Existence of solution to Laplace equation
$begingroup$
Given condition: Let $Omega$ is a smooth bounded domain in $mathbb{R}^2$.
Suppose there exist $w(>0)in H_{0}^1(Omega)$ satisfying the inequality $-Delta wleq C e^w$ in $Omega$ for some positive constant $C$.
Can you please help me with the following question.
Does there exist some function $vin H_{0}^1(Omega)cap C^1(Omega)$ such that $wleq v$ and satisfying $-Delta v=Ce^{v}$ in $Omega$.
pde sobolev-spaces regularity-theory-of-pdes linear-pde
asked Dec 18 '18 at 16:46
MathloverMathlover
1558
$endgroup$
add a comment |
$begingroup$
Given condition: Let $Omega$ is a smooth bounded domain in $mathbb{R}^2$.
Suppose there exist $w(>0)in H_{0}^1(Omega)$ satisfying the inequality $-Delta wleq C e^w$ in $Omega$ for some positive constant $C$.
Can you please help me with the following question.
Does there exist some function $vin H_{0}^1(Omega)cap C^1(Omega)$ such that $wleq v$ and satisfying $-Delta v=Ce^{v}$ in $Omega$.
pde sobolev-spaces regularity-theory-of-pdes linear-pde
asked Dec 18 '18 at 16:46
MathloverMathlover
1558
$endgroup$
add a comment |
$begingroup$
Given condition: Let $Omega$ is a smooth bounded domain in $mathbb{R}^2$.
Suppose there exist $w(>0)in H_{0}^1(Omega)$ satisfying the inequality $-Delta wleq C e^w$ in $Omega$ for some positive constant $C$.
Can you please help me with the following question.
Does there exist some function $vin H_{0}^1(Omega)cap C^1(Omega)$ such that $wleq v$ and satisfying $-Delta v=Ce^{v}$ in $Omega$.
pde sobolev-spaces regularity-theory-of-pdes linear-pde
asked Dec 18 '18 at 16:46
MathloverMathlover
1558
$endgroup$
Given condition: Let $Omega$ is a smooth bounded domain in $mathbb{R}^2$.
Suppose there exist $w(>0)in H_{0}^1(Omega)$ satisfying the inequality $-Delta wleq C e^w$ in $Omega$ for some positive constant $C$.
Can you please help me with the following question.
Does there exist some function $vin H_{0}^1(Omega)cap C^1(Omega)$ such that $wleq v$ and satisfying $-Delta v=Ce^{v}$ in $Omega$.
pde sobolev-spaces regularity-theory-of-pdes linear-pde
pde sobolev-spaces regularity-theory-of-pdes linear-pde
asked Dec 18 '18 at 16:46
MathloverMathlover
1558
asked Dec 18 '18 at 16:46
MathloverMathlover
1558
asked Dec 18 '18 at 16:46
MathloverMathlover
1558
asked Dec 18 '18 at 16:46
MathloverMathlover
1558
asked Dec 18 '18 at 16:46
MathloverMathlover
1558
1558
add a comment |
add a comment |
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