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If $u$ subharmonic, why $U_s=xmid u(x)=s$ is open?
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Proving the maximum principle for harmonic real valued functions in $mathbbR^n$Properties of subharmonic functionsEstimates of derivatives of harmonic functionA nonlinear Poisson equation problem. (Related to Laplace equation)Prove or disprove a claim related to $L^p$ spaceAn upper semicontinous function which is not subharmonic.If $fin S(mathbb R^n)$ (schwarz space), why $fin L^p(mathbb R^n)$?Questions on Harmonic function.Harmonic function : if $u=0$ on $Omega $ then $u=0$ on $partial Omega $.Poisson and subharmonic equation
$begingroup$
Let $U$ a domain and $uin mathcal C^2(U)cap mathcal C^1(bar U)$ and let $s=sup_Uu$. Let $U_s=xmid u(x)=s$. I want to prove that $U$ is clopen. I proved that it's closed (because $U=u^-1s$ and $U$ continuous), but I don't understand why for all $xin U$, there is $r>0$ s.t. $mathcal B_r(x)subset U_s$. Any idea ?
harmonic-analysis
asked Mar 26 at 7:35
user657607user657607
245
$endgroup$
add a comment |
$begingroup$
Let $U$ a domain and $uin mathcal C^2(U)cap mathcal C^1(bar U)$ and let $s=sup_Uu$. Let $U_s=xmid u(x)=s$. I want to prove that $U$ is clopen. I proved that it's closed (because $U=u^-1s$ and $U$ continuous), but I don't understand why for all $xin U$, there is $r>0$ s.t. $mathcal B_r(x)subset U_s$. Any idea ?
harmonic-analysis
asked Mar 26 at 7:35
user657607user657607
245
$endgroup$
add a comment |
$begingroup$
Let $U$ a domain and $uin mathcal C^2(U)cap mathcal C^1(bar U)$ and let $s=sup_Uu$. Let $U_s=xmid u(x)=s$. I want to prove that $U$ is clopen. I proved that it's closed (because $U=u^-1s$ and $U$ continuous), but I don't understand why for all $xin U$, there is $r>0$ s.t. $mathcal B_r(x)subset U_s$. Any idea ?
harmonic-analysis
asked Mar 26 at 7:35
user657607user657607
245
$endgroup$
Let $U$ a domain and $uin mathcal C^2(U)cap mathcal C^1(bar U)$ and let $s=sup_Uu$. Let $U_s=xmid u(x)=s$. I want to prove that $U$ is clopen. I proved that it's closed (because $U=u^-1s$ and $U$ continuous), but I don't understand why for all $xin U$, there is $r>0$ s.t. $mathcal B_r(x)subset U_s$. Any idea ?
harmonic-analysis
harmonic-analysis
asked Mar 26 at 7:35
user657607user657607
245
asked Mar 26 at 7:35
user657607user657607
245
asked Mar 26 at 7:35
user657607user657607
245
asked Mar 26 at 7:35
user657607user657607
245
asked Mar 26 at 7:35
user657607user657607
245
245
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Hint
Apply mean value property to $xmapsto u(x)-s$ that is subharmonic on $U$.
answered Mar 26 at 7:37
user657324user657324
60110
$endgroup$
add a comment |
$begingroup$
By the maximum principle if $u(x)=s$ then $u(y)=s$ for all $y$ in some disk around $x$.
answered Mar 26 at 7:38
Kavi Rama MurthyKavi Rama Murthy
74.9k53270
$endgroup$
add a comment |
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2 Answers
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2 Answers
2
active
oldest
votes
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votes
$begingroup$
Hint
Apply mean value property to $xmapsto u(x)-s$ that is subharmonic on $U$.
answered Mar 26 at 7:37
user657324user657324
60110
$endgroup$
add a comment |
$begingroup$
Hint
Apply mean value property to $xmapsto u(x)-s$ that is subharmonic on $U$.
answered Mar 26 at 7:37
user657324user657324
60110
$endgroup$
add a comment |
$begingroup$
Hint
Apply mean value property to $xmapsto u(x)-s$ that is subharmonic on $U$.
answered Mar 26 at 7:37
user657324user657324
60110
$endgroup$
Hint
Apply mean value property to $xmapsto u(x)-s$ that is subharmonic on $U$.
answered Mar 26 at 7:37
user657324user657324
60110
answered Mar 26 at 7:37
user657324user657324
60110
answered Mar 26 at 7:37
user657324user657324
60110
answered Mar 26 at 7:37
user657324user657324
60110
60110
add a comment |
add a comment |
$begingroup$
By the maximum principle if $u(x)=s$ then $u(y)=s$ for all $y$ in some disk around $x$.
answered Mar 26 at 7:38
Kavi Rama MurthyKavi Rama Murthy
74.9k53270
$endgroup$
add a comment |
$begingroup$
By the maximum principle if $u(x)=s$ then $u(y)=s$ for all $y$ in some disk around $x$.
answered Mar 26 at 7:38
Kavi Rama MurthyKavi Rama Murthy
74.9k53270
$endgroup$
add a comment |
$begingroup$
By the maximum principle if $u(x)=s$ then $u(y)=s$ for all $y$ in some disk around $x$.
answered Mar 26 at 7:38
Kavi Rama MurthyKavi Rama Murthy
74.9k53270
$endgroup$
By the maximum principle if $u(x)=s$ then $u(y)=s$ for all $y$ in some disk around $x$.
answered Mar 26 at 7:38
Kavi Rama MurthyKavi Rama Murthy
74.9k53270
answered Mar 26 at 7:38
Kavi Rama MurthyKavi Rama Murthy
74.9k53270
answered Mar 26 at 7:38
Kavi Rama MurthyKavi Rama Murthy
74.9k53270
answered Mar 26 at 7:38
Kavi Rama MurthyKavi Rama Murthy
74.9k53270
74.9k53270
add a comment |
add a comment |
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