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minimization problem of integral functional for a given function
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Let $Omega subset mathbbR^n$ ($ngeq 2)$ be a bounded domain satisfying an exterior ball condition for any $xi_0 in partialOmega$ and $forall x in overlineOmega,, u_xi_0(x) = lnfracR$. Here $x_0 not in overlineOmega$ and $textdist(x_0,xi_0)=R$. Now, we fix $xi_0$, $x_1 in Omega$, $varepsilon > 0$, and $x_0$ first such that $B(x_1,varepsilon) subset Omega$. My question is how to find $mu in mathbbR$ satisfying
$$ inflimits_muinmathbbRint_B(x_1,varepsilon)|u_xi_0(y)-mu|^n-2(u_xi_0(y)-mu)dy=0$$
Any help is much appreciated. I've been thinking over this problem and tried to define $F(mu) = int_B(x_1,varepsilon)|u_xi_0(y)-mu|^n(u_xi_0(y)-mu)dy$ to find the derivative where $F'(mu) = 0$ but it didn't work. Of course for the case $n=2$ it is rather not difficult, so I am asking for $n>2$.
functional-analysis maxima-minima variational-analysis
asked Jul 15 at 4:57
Evan William Chandra
392213
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up vote
0
down vote
favorite
Let $Omega subset mathbbR^n$ ($ngeq 2)$ be a bounded domain satisfying an exterior ball condition for any $xi_0 in partialOmega$ and $forall x in overlineOmega,, u_xi_0(x) = lnfracR$. Here $x_0 not in overlineOmega$ and $textdist(x_0,xi_0)=R$. Now, we fix $xi_0$, $x_1 in Omega$, $varepsilon > 0$, and $x_0$ first such that $B(x_1,varepsilon) subset Omega$. My question is how to find $mu in mathbbR$ satisfying
$$ inflimits_muinmathbbRint_B(x_1,varepsilon)|u_xi_0(y)-mu|^n-2(u_xi_0(y)-mu)dy=0$$
Any help is much appreciated. I've been thinking over this problem and tried to define $F(mu) = int_B(x_1,varepsilon)|u_xi_0(y)-mu|^n(u_xi_0(y)-mu)dy$ to find the derivative where $F'(mu) = 0$ but it didn't work. Of course for the case $n=2$ it is rather not difficult, so I am asking for $n>2$.
functional-analysis maxima-minima variational-analysis
asked Jul 15 at 4:57
Evan William Chandra
392213
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $Omega subset mathbbR^n$ ($ngeq 2)$ be a bounded domain satisfying an exterior ball condition for any $xi_0 in partialOmega$ and $forall x in overlineOmega,, u_xi_0(x) = lnfracR$. Here $x_0 not in overlineOmega$ and $textdist(x_0,xi_0)=R$. Now, we fix $xi_0$, $x_1 in Omega$, $varepsilon > 0$, and $x_0$ first such that $B(x_1,varepsilon) subset Omega$. My question is how to find $mu in mathbbR$ satisfying
$$ inflimits_muinmathbbRint_B(x_1,varepsilon)|u_xi_0(y)-mu|^n-2(u_xi_0(y)-mu)dy=0$$
Any help is much appreciated. I've been thinking over this problem and tried to define $F(mu) = int_B(x_1,varepsilon)|u_xi_0(y)-mu|^n(u_xi_0(y)-mu)dy$ to find the derivative where $F'(mu) = 0$ but it didn't work. Of course for the case $n=2$ it is rather not difficult, so I am asking for $n>2$.
functional-analysis maxima-minima variational-analysis
asked Jul 15 at 4:57
Evan William Chandra
392213
Let $Omega subset mathbbR^n$ ($ngeq 2)$ be a bounded domain satisfying an exterior ball condition for any $xi_0 in partialOmega$ and $forall x in overlineOmega,, u_xi_0(x) = lnfracR$. Here $x_0 not in overlineOmega$ and $textdist(x_0,xi_0)=R$. Now, we fix $xi_0$, $x_1 in Omega$, $varepsilon > 0$, and $x_0$ first such that $B(x_1,varepsilon) subset Omega$. My question is how to find $mu in mathbbR$ satisfying
$$ inflimits_muinmathbbRint_B(x_1,varepsilon)|u_xi_0(y)-mu|^n-2(u_xi_0(y)-mu)dy=0$$
Any help is much appreciated. I've been thinking over this problem and tried to define $F(mu) = int_B(x_1,varepsilon)|u_xi_0(y)-mu|^n(u_xi_0(y)-mu)dy$ to find the derivative where $F'(mu) = 0$ but it didn't work. Of course for the case $n=2$ it is rather not difficult, so I am asking for $n>2$.
functional-analysis maxima-minima variational-analysis
asked Jul 15 at 4:57
Evan William Chandra
392213
edited Jul 15 at 5:06
asked Jul 15 at 4:57
Evan William Chandra
392213
asked Jul 15 at 4:57
Evan William Chandra
392213
asked Jul 15 at 4:57
Evan William Chandra
392213
392213
add a comment |Â
add a comment |Â
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