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My question is, essentially, suppose I have two simply connected subset of $\mathbb{R}^n$, if I know that the boundaries of both are very close, how can I bound the determinant of the Jacobian between them. I can probably assume anything reasonable on $\Omega$, and possibly $\partial \Omega$.

More precisely:

Suppose I have a bounded simply connected set $\Omega \subset \mathbb{R}^2$, with a boundary $\partial \Omega$ that is a smooth as I like. Think of $\Omega$ as some kind of distorted circle. $\partial \Omega$ is parametrised by $X : [0, 2 \pi ] \rightarrow \partial \Omega$.

Suppose I also have an approximation to this, given by $\Omega_h$ where $\Omega_h$ is a polyhedral shape. It has a boundary $\partial \Omega_h$ that is parametrised by $X_h : [0, 2 \pi ] \rightarrow \partial \Omega_h$.

Since $\Omega_h$ approximates $\Omega$, we have that $\| X - X_h \|^2_{L^2 [0, 2 \pi] } \leq C_1 h^2$, where $C_1$ is a positive constant and $h$ is a small positive number that I control.

If I have a mapping $ \omega : \Omega_h \rightarrow \Omega $, let $J$ be the determinant of the Jacobian of $\omega$. Are there any theorems that state something like:

$ \| X_h - X \|^2_{L^2 [0, 2 \pi]} \leq C_1 h^2 \implies |J - 1|^2 \leq C_2 h^2 $,

where $C_2$ is once again a positive constant. Anything similar would be greatly appreciated.

Edit: Perhaps my question is more about whether such an $\omega$ exists.

My question is, essentially, suppose I have two simply connected subset of $\mathbb{R}^n$, if I know that the boundaries of both are very close, how can I bound the determinant of the Jacobian between them. I can probably assume anything reasonable on $\Omega$, and possibly $\partial \Omega$.

More precisely:

Suppose I have a bounded simply connected set $\Omega \subset \mathbb{R}^2$, with a boundary $\partial \Omega$ that is as smooth as I like. Think of $\Omega$ as some kind of distorted circle. $\partial \Omega$ is parametrised by $X : [0, 2 \pi ] \rightarrow \partial \Omega$.

Suppose I also have an approximation to this, given by $\Omega_h$ where $\Omega_h$ is a polyhedral shape. It has a boundary $\partial \Omega_h$ that is parametrised by $X_h : [0, 2 \pi ] \rightarrow \partial \Omega_h$.

Since $\Omega_h$ approximates $\Omega$, we have that $\| X - X_h \|^2_{L^2 [0, 2 \pi] } \leq C_1 h^2$, where $C_1$ is a positive constant and $h$ is a small positive number that I control.

If I have a mapping $ \omega : \Omega_h \rightarrow \Omega $, let $J$ be the determinant of the Jacobian of $\omega$. Are there any theorems that state something like:

$ \| X_h - X \|^2_{L^2 [0, 2 \pi]} \leq C_1 h^2 \implies |J - 1|^2 \leq C_2 h^2 $,

where $C_2$ is once again a positive constant. Anything similar would be greatly appreciated.

Edit: Perhaps my question is more about whether such an $\omega$ exists.

My question is, essentially, suppose I have two simply connected subset of $R^n$, if I know that the boundary's of both are very close, how can I bound the determinant of the Jacobian between them. I can probably assume anything reasonable on $\Omega$, and possibly $\delta \Omega$.

More precisely:

Suppose I have a bounded simply connected set $\Omega \subset R^2$, with a boundary $\delta \Omega$ that is a smooth as I like. Think of $\Omega$ as some kind of distorted circle. $\delta \Omega$ is parametrised by $X : [0, 2 \pi ] \rightarrow \delta \Omega$.

Suppose I also have an approximation to this, given by $\Omega_h$ where $\Omega_h$ is a polyhedral shape. It has a boundary $\delta \Omega_h$ that is parametrised by $X_h : [0, 2 \pi ] \rightarrow \delta \Omega_h$.

Since $\Omega_h$ approximates $\Omega$, we have that $|| X - X_h ||^2_{L^2 [0, 2 \pi] } \leq C_1 h^2$, where $C_1$ is a positive constant and h is a small positive number that I control.

If I have a mapping $ \omega : \Omega_h \rightarrow \Omega $. Let $J$ be the determinant of the Jacobian of $\omega$. Are there any theorems that state something like:

$ || X_h - X ||^2_{L^2 [0, 2 \pi]} \leq C_1 h^2 \implies |J - 1|^2 \leq C_2 h^2 $,

where $C_2$ is once again a positive constant. Anything similar would be greatly appreciated.

