Continuity of an upper envelope












1












$begingroup$


I'm now approaching for the first time to the definition of upper envelope of a family of functions, and I just wonder to know some basic properties.



Suppose $Omega $ is an open subset of $mathbb{R}^n$. Let $A$ a family of index. For each $ain A$ we have two functions $f_a:Omegato mathbb{R}^n$, $l_a:Omegatomathbb{R}$ and a scalar number $c_ain mathbb{R}$.



For each fixed $a$ we consider the real valued function: $$g_a:Omegatimesmathbb{R}timesmathbb{R}^nto mathbb{R}$$ such that $$g_a(x,z,p):=c_a z+f_a(x)cdot p+l_a(x)$$ for all $(x,z,p)in Omegatimesmathbb{R}timesmathbb{R}^n$, where $(cdot)$ is the usual scalar product on $mathbb{R}^n$.



We can now consider the upper envelope of the family $(g_a)_{ain A}$, namely $$F(x,z,p):=sup_{ain A}g_a(x,z,p)=sup_{ain A}{c_a z+f_a(x)cdot p+l_a(x)}.$$



My question is the following:



What are sufficient conditions to have:



1) $|F(x,z,p)|neq infty$ for every $(x,z,p)$



2) $F$ is continuous on $Omegatimesmathbb{R}timesmathbb{R}^n$??



Any suggestions will be really appreciated.






real-analysis supremum-and-infimum envelope







share|cite|improve this question









$endgroup$












  • $begingroup$
    Translate: family of index.
    $endgroup$
    – William Elliot
    Jan 10 at 11:35










  • $begingroup$
    We can think $A$ as a compact subset of $mathbb{R}^m$.
    $endgroup$
    – eleguitar
    Jan 10 at 11:41
















1












$begingroup$


I'm now approaching for the first time to the definition of upper envelope of a family of functions, and I just wonder to know some basic properties.



Suppose $Omega $ is an open subset of $mathbb{R}^n$. Let $A$ a family of index. For each $ain A$ we have two functions $f_a:Omegato mathbb{R}^n$, $l_a:Omegatomathbb{R}$ and a scalar number $c_ain mathbb{R}$.



For each fixed $a$ we consider the real valued function: $$g_a:Omegatimesmathbb{R}timesmathbb{R}^nto mathbb{R}$$ such that $$g_a(x,z,p):=c_a z+f_a(x)cdot p+l_a(x)$$ for all $(x,z,p)in Omegatimesmathbb{R}timesmathbb{R}^n$, where $(cdot)$ is the usual scalar product on $mathbb{R}^n$.



We can now consider the upper envelope of the family $(g_a)_{ain A}$, namely $$F(x,z,p):=sup_{ain A}g_a(x,z,p)=sup_{ain A}{c_a z+f_a(x)cdot p+l_a(x)}.$$



My question is the following:



What are sufficient conditions to have:



1) $|F(x,z,p)|neq infty$ for every $(x,z,p)$



2) $F$ is continuous on $Omegatimesmathbb{R}timesmathbb{R}^n$??



Any suggestions will be really appreciated.






real-analysis supremum-and-infimum envelope







share|cite|improve this question









$endgroup$












  • $begingroup$
    Translate: family of index.
    $endgroup$
    – William Elliot
    Jan 10 at 11:35










  • $begingroup$
    We can think $A$ as a compact subset of $mathbb{R}^m$.
    $endgroup$
    – eleguitar
    Jan 10 at 11:41














1












1








1





$begingroup$


I'm now approaching for the first time to the definition of upper envelope of a family of functions, and I just wonder to know some basic properties.



Suppose $Omega $ is an open subset of $mathbb{R}^n$. Let $A$ a family of index. For each $ain A$ we have two functions $f_a:Omegato mathbb{R}^n$, $l_a:Omegatomathbb{R}$ and a scalar number $c_ain mathbb{R}$.



