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It seems I am too fast to ask the question without thinking carefully.

The equation I consider is find $w:\Omega \to R$ satisfied : $$\frac{Du}{\sqrt{1+|\nabla u|^2}}=Dw. \tag{*}$$ Where $u:\Omega \to R$ is given and $u=0$ on $\partial \Omega$.

It is obvious this equation could not be solved for general $u$, the central obstacle is the frobenius condition(integrable condition).But I still wish to get some more information about this equation, in particular my eager could divide into two part:

1.Could the $u$ for which $(*)$ is solvable is dense in some suitable space, maybe $C^{2}(\Omega)$?

2.For the $u$ which make $(*)$ locally solvable, what information of $w$ could we get from $u$?

Background: I am try to get some geometric explanation for certain PDE similar to mean curvature equation.

I will be appreciate to any related answer and remark.

It seems I am too fast to ask the question without thinking carefully.

The equation I consider is find $w:\Omega \to R$ satisfying $$\frac{Du}{\sqrt{1+|\nabla u|^2}}=Dw. \tag{*}$$ Where $u:\Omega \to R$ is given and $u=0$ on $\partial \Omega$.

It is obvious this equation could not be solved for general $u$, the central obstacle is the Frobenius condition  (integrable condition). But I still wish to get some more information about this equation, in particular my eager could divide into two parts:

1. Could the $u$ for which $(*)$ is solvable be dense in some suitable space, maybe $C^{2}(\Omega)$?

2. For the $u$ which make $(*)$ locally solvable, what information of $w$ could we get from $u$?

Background: I am try to get some geometric explanation for certain PDE similar to mean curvature equation.

I will be appreciate to any related answer and remark.

It seems I am too fast to ask the question without thinking carefully.

The equation I consider is find $w:\Omega \to R$ satisfied : $$\frac{Du}{\sqrt{1+|\nabla u|^2}}=Dw. \tag{*}$$ Where $u:\Omega \to R$ is given and $u=0$ on $\partial \Omega$.

It is obvious this equation could not be solved for general $u$, the central obstacle is the frobenius condition(integrable condition).But I still wish to get some more information about this equation, in particular my eager could divide into two part:

1.Could the $u$ for which $(*)$ is solvable is dense in some suitable space, maybe $C^{2}(\Omega)$?

2.For the $u$ which make $(*)$, what information of $w$ could we get from $u$?

Background: I am try to get some geometric explanation for certain PDE similar to mean curvature equation.

I will be appreciate to any related answer and remark.

It seems I am too fast to ask the question without thinking carefully.

The equation I consider is find $w:\Omega \to R$ satisfied : $$\frac{Du}{\sqrt{1+|\nabla u|^2}}=Dw. \tag{*}$$ Where $u:\Omega \to R$ is given and $u=0$ on $\partial \Omega$.

It is obvious this equation could not be solved for general $u$, the central obstacle is the frobenius condition(integrable condition).But I still wish to get some more information about this equation, in particular my eager could divide into two part:

1.Could the $u$ for which $(*)$ is solvable is dense in some suitable space, maybe $C^{2}(\Omega)$?

2.For the $u$ which make $(*)$ locally solvable, what information of $w$ could we get from $u$?

Background: I am try to get some geometric explanation for certain PDE similar to mean curvature equation.

I will be appreciate to any related answer and remark.

It seems I am too fast to ask the question without thinking carefully.

The equation I consider is find $w:\Omega \to R$ satisfied : $$\frac{Du}{\sqrt{1+|\nabla u|^2}}=Dw. ... (*)$$ Where $u:\Omega \to R$ is given and $u=0$ on $\partial \Omega$.

It is obvious this equation could not be solved for general $u$, the central obstacle is the frobenius condition(integrable condition).But I still wish to get some more information about this equation, in particular my eager could divide into two part:

1.Could the $u$ for which $(*)$ is solvable is dense in some suitable space, maybe $C^{2}(\Omega)$?

2.For the $u$ which make $(*)$, what information of $w$ could we get from $u$?

Background: I am try to get some geometric explanation for certain PDE similar to mean curvature equation.

I will be appreciate to any related answer and remark.

It seems I am too fast to ask the question without thinking carefully.

The equation I consider is find $w:\Omega \to R$ satisfied : $$\frac{Du}{\sqrt{1+|\nabla u|^2}}=Dw. \tag{*}$$ Where $u:\Omega \to R$ is given and $u=0$ on $\partial \Omega$.

It is obvious this equation could not be solved for general $u$, the central obstacle is the frobenius condition(integrable condition).But I still wish to get some more information about this equation, in particular my eager could divide into two part:

1.Could the $u$ for which $(*)$ is solvable is dense in some suitable space, maybe $C^{2}(\Omega)$?

2.For the $u$ which make $(*)$, what information of $w$ could we get from $u$?

Background: I am try to get some geometric explanation for certain PDE similar to mean curvature equation.

I will be appreciate to any related answer and remark.