Mixing
Can there exist such a harmonic function?
0
$begingroup$
If $D={z:Im(z)>0, |Re(z)|<pi/2}$ be a half strip. Let $L_1$ be the left boundary:
$L_1={z:Re(z)=-pi/2,Im(z)>0}$. Similarly, let $L_2={z:Re(z)=pi/2,Im(z)>0}$, $L_3={z:|Re(z)|<pi/2,Im(z)=0}$. Can a harmonic function be found if $u|_{L_1}=1$, $u|_{L_2}=3$, $u|_{L_3}=5$?.
complex-analysis
asked Jan 31 at 19:13
AlexanderAlexander
333111
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show 1 more comment
0
$begingroup$
If $D={z:Im(z)>0, |Re(z)|<pi/2}$ be a half strip. Let $L_1$ be the left boundary:
$L_1={z:Re(z)=-pi/2,Im(z)>0}$. Similarly, let $L_2={z:Re(z)=pi/2,Im(z)>0}$, $L_3={z:|Re(z)|<pi/2,Im(z)=0}$. Can a harmonic function be found if $u|_{L_1}=1$, $u|_{L_2}=3$, $u|_{L_3}=5$?.
complex-analysis
asked Jan 31 at 19:13
AlexanderAlexander
333111
$endgroup$
$begingroup$
You should be clearer about the properties of $u.$ Is $u$ continuous on $D cup L_1 cup L_2cup L_3?$ Also, what does this have to do with several complex variables?
$endgroup$
– zhw.
Jan 31 at 21:51
$begingroup$
@zhw when I am seeking for a harmonic function I presume the function to be at least twice continuously differentiable in $D$ in the regular sense and not in the weak sense in $D$. Of course, I need continuity on $L_1bigcup L_2bigcup L_3$.
$endgroup$
– Alexander
Feb 1 at 8:06
$begingroup$
Continuity on $L_1cup L_2cup L_3$ is obvious. It's continuity on $Dcup L_1cup L_2cup L_3$ that we need.
$endgroup$
– zhw.
Feb 1 at 16:18
$begingroup$
@zhw...Well, being twice differentiable in the interior and continuous on the boundary obviously implies continuity of the function in $Dbigcup L_1bigcup L_2bigcup L_3$. I don't see what is the relevance of your answer to this question. Could you please elaborate?.
$endgroup$
– Alexander
Feb 1 at 18:45
$begingroup$
No, that's not true. Being continuous on $L_1cup L_2 cup L_3$ does not imply continuity on $Dcup L_1cup L_2 cup L_3$ For example, the function that equals $0$ on $D$ and equals $1$ on $L_1cup L_2 cup L_3$ is continuous on $L_1cup L_2 cup L_3$ but not on $Dcup L_1cup L_2 cup L_3$
$endgroup$
– zhw.
Feb 1 at 18:55
|
show 1 more comment
0
0
0
$begingroup$
If $D={z:Im(z)>0, |Re(z)|<pi/2}$ be a half strip. Let $L_1$ be the left boundary:
$L_1={z:Re(z)=-pi/2,Im(z)>0}$. Similarly, let $L_2={z:Re(z)=pi/2,Im(z)>0}$, $L_3={z:|Re(z)|<pi/2,Im(z)=0}$. Can a harmonic function be found if $u|_{L_1}=1$, $u|_{L_2}=3$, $u|_{L_3}=5$?.
complex-analysis
asked Jan 31 at 19:13
AlexanderAlexander
333111
$endgroup$
If $D={z:Im(z)>0, |Re(z)|<pi/2}$ be a half strip. Let $L_1$ be the left boundary:
$L_1={z:Re(z)=-pi/2,Im(z)>0}$. Similarly, let $L_2={z:Re(z)=pi/2,Im(z)>0}$, $L_3={z:|Re(z)|<pi/2,Im(z)=0}$. Can a harmonic function be found if $u|_{L_1}=1$, $u|_{L_2}=3$, $u|_{L_3}=5$?.
complex-analysis
complex-analysis
asked Jan 31 at 19:13
AlexanderAlexander
333111
asked Jan 31 at 19:13
AlexanderAlexander
333111
edited Feb 1 at 8:07
Alexander
asked Jan 31 at 19:13
AlexanderAlexander
333111
asked Jan 31 at 19:13
AlexanderAlexander
333111
asked Jan 31 at 19:13
AlexanderAlexander
333111
333111
$begingroup$
You should be clearer about the properties of $u.$ Is $u$ continuous on $D cup L_1 cup L_2cup L_3?$ Also, what does this have to do with several complex variables?