For each fixed $a$ we consider the real valued function: $$g_a:Omegatimesmathbb{R}timesmathbb{R}^nto mathbb{R}$$ such that $$g_a(x,z,p):=c_a z+f_a(x)cdot p+l_a(x)$$ for all $(x,z,p)in Omegatimesmathbb{R}timesmathbb{R}^n$, where $(cdot)$ is the usual scalar product on $mathbb{R}^n$.



We can now consider the upper envelope of the family $(g_a)_{ain A}$, namely $$F(x,z,p):=sup_{ain A}g_a(x,z,p)=sup_{ain A}{c_a z+f_a(x)cdot p+l_a(x)}.$$



My question is the following:



What are sufficient conditions to have:



1) $|F(x,z,p)|neq infty$ for every $(x,z,p)$



2) $F$ is continuous on $Omegatimesmathbb{R}timesmathbb{R}^n$??



Any suggestions will be really appreciated.






real-analysis supremum-and-infimum envelope







share|cite|improve this question









$endgroup$




I'm now approaching for the first time to the definition of upper envelope of a family of functions, and I just wonder to know some basic properties.



Suppose $Omega $ is an open subset of $mathbb{R}^n$. Let $A$ a family of index. For each $ain A$ we have two functions $f_a:Omegato mathbb{R}^n$, $l_a:Omegatomathbb{R}$ and a scalar number $c_ain mathbb{R}$.



For each fixed $a$ we consider the real valued function: $$g_a:Omegatimesmathbb{R}timesmathbb{R}^nto mathbb{R}$$ such that $$g_a(x,z,p):=c_a z+f_a(x)cdot p+l_a(x)$$ for all $(x,z,p)in Omegatimesmathbb{R}timesmathbb{R}^n$, where $(cdot)$ is the usual scalar product on $mathbb{R}^n$.



We can now consider the upper envelope of the family $(g_a)_{ain A}$, namely $$F(x,z,p):=sup_{ain A}g_a(x,z,p)=sup_{ain A}{c_a z+f_a(x)cdot p+l_a(x)}.$$



My question is the following:



What are sufficient conditions to have:



1) $|F(x,z,p)|neq infty$ for every $(x,z,p)$



2) $F$ is continuous on $Omegatimesmathbb{R}timesmathbb{R}^n$??



Any suggestions will be really appreciated.






real-analysis supremum-and-infimum envelope




real-analysis supremum-and-infimum envelope






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 10 at 11:20









eleguitareleguitar

140114




140114












  • $begingroup$
    Translate: family of index.
    $endgroup$
    – William Elliot
    Jan 10 at 11:35










  • $begingroup$
    We can think $A$ as a compact subset of $mathbb{R}^m$.
    $endgroup$
    – eleguitar
    Jan 10 at 11:41


















  • $begingroup$
    Translate: family of index.
    $endgroup$
    – William Elliot
    Jan 10 at 11:35










  • $begingroup$
    We can think $A$ as a compact subset of $mathbb{R}^m$.
    $endgroup$
    – eleguitar
    Jan 10 at 11:41
















$begingroup$
Translate: family of index.
$endgroup$
– William Elliot
Jan 10 at 11:35




$begingroup$
Translate: family of index.
$endgroup$
– William Elliot
Jan 10 at 11:35












$begingroup$
We can think $A$ as a compact subset of $mathbb{R}^m$.
$endgroup$
– eleguitar
Jan 10 at 11:41




$begingroup$
We can think $A$ as a compact subset of $mathbb{R}^m$.
$endgroup$
– eleguitar
Jan 10 at 11:41










1 Answer
1






active

oldest

votes


















1












$begingroup$

I would suggest the following conditions:



1) $c_a$ is bounded, and $f_a(x)$ and $l_a(x)$ as well for each $x$.



2) $f_a$ and $l_a$ are locally uniformly bounded, ie for all $x in Omega$, there exists some neighborhood $U$ of $x$ and some $M_U > 0$ such that for every $y in U$, $|f_a(y)| +|l_a(y)| leq M_U$.



