Hello:
This is my first time posting on mathoverflow. It is a fairly difficult question. I've posted it on Math Stack Exchange, and got one upvote, no comments and no answers. So here goes:
In 1-2 papers, Sternberg, Williams, and Ziemer proved the following result: if $\Omega$ is a bounded connected open set in $\mathbb{R}^n$ whose boundary is smooth and has positive mean curvature almost everywhere (actually a weaker condition stated in geometric measure-theoretic terms), and $f \in C(\partial \Omega)$, then there exists $u\in BV(\Omega)$ such that
\begin{equation} \int_\Omega |Du| = \inf {\int_\Omega |Dw| \mid w \in BV(\Omega),\ w|_{\partial \Omega} =f }, \end{equation}
and this $u$ is continuous. By $w|_{\partial\Omega}=f$, I mean that the trace of $w$ on $\partial\Omega$ is $f$. I think they also proved uniqueness. I don't have the information at hand right now, but if anyone asks, I will edit my question to provide specifics, such as the papers' titles, journals, dates, and whether and where they proved uniqueness.
I am pretty sure I can show that this kind of result is impossible if you assume merely $f\in L^1 (\partial\Omega)$. Specifically, letting $D$ be the unit disc in $\mathbb{R}^2$, I have found a function $f\in L^1(\partial D)$ such that there does not exist $u \in BV(D)$ with $\int_D |Du|=\inf{\int_D |Dw|\mid w\in BV(D),\ w|_{\partial D} =f}$. It is fairly difficult to verify that $f$ has this property because the entire boundary of $D$ has positive mean curvature. Working with other domains, such as a non-convex domain or a square, it is much easier to construct such an $f$ and prove that it has this property.
I apologize that the equations did not come out 100% correct above, although the exact same $\LaTeX$ code worked in Math Stack Exchange. It should be apparent what I mean.
My question is, is this result worth taking the time to write up and try to publish? I think it's interesting, but frankly, I don't know if anyone else will. I don't want to take the time to write and typeset a proof unless the paper has a chance of being accepted by a good journal. I am not an expert in geometric measure theory and I don't know if this would be considered significant.
Any opinions? If anyone has an affirmative opinion, can you suggest a journal to consider?
I added the tag geometric-measure-theory. There does not seem to be a bounded-variation tag like there is in Math Stack Exchange.
Stefan