$endgroup$
– zhw.
Jan 31 at 21:51
$begingroup$
@zhw when I am seeking for a harmonic function I presume the function to be at least twice continuously differentiable in $D$ in the regular sense and not in the weak sense in $D$. Of course, I need continuity on $L_1bigcup L_2bigcup L_3$.
$endgroup$
– Alexander
Feb 1 at 8:06
$begingroup$
Continuity on $L_1cup L_2cup L_3$ is obvious. It's continuity on $Dcup L_1cup L_2cup L_3$ that we need.
$endgroup$
– zhw.
Feb 1 at 16:18
$begingroup$
@zhw...Well, being twice differentiable in the interior and continuous on the boundary obviously implies continuity of the function in $Dbigcup L_1bigcup L_2bigcup L_3$. I don't see what is the relevance of your answer to this question. Could you please elaborate?.
$endgroup$
– Alexander
Feb 1 at 18:45
$begingroup$
No, that's not true. Being continuous on $L_1cup L_2 cup L_3$ does not imply continuity on $Dcup L_1cup L_2 cup L_3$ For example, the function that equals $0$ on $D$ and equals $1$ on $L_1cup L_2 cup L_3$ is continuous on $L_1cup L_2 cup L_3$ but not on $Dcup L_1cup L_2 cup L_3$
$endgroup$
– zhw.
Feb 1 at 18:55
|
show 1 more comment
$begingroup$
You should be clearer about the properties of $u.$ Is $u$ continuous on $D cup L_1 cup L_2cup L_3?$ Also, what does this have to do with several complex variables?
$endgroup$
– zhw.
Jan 31 at 21:51
$begingroup$
@zhw when I am seeking for a harmonic function I presume the function to be at least twice continuously differentiable in $D$ in the regular sense and not in the weak sense in $D$. Of course, I need continuity on $L_1bigcup L_2bigcup L_3$.
$endgroup$
– Alexander
Feb 1 at 8:06
$begingroup$
Continuity on $L_1cup L_2cup L_3$ is obvious. It's continuity on $Dcup L_1cup L_2cup L_3$ that we need.
$endgroup$
– zhw.
Feb 1 at 16:18
$begingroup$
@zhw...Well, being twice differentiable in the interior and continuous on the boundary obviously implies continuity of the function in $Dbigcup L_1bigcup L_2bigcup L_3$. I don't see what is the relevance of your answer to this question. Could you please elaborate?.
$endgroup$
– Alexander
Feb 1 at 18:45
$begingroup$
No, that's not true. Being continuous on $L_1cup L_2 cup L_3$ does not imply continuity on $Dcup L_1cup L_2 cup L_3$ For example, the function that equals $0$ on $D$ and equals $1$ on $L_1cup L_2 cup L_3$ is continuous on $L_1cup L_2 cup L_3$ but not on $Dcup L_1cup L_2 cup L_3$
$endgroup$
– zhw.
Feb 1 at 18:55
$begingroup$
You should be clearer about the properties of $u.$ Is $u$ continuous on $D cup L_1 cup L_2cup L_3?$ Also, what does this have to do with several complex variables?
$endgroup$
– zhw.
Jan 31 at 21:51
$begingroup$
You should be clearer about the properties of $u.$ Is $u$ continuous on $D cup L_1 cup L_2cup L_3?$ Also, what does this have to do with several complex variables?
$endgroup$
– zhw.