3) The $f_a$ and $l_a$ are equicontinuous, ie: for all $x in Omega$ and $epsilon >0$, there exists some neighborhood $U subset Omega$ of $x$ such that for all $a in A$, $y in U$, $|f_a(x)-f_a(y)|+|l_a(x)-l_a(y)| leq epsilon$.



1) ensures that $F$ is well-defined.



2) ensures that each $g_a$ is locally (wrt $x$) Lipschitz-continuous wrt $(p,z)$ and that the constants and neighborhoods can be chosen independently from the index, thus the property goes on to $F$.



3) Gives the same kind of continuity wrt $x$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks a lot. Really helpful. Could you give me some good references to learn more about this kind of functions?
    $endgroup$
    – eleguitar
    Jan 10 at 14:51






  • 1




    $begingroup$
    I am sorry, I do not know anything specific. For, say, properties of supremum of functions, you could try and check out viscosity solutions for PDE — but all in all this is fairly standard analysis.
    $endgroup$
    – Mindlack
    Jan 10 at 16:34












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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









1












$begingroup$

I would suggest the following conditions:



1) $c_a$ is bounded, and $f_a(x)$ and $l_a(x)$ as well for each $x$.



2) $f_a$ and $l_a$ are locally uniformly bounded, ie for all $x in Omega$, there exists some neighborhood $U$ of $x$ and some $M_U > 0$ such that for every $y in U$, $|f_a(y)| +|l_a(y)| leq M_U$.



3) The $f_a$ and $l_a$ are equicontinuous, ie: for all $x in Omega$ and $epsilon >0$, there exists some neighborhood $U subset Omega$ of $x$ such that for all $a in A$, $y in U$, $|f_a(x)-f_a(y)|+|l_a(x)-l_a(y)| leq epsilon$.



1) ensures that $F$ is well-defined.



2) ensures that each $g_a$ is locally (wrt $x$) Lipschitz-continuous wrt $(p,z)$ and that the constants and neighborhoods can be chosen independently from the index, thus the property goes on to $F$.



3) Gives the same kind of continuity wrt $x$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks a lot. Really helpful. Could you give me some good references to learn more about this kind of functions?
    $endgroup$
    – eleguitar
    Jan 10 at 14:51






  • 1




    $begingroup$
    I am sorry, I do not know anything specific. For, say, properties of supremum of functions, you could try and check out viscosity solutions for PDE — but all in all this is fairly standard analysis.
    $endgroup$
    – Mindlack
    Jan 10 at 16:34
















1












$begingroup$

I would suggest the following conditions:



1) $c_a$ is bounded, and $f_a(x)$ and $l_a(x)$ as well for each $x$.



2) $f_a$ and $l_a$ are locally uniformly bounded, ie for all $x in Omega$, there exists some neighborhood $U$ of $x$ and some $M_U > 0$ such that for every $y in U$, $|f_a(y)| +|l_a(y)| leq M_U$.



3) The $f_a$ and $l_a$ are equicontinuous, ie: for all $x in Omega$ and $epsilon >0$, there exists some neighborhood $U subset Omega$ of $x$ such that for all $a in A$, $y in U$, $|f_a(x)-f_a(y)|+|l_a(x)-l_a(y)| leq epsilon$.



1) ensures that $F$ is well-defined.



2) ensures that each $g_a$ is locally (wrt $x$) Lipschitz-continuous wrt $(p,z)$ and that the constants and neighborhoods can be chosen independently from the index, thus the property goes on to $F$.



3) Gives the same kind of continuity wrt $x$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks a lot. Really helpful. Could you give me some good references to learn more about this kind of functions?
    $endgroup$
    – eleguitar
    Jan 10 at 14:51






  • 1




    $begingroup$
    I am sorry, I do not know anything specific. For, say, properties of supremum of functions, you could try and check out viscosity solutions for PDE — but all in all this is fairly standard analysis.
    $endgroup$
    – Mindlack
    Jan 10 at 16:34














1












1








1





$begingroup$

I would suggest the following conditions:



1) $c_a$ is bounded, and $f_a(x)$ and $l_a(x)$ as well for each $x$.



