User eugene - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T02:13:46Z http://mathoverflow.net/feeds/user/12710 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54265/existence-uniqueness-and-smoothness-of-a-solution-to-a-first-order-pde-on-riema Existence, uniqueness, and smoothness of a solution to a first order PDE on Riemannian M Eugene 2011-02-03T23:54:35Z 2011-02-04T00:49:49Z <p>Let $\mathcal{M}$ be an $n$-dimensional compact Riemannian manifold, and let $\mathcal{A} \subset \mathcal{M}$ be an $n$-dimensional Riemannian submanifold. I wish to determine the local existence, uniqueness, and smoothness of the solution $Z$ to the following PDE:</p> <p>$$Z(x) \Delta d_{\mathcal{A}}(x) + 2 \nabla d_{\mathcal{A}}(x) \cdot \nabla Z(x) = 0$$ for $x \in \mathcal{M} \backslash \mathcal{A}$. Here $d_{\mathcal{A}}$ is the distance from point $x$ to $\mathcal{A}$. The Dirichlet boundary conditions are $$Z(x) = W(x)$$ for $x \in \partial \mathcal{A}$, where $W \in C^{\infty}(\mathcal{A})$.</p> <p>In general, $\nabla d_{\mathcal{A}}$ is locally of special bounded variation, and absolutely continuous for small enough open balls around $\partial \mathcal{A}$. </p> <p>The PDE above does not admit a global continuous solution. None of the methods for viscosity solutions of Hamilton-Jacobi problems apply even for open sets $U$ near $\partial \mathcal{A}$. Can anything can be said about the existence, uniqueness, continuity, and possibly more general smoothness of $Z$ for open balls $U$ within the singularity set (the generalized "cut locus"), i.e. near, $\partial \mathcal{A}$?</p> http://mathoverflow.net/questions/54270/why-does-excel-wolfram-come-up-with-differnt-stdev-than-i-do/54272#54272 Answer by Eugene for Why does Excel/Wolfram come up with differnt stdev than I do? Eugene 2011-02-04T00:20:00Z 2011-02-04T00:20:00Z <p>The unbiased variance estimator normalizes by $n-1$, instead of $n$. This may be the difference you're seeing.</p> http://mathoverflow.net/questions/54265/existence-uniqueness-and-smoothness-of-a-solution-to-a-first-order-pde-on-riema Comment by Eugene Eugene 2011-02-24T04:55:54Z 2011-02-24T04:55:54Z To follow up: I'm able to construct a solution around $\partial \mathcal A$ by applying the method of characteristics within each chart, and (by compactness) patching a finite number of them together. However, this only works locally around the boundary. I have tried to extend the solution towards $\mathcal C$, as you suggested can be done. However, I'm running into difficulties. Is there a direct approach (possibly given in a book or paper you can cite) for extending the method of characteristics to give a solution all the way to $\mathcal C$? http://mathoverflow.net/questions/54265/existence-uniqueness-and-smoothness-of-a-solution-to-a-first-order-pde-on-riema Comment by Eugene Eugene 2011-02-04T15:39:26Z 2011-02-04T15:39:26Z Sorry: to clarify, I agree that $d_{\mathcal{A}}$ is smooth near $\partial \mathcal{A}$; I just didn't know the solution method to use here. You provided me with the solution, thank you!