Edit: Perhaps my question is more about whether such an $\omega$ exists.

My question is, essentially, suppose I have two simply connected subset of $\mathbb{R}^n$, if I know that the boundaries of both are very close, how can I bound the determinant of the Jacobian between them. I can probably assume anything reasonable on $\Omega$, and possibly $\partial \Omega$.

More precisely:

Suppose I have a bounded simply connected set $\Omega \subset \mathbb{R}^2$, with a boundary $\partial \Omega$ that is a smooth as I like. Think of $\Omega$ as some kind of distorted circle. $\partial \Omega$ is parametrised by $X : [0, 2 \pi ] \rightarrow \partial \Omega$.

Suppose I also have an approximation to this, given by $\Omega_h$ where $\Omega_h$ is a polyhedral shape. It has a boundary $\partial \Omega_h$ that is parametrised by $X_h : [0, 2 \pi ] \rightarrow \partial \Omega_h$.

Since $\Omega_h$ approximates $\Omega$, we have that $\| X - X_h \|^2_{L^2 [0, 2 \pi] } \leq C_1 h^2$, where $C_1$ is a positive constant and $h$ is a small positive number that I control.

If I have a mapping $ \omega : \Omega_h \rightarrow \Omega $, let $J$ be the determinant of the Jacobian of $\omega$. Are there any theorems that state something like:

$ \| X_h - X \|^2_{L^2 [0, 2 \pi]} \leq C_1 h^2 \implies |J - 1|^2 \leq C_2 h^2 $,

where $C_2$ is once again a positive constant. Anything similar would be greatly appreciated.

Edit: Perhaps my question is more about whether such an $\omega$ exists.

My question is, essentially, suppose I have two simply connected subset of $R^n$, if I know that the boundary's of both are very close, how can I bound the determinant of the Jacobian between them. I can probably assume anything reasonable on $\Omega$, and possibly $\delta \Omega$.

More precisely:

Suppose I have a bounded simply connected set $\Omega \subset R^2$, with a boundary $\delta \Omega$ that is a smooth as I like. Think of $\Omega$ as some kind of distorted circle. $\delta \Omega$ is parametrised by $X : [0, 2 \pi ] \rightarrow \delta \Omega$.

Suppose I also have an approximation to this, given by $\Omega_h$ where $\Omega_h$ is a polyhedral shape. It has a boundary $\delta \Omega_h$ that is parametrised by $X_h : [0, 2 \pi ] \rightarrow \delta \Omega_h$.

Since $\Omega_h$ approximates $\Omega$, we have that $|| X - X_h ||^2_{L^2 [0, 2 \pi] } \leq C_1 h^2$, where $C_1$ is a positive constant and h is a small positive number that I control.

If I have a mapping $ \omega : \Omega_h \rightarrow \Omega $. Let $J$ be the determinant of the Jacobian of $\omega$. Are there any theorems that state something like:

$ || X_h - X ||^2_{L^2 [0, 2 \pi]} \leq C_1 h^2 \implies |J - 1|^2 \leq C_2 h^2 $,

where $C_2$ is once again a positive constant. Anything similar would be greatly appreciated.

My question is, essentially, suppose I have two simply connected subset of $R^n$, if I know that the boundary's of both are very close, how can I bound the determinant of the Jacobian between them. I can probably assume anything reasonable on $\Omega$, and possibly $\delta \Omega$.

More precisely:

Suppose I have a bounded simply connected set $\Omega \subset R^2$, with a boundary $\delta \Omega$ that is a smooth as I like. Think of $\Omega$ as some kind of distorted circle. $\delta \Omega$ is parametrised by $X : [0, 2 \pi ] \rightarrow \delta \Omega$.

Suppose I also have an approximation to this, given by $\Omega_h$ where $\Omega_h$ is a polyhedral shape. It has a boundary $\delta \Omega_h$ that is parametrised by $X_h : [0, 2 \pi ] \rightarrow \delta \Omega_h$.

Since $\Omega_h$ approximates $\Omega$, we have that $|| X - X_h ||^2_{L^2 [0, 2 \pi] } \leq C_1 h^2$, where $C_1$ is a positive constant and h is a small positive number that I control.

If I have a mapping $ \omega : \Omega_h \rightarrow \Omega $. Let $J$ be the determinant of the Jacobian of $\omega$. Are there any theorems that state something like:

$ || X_h - X ||^2_{L^2 [0, 2 \pi]} \leq C_1 h^2 \implies |J - 1|^2 \leq C_2 h^2 $,

where $C_2$ is once again a positive constant. Anything similar would be greatly appreciated.

Edit: Perhaps my question is more about whether such an $\omega$ exists.