2) $f_a$ and $l_a$ are locally uniformly bounded, ie for all $x in Omega$, there exists some neighborhood $U$ of $x$ and some $M_U > 0$ such that for every $y in U$, $|f_a(y)| +|l_a(y)| leq M_U$.



3) The $f_a$ and $l_a$ are equicontinuous, ie: for all $x in Omega$ and $epsilon >0$, there exists some neighborhood $U subset Omega$ of $x$ such that for all $a in A$, $y in U$, $|f_a(x)-f_a(y)|+|l_a(x)-l_a(y)| leq epsilon$.



1) ensures that $F$ is well-defined.



2) ensures that each $g_a$ is locally (wrt $x$) Lipschitz-continuous wrt $(p,z)$ and that the constants and neighborhoods can be chosen independently from the index, thus the property goes on to $F$.



3) Gives the same kind of continuity wrt $x$.






share|cite|improve this answer









$endgroup$



I would suggest the following conditions:



1) $c_a$ is bounded, and $f_a(x)$ and $l_a(x)$ as well for each $x$.



2) $f_a$ and $l_a$ are locally uniformly bounded, ie for all $x in Omega$, there exists some neighborhood $U$ of $x$ and some $M_U > 0$ such that for every $y in U$, $|f_a(y)| +|l_a(y)| leq M_U$.



3) The $f_a$ and $l_a$ are equicontinuous, ie: for all $x in Omega$ and $epsilon >0$, there exists some neighborhood $U subset Omega$ of $x$ such that for all $a in A$, $y in U$, $|f_a(x)-f_a(y)|+|l_a(x)-l_a(y)| leq epsilon$.



1) ensures that $F$ is well-defined.



2) ensures that each $g_a$ is locally (wrt $x$) Lipschitz-continuous wrt $(p,z)$ and that the constants and neighborhoods can be chosen independently from the index, thus the property goes on to $F$.



3) Gives the same kind of continuity wrt $x$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 10 at 11:52









MindlackMindlack

4,900211




4,900211












  • $begingroup$
    Thanks a lot. Really helpful. Could you give me some good references to learn more about this kind of functions?
    $endgroup$
    – eleguitar
    Jan 10 at 14:51






  • 1




    $begingroup$
    I am sorry, I do not know anything specific. For, say, properties of supremum of functions, you could try and check out viscosity solutions for PDE — but all in all this is fairly standard analysis.
    $endgroup$
    – Mindlack
    Jan 10 at 16:34


















  • $begingroup$
    Thanks a lot. Really helpful. Could you give me some good references to learn more about this kind of functions?
    $endgroup$
    – eleguitar
    Jan 10 at 14:51






  • 1




    $begingroup$
    I am sorry, I do not know anything specific. For, say, properties of supremum of functions, you could try and check out viscosity solutions for PDE — but all in all this is fairly standard analysis.
    $endgroup$
    – Mindlack
    Jan 10 at 16:34
















$begingroup$
Thanks a lot. Really helpful. Could you give me some good references to learn more about this kind of functions?
$endgroup$
– eleguitar
Jan 10 at 14:51




$begingroup$
Thanks a lot. Really helpful. Could you give me some good references to learn more about this kind of functions?
$endgroup$
– eleguitar
Jan 10 at 14:51




1




1




$begingroup$
I am sorry, I do not know anything specific. For, say, properties of supremum of functions, you could try and check out viscosity solutions for PDE — but all in all this is fairly standard analysis.
$endgroup$
– Mindlack
Jan 10 at 16:34




$begingroup$
I am sorry, I do not know anything specific. For, say, properties of supremum of functions, you could try and check out viscosity solutions for PDE — but all in all this is fairly standard analysis.
$endgroup$
– Mindlack
Jan 10 at 16:34


















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