$ then u is called a distributional solution for (1) on Omega .$$
$$ \ $$
I'm not really sure , but I think u is a distributional solution for (1) . Using a few times partial Integration and keeping in mind that the integral on the boundary disappears and that u is continuously differentiable I get the following :
$$ -int_B_1(0)uDeltaphi dx=-int_partial B_1(0)unablaphinu dsigma_x+int_B_1(0)nabla u nabla phi dx\
=int_partial B_1(0)nabla u phinu dsigma_x-int_B_1(0)Delta uphi dx=-int_B_1(0)Delta uphi dx=int_B_1(0)fphi dx . $$
I forgot something to mention $$ u(x_1,x_2)= x_1x_2(1-sqrtx_1^2+x_2^2) , where (x_1,x_2)in overlineB_1(0)subset mathbbR^2$$
pde
asked Sep 10 at 18:14
Matillo
114
add a comment |Â
up vote
1
down vote
favorite
Consider $$ beginequation
begincases
-Deltaomega=f on Omega \
omega=0 on partialOmega
endcases (1)endequation $$
Definition 1 : $$ Let uin W^1,2_0(Omega) such that \
int_Omega(nabla u*nablaphi-fphi)dx=0 for all phiin W^1,2_0(Omega) , $$
$$ then u is called a weak solution for (1) on Omega .$$
$$ \ $$
Definition 2 :$$ Let uin C_0(overline Omega) such that \
int_Omega(- u*Deltaphi-fphi)dx=0 for all phiin C^infty_0(Omega) ,$$
$$ then u is called a distributional solution for (1) on Omega .$$
$$ \ $$
I'm not really sure , but I think u is a distributional solution for (1) . Using a few times partial Integration and keeping in mind that the integral on the boundary disappears and that u is continuously differentiable I get the following :
$$ -int_B_1(0)uDeltaphi dx=-int_partial B_1(0)unablaphinu dsigma_x+int_B_1(0)nabla u nabla phi dx\
=int_partial B_1(0)nabla u phinu dsigma_x-int_B_1(0)Delta uphi dx=-int_B_1(0)Delta uphi dx=int_B_1(0)fphi dx . $$
I forgot something to mention $$ u(x_1,x_2)= x_1x_2(1-sqrtx_1^2+x_2^2) , where (x_1,x_2)in overlineB_1(0)subset mathbbR^2$$
pde
asked Sep 10 at 18:14
Matillo
114
I did a quick calculation, and it appears that $uin W^1,2_0$ for the $u$ that you provided. This may mean it is a weak solution. Someone may need to confirm, though.
– MasterYoda
Sep 10 at 20:46
I don't really understand what I've to do if $$ u in W^1,2_0 $$ . Well then it follows that $$ u in B_1(0)0 $$ is a distributional solution for (1) .
– Matillo
Sep 11 at 6:17
I don't really understand what I've to do if $$ u in W^1,2_0 $$ . In fact I computed that u is in $$ W^3,p(B_1(0)) cap W^4,1(B_1(0)) $$ .
– Matillo
Sep 11 at 6:36
Can someone explain me how I can show that u is a distributional solution for (1) on $$ B_1(0)diagdown (0,0) $$ ? I know that $$ uin C^infty(B_1(0)diagdown(0,0)) $$ .
– Matillo
Sep 11 at 14:11
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Consider $$ beginequation
begincases
-Deltaomega=f on Omega \
omega=0 on partialOmega
endcases (1)endequation $$
Definition 1 : $$ Let uin W^1,2_0(Omega) such that \
int_Omega(nabla u*nablaphi-fphi)dx=0 for all phiin W^1,2_0(Omega) , $$
$$ then u is called a weak solution for (1) on Omega .$$
$$ \ $$
Definition 2 :$$ Let uin C_0(overline Omega) such that \
int_Omega(- u*Deltaphi-fphi)dx=0 for all phiin C^infty_0(Omega) ,$$
$$ then u is called a distributional solution for (1) on Omega .$$
$$ \ $$
I'm not really sure , but I think u is a distributional solution for (1) . Using a few times partial Integration and keeping in mind that the integral on the boundary disappears and that u is continuously differentiable I get the following :
$$ -int_B_1(0)uDeltaphi dx=-int_partial B_1(0)unablaphinu dsigma_x+int_B_1(0)nabla u nabla phi dx\
=int_partial B_1(0)nabla u phinu dsigma_x-int_B_1(0)Delta uphi dx=-int_B_1(0)Delta uphi dx=int_B_1(0)fphi dx . $$
I forgot something to mention $$ u(x_1,x_2)= x_1x_2(1-sqrtx_1^2+x_2^2) , where (x_1,x_2)in overlineB_1(0)subset mathbbR^2$$
pde
asked Sep 10 at 18:14
Matillo
114
Consider $$ beginequation
begincases
-Deltaomega=f on Omega \
omega=0 on partialOmega
endcases (1)endequation $$
Definition 1 : $$ Let uin W^1,2_0(Omega) such that \
int_Omega(nabla u*nablaphi-fphi)dx=0 for all phiin W^1,2_0(Omega) , $$
$$ then u is called a weak solution for (1) on Omega .$$
$$ \ $$
Definition 2 :$$ Let uin C_0(overline Omega) such that \
int_Omega(- u*Deltaphi-fphi)dx=0 for all phiin C^infty_0(Omega) ,$$
$$ then u is called a distributional solution for (1) on Omega .$$
$$ \ $$
I'm not really sure , but I think u is a distributional solution for (1) . Using a few times partial Integration and keeping in mind that the integral on the boundary disappears and that u is continuously differentiable I get the following :
$$ -int_B_1(0)uDeltaphi dx=-int_partial B_1(0)unablaphinu dsigma_x+int_B_1(0)nabla u nabla phi dx\
=int_partial B_1(0)nabla u phinu dsigma_x-int_B_1(0)Delta uphi dx=-int_B_1(0)Delta uphi dx=int_B_1(0)fphi dx . $$
I forgot something to mention $$ u(x_1,x_2)= x_1x_2(1-sqrtx_1^2+x_2^2) , where (x_1,x_2)in overlineB_1(0)subset mathbbR^2$$
pde
pde
asked Sep 10 at 18:14
Matillo
114
asked Sep 10 at 18:14
Matillo
114
edited Sep 10 at 18:44
asked Sep 10 at 18:14
Matillo
114
asked Sep 10 at 18:14
Matillo
114
asked Sep 10 at 18:14
Matillo
114
114
I did a quick calculation, and it appears that $uin W^1,2_0$ for the $u$ that you provided. This may mean it is a weak solution. Someone may need to confirm, though.
