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edited Mar 10, 2015 at 15:51

Joonas Ilmavirta

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I would like to know the following: Let $M$ be a smooth surface with connected boundary. Let $f: M \rightarrow \mathbb{R}^3$ be an embedding such that the boundary $\partial M$ of $M$ is mapped onto a Jordan curve $\gamma$. We let furthermore $\nu$ be the choice of an unit normal for such an embedding and we denote by $H$ the mean curvature with respect to $\nu$. \ Now

Now we consider the following: Let $f_t$ be a smooth family of embeddings $f_t: M \rightarrow \mathbb{R}^3$ which fulfills the equation $$ \frac{\partial f_t}{\partial t}= - H \cdot \nu$$ in the interior of $M$ and satisfies $f_t(x)=f(x)$ for $x \in \partial M$. We call such a family a solution to the Dirichlet problem. \ I

I would now like to know about the existence of such families $(f_t)$. I know that usually there is no smooth solution for times $t \in [0, T)$ for some $T>0$, since for Dirichlet problems we usually do not have smoothness at the boundary for $t=0$. \ But

But I think that the following should be true: There exist $T_1, T_2 >0$ and a smooth family $(f_t)$ for $t \in (T_1, T_2)$ solving the above Dirichlet problem. Do you agree with me? If yes, do you maybe know a reference, from which the supposed result follows?

I would like to know the following: Let $M$ be a smooth surface with connected boundary. Let $f: M \rightarrow \mathbb{R}^3$ be an embedding such that the boundary $\partial M$ of $M$ is mapped onto a Jordan curve $\gamma$. We let furthermore $\nu$ be the choice of an unit normal for such an embedding and we denote by $H$ the mean curvature with respect to $\nu$. \ Now we consider the following: Let $f_t$ be a smooth family of embeddings $f_t: M \rightarrow \mathbb{R}^3$ which fulfills the equation $$ \frac{\partial f_t}{\partial t}= - H \cdot \nu$$ in the interior of $M$ and satisfies $f_t(x)=f(x)$ for $x \in \partial M$. We call such a family a solution to the Dirichlet problem. \ I would now like to know about the existence of such families $(f_t)$. I know that usually there is no smooth solution for times $t \in [0, T)$ for some $T>0$, since for Dirichlet problems we usually do not have smoothness at the boundary for $t=0$. \ But I think that the following should be true: There exist $T_1, T_2 >0$ and a smooth family $(f_t)$ for $t \in (T_1, T_2)$ solving the above Dirichlet problem. Do you agree with me? If yes, do you maybe know a reference, from which the supposed result follows?

I would like to know the following: Let $M$ be a smooth surface with connected boundary. Let $f: M \rightarrow \mathbb{R}^3$ be an embedding such that the boundary $\partial M$ of $M$ is mapped onto a Jordan curve $\gamma$. We let furthermore $\nu$ be the choice of an unit normal for such an embedding and we denote by $H$ the mean curvature with respect to $\nu$.

Now we consider the following: Let $f_t$ be a smooth family of embeddings $f_t: M \rightarrow \mathbb{R}^3$ which fulfills the equation $$ \frac{\partial f_t}{\partial t}= - H \cdot \nu$$ in the interior of $M$ and satisfies $f_t(x)=f(x)$ for $x \in \partial M$. We call such a family a solution to the Dirichlet problem.

I would now like to know about the existence of such families $(f_t)$. I know that usually there is no smooth solution for times $t \in [0, T)$ for some $T>0$, since for Dirichlet problems we usually do not have smoothness at the boundary for $t=0$.

But I think that the following should be true: There exist $T_1, T_2 >0$ and a smooth family $(f_t)$ for $t \in (T_1, T_2)$ solving the above Dirichlet problem. Do you agree with me? If yes, do you maybe know a reference, from which the supposed result follows?

dg.differential-geometry ap.analysis-of-pdes

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asked Mar 10, 2015 at 15:47

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Existence for special Dirichlet problem

I would like to know the following: Let $M$ be a smooth surface with connected boundary. Let $f: M \rightarrow \mathbb{R}^3$ be an embedding such that the boundary $\partial M$ of $M$ is mapped onto a Jordan curve $\gamma$. We let furthermore $\nu$ be the choice of an unit normal for such an embedding and we denote by $H$ the mean curvature with respect to $\nu$. \ Now we consider the following: Let $f_t$ be a smooth family of embeddings $f_t: M \rightarrow \mathbb{R}^3$ which fulfills the equation $$ \frac{\partial f_t}{\partial t}= - H \cdot \nu$$ in the interior of $M$ and satisfies $f_t(x)=f(x)$ for $x \in \partial M$. We call such a family a solution to the Dirichlet problem. \ I would now like to know about the existence of such families $(f_t)$. I know that usually there is no smooth solution for times $t \in [0, T)$ for some $T>0$, since for Dirichlet problems we usually do not have smoothness at the boundary for $t=0$. \ But I think that the following should be true: There exist $T_1, T_2 >0$ and a smooth family $(f_t)$ for $t \in (T_1, T_2)$ solving the above Dirichlet problem. Do you agree with me? If yes, do you maybe know a reference, from which the supposed result follows?

ap.analysis-of-pdes