Skip to main content
MathOverflow

Return to Question

Added details based on the request of a user.
Source Link
RWien
RWien
  • 245
  • 1
  • 9

I'm researching on discrete/semi-discrete/smooth differential geometry. Recently, I transformed one of my surface theory problems (in the smooth scenario) into the following Moutard PDE

$$h_{uv} = q\,h$$

where $h(u,v)$ and $q(u,v): [0,a]\times [0,b] \subset \mathbb{R}^2 \rightarrow \mathbb{R}$ are smooth functions on $(0,a)\times (0,b)$. I have the functions $h(u,0)$ and $h(0,v)$ and $q(u,v)$ as well. By observing its discrete/semi-discrete counterparts I believe that it should have a unique solution by the boundaries that I provided. However, since my knowledge of PDE is very limited I don't know how to solve it in the smooth case.

I would very much appreciate if you can direct me on how to solve it. I would specially like to know if $h(u,0)$ and $h(0,v)$ being smooth is enough for getting the unique solution. Or should they be specifically analytic?

Extra Info: In the special case that I am considering $q(u,v) = \cos(\omega(u,v))$ where $\omega(u,v)$ is a solution of the famous sine-Gordon equation:

$$\omega_{uv} = \sin{\omega}$$

I'm researching on discrete/semi-discrete/smooth differential geometry. Recently, I transformed one of my surface theory problems (in the smooth scenario) into the following Moutard PDE

$$h_{uv} = q\,h$$

where $h(u,v)$ and $q(u,v): [0,a]\times [0,b] \subset \mathbb{R}^2 \rightarrow \mathbb{R}$ are smooth functions on $(0,a)\times (0,b)$. I have the functions $h(u,0)$ and $h(0,v)$ and $q(u,v)$ as well. By observing its discrete/semi-discrete counterparts I believe that it should have a unique solution by the boundaries that I provided. However, since my knowledge of PDE is very limited I don't know how to solve it in the smooth case.

I would very much appreciate if you can direct me on how to solve it. I would specially like to know if $h(u,0)$ and $h(0,v)$ being smooth is enough for getting the unique solution. Or should they be specifically analytic?

I'm researching on discrete/semi-discrete/smooth differential geometry. Recently, I transformed one of my surface theory problems (in the smooth scenario) into the following Moutard PDE

$$h_{uv} = q\,h$$

where $h(u,v)$ and $q(u,v): [0,a]\times [0,b] \subset \mathbb{R}^2 \rightarrow \mathbb{R}$ are smooth functions on $(0,a)\times (0,b)$. I have the functions $h(u,0)$ and $h(0,v)$ and $q(u,v)$ as well. By observing its discrete/semi-discrete counterparts I believe that it should have a unique solution by the boundaries that I provided. However, since my knowledge of PDE is very limited I don't know how to solve it in the smooth case.

I would very much appreciate if you can direct me on how to solve it. I would specially like to know if $h(u,0)$ and $h(0,v)$ being smooth is enough for getting the unique solution. Or should they be specifically analytic?

Extra Info: In the special case that I am considering $q(u,v) = \cos(\omega(u,v))$ where $\omega(u,v)$ is a solution of the famous sine-Gordon equation:

$$\omega_{uv} = \sin{\omega}$$

Source Link
RWien
RWien
  • 245
  • 1
  • 9

Solving the Moutard PDE

I'm researching on discrete/semi-discrete/smooth differential geometry. Recently, I transformed one of my surface theory problems (in the smooth scenario) into the following Moutard PDE

$$h_{uv} = q\,h$$

where $h(u,v)$ and $q(u,v): [0,a]\times [0,b] \subset \mathbb{R}^2 \rightarrow \mathbb{R}$ are smooth functions on $(0,a)\times (0,b)$. I have the functions $h(u,0)$ and $h(0,v)$ and $q(u,v)$ as well. By observing its discrete/semi-discrete counterparts I believe that it should have a unique solution by the boundaries that I provided. However, since my knowledge of PDE is very limited I don't know how to solve it in the smooth case.

I would very much appreciate if you can direct me on how to solve it. I would specially like to know if $h(u,0)$ and $h(0,v)$ being smooth is enough for getting the unique solution. Or should they be specifically analytic?