– MasterYoda
Sep 10 at 20:46
I don't really understand what I've to do if $$ u in W^1,2_0 $$ . Well then it follows that $$ u in B_1(0)0 $$ is a distributional solution for (1) .
– Matillo
Sep 11 at 6:17
I don't really understand what I've to do if $$ u in W^1,2_0 $$ . In fact I computed that u is in $$ W^3,p(B_1(0)) cap W^4,1(B_1(0)) $$ .
– Matillo
Sep 11 at 6:36
Can someone explain me how I can show that u is a distributional solution for (1) on $$ B_1(0)diagdown (0,0) $$ ? I know that $$ uin C^infty(B_1(0)diagdown(0,0)) $$ .
– Matillo
Sep 11 at 14:11
add a comment |Â
I did a quick calculation, and it appears that $uin W^1,2_0$ for the $u$ that you provided. This may mean it is a weak solution. Someone may need to confirm, though.
– MasterYoda
Sep 10 at 20:46
I don't really understand what I've to do if $$ u in W^1,2_0 $$ . Well then it follows that $$ u in B_1(0)0 $$ is a distributional solution for (1) .
– Matillo
Sep 11 at 6:17
I don't really understand what I've to do if $$ u in W^1,2_0 $$ . In fact I computed that u is in $$ W^3,p(B_1(0)) cap W^4,1(B_1(0)) $$ .
– Matillo
Sep 11 at 6:36
Can someone explain me how I can show that u is a distributional solution for (1) on $$ B_1(0)diagdown (0,0) $$ ? I know that $$ uin C^infty(B_1(0)diagdown(0,0)) $$ .
– Matillo
Sep 11 at 14:11
I did a quick calculation, and it appears that $uin W^1,2_0$ for the $u$ that you provided. This may mean it is a weak solution. Someone may need to confirm, though.
– MasterYoda
Sep 10 at 20:46
I did a quick calculation, and it appears that $uin W^1,2_0$ for the $u$ that you provided. This may mean it is a weak solution. Someone may need to confirm, though.
– MasterYoda
Sep 10 at 20:46
I don't really understand what I've to do if $$ u in W^1,2_0 $$ . Well then it follows that $$ u in B_1(0)0 $$ is a distributional solution for (1) .
– Matillo
Sep 11 at 6:17
I don't really understand what I've to do if $$ u in W^1,2_0 $$ . Well then it follows that $$ u in B_1(0)0 $$ is a distributional solution for (1) .
– Matillo
Sep 11 at 6:17
I don't really understand what I've to do if $$ u in W^1,2_0 $$ . In fact I computed that u is in $$ W^3,p(B_1(0)) cap W^4,1(B_1(0)) $$ .
– Matillo
Sep 11 at 6:36
I don't really understand what I've to do if $$ u in W^1,2_0 $$ . In fact I computed that u is in $$ W^3,p(B_1(0)) cap W^4,1(B_1(0)) $$ .
– Matillo
Sep 11 at 6:36
Can someone explain me how I can show that u is a distributional solution for (1) on $$ B_1(0)diagdown (0,0) $$ ? I know that $$ uin C^infty(B_1(0)diagdown(0,0)) $$ .
– Matillo
Sep 11 at 14:11
Can someone explain me how I can show that u is a distributional solution for (1) on $$ B_1(0)diagdown (0,0) $$ ? I know that $$ uin C^infty(B_1(0)diagdown(0,0)) $$ .
– Matillo
Sep 11 at 14:11
add a comment |Â
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I did a quick calculation, and it appears that $uin W^1,2_0$ for the $u$ that you provided. This may mean it is a weak solution. Someone may need to confirm, though.
– MasterYoda
Sep 10 at 20:46
I don't really understand what I've to do if $$ u in W^1,2_0 $$ . Well then it follows that $$ u in B_1(0)0 $$ is a distributional solution for (1) .
– Matillo
Sep 11 at 6:17
I don't really understand what I've to do if $$ u in W^1,2_0 $$ . In fact I computed that u is in $$ W^3,p(B_1(0)) cap W^4,1(B_1(0)) $$ .
– Matillo
Sep 11 at 6:36
Can someone explain me how I can show that u is a distributional solution for (1) on $$ B_1(0)diagdown (0,0) $$ ? I know that $$ uin C^infty(B_1(0)diagdown(0,0)) $$ .
– Matillo
Sep 11 at 14:11