Jan 31 at 21:51
$begingroup$
@zhw when I am seeking for a harmonic function I presume the function to be at least twice continuously differentiable in $D$ in the regular sense and not in the weak sense in $D$. Of course, I need continuity on $L_1bigcup L_2bigcup L_3$.
$endgroup$
– Alexander
Feb 1 at 8:06
$begingroup$
@zhw when I am seeking for a harmonic function I presume the function to be at least twice continuously differentiable in $D$ in the regular sense and not in the weak sense in $D$. Of course, I need continuity on $L_1bigcup L_2bigcup L_3$.
$endgroup$
– Alexander
Feb 1 at 8:06
$begingroup$
Continuity on $L_1cup L_2cup L_3$ is obvious. It's continuity on $Dcup L_1cup L_2cup L_3$ that we need.
$endgroup$
– zhw.
Feb 1 at 16:18
$begingroup$
Continuity on $L_1cup L_2cup L_3$ is obvious. It's continuity on $Dcup L_1cup L_2cup L_3$ that we need.
$endgroup$
– zhw.
Feb 1 at 16:18
$begingroup$
@zhw...Well, being twice differentiable in the interior and continuous on the boundary obviously implies continuity of the function in $Dbigcup L_1bigcup L_2bigcup L_3$. I don't see what is the relevance of your answer to this question. Could you please elaborate?.
$endgroup$
– Alexander
Feb 1 at 18:45
$begingroup$
@zhw...Well, being twice differentiable in the interior and continuous on the boundary obviously implies continuity of the function in $Dbigcup L_1bigcup L_2bigcup L_3$. I don't see what is the relevance of your answer to this question. Could you please elaborate?.
$endgroup$
– Alexander
Feb 1 at 18:45
$begingroup$
No, that's not true. Being continuous on $L_1cup L_2 cup L_3$ does not imply continuity on $Dcup L_1cup L_2 cup L_3$ For example, the function that equals $0$ on $D$ and equals $1$ on $L_1cup L_2 cup L_3$ is continuous on $L_1cup L_2 cup L_3$ but not on $Dcup L_1cup L_2 cup L_3$
$endgroup$
– zhw.
Feb 1 at 18:55
$begingroup$
No, that's not true. Being continuous on $L_1cup L_2 cup L_3$ does not imply continuity on $Dcup L_1cup L_2 cup L_3$ For example, the function that equals $0$ on $D$ and equals $1$ on $L_1cup L_2 cup L_3$ is continuous on $L_1cup L_2 cup L_3$ but not on $Dcup L_1cup L_2 cup L_3$
$endgroup$
– zhw.
Feb 1 at 18:55
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$begingroup$
You should be clearer about the properties of $u.$ Is $u$ continuous on $D cup L_1 cup L_2cup L_3?$ Also, what does this have to do with several complex variables?
$endgroup$
– zhw.
Jan 31 at 21:51
$begingroup$
@zhw when I am seeking for a harmonic function I presume the function to be at least twice continuously differentiable in $D$ in the regular sense and not in the weak sense in $D$. Of course, I need continuity on $L_1bigcup L_2bigcup L_3$.
$endgroup$
– Alexander
Feb 1 at 8:06
$begingroup$
Continuity on $L_1cup L_2cup L_3$ is obvious. It's continuity on $Dcup L_1cup L_2cup L_3$ that we need.
$endgroup$
– zhw.
Feb 1 at 16:18
$begingroup$
@zhw...Well, being twice differentiable in the interior and continuous on the boundary obviously implies continuity of the function in $Dbigcup L_1bigcup L_2bigcup L_3$. I don't see what is the relevance of your answer to this question. Could you please elaborate?.
$endgroup$
– Alexander
Feb 1 at 18:45
$begingroup$
No, that's not true. Being continuous on $L_1cup L_2 cup L_3$ does not imply continuity on $Dcup L_1cup L_2 cup L_3$ For example, the function that equals $0$ on $D$ and equals $1$ on $L_1cup L_2 cup L_3$ is continuous on $L_1cup L_2 cup L_3$ but not on $Dcup L_1cup L_2 cup L_3$
$endgroup$
– zhw.
Feb 1 at 18:55