Non-Normal derivative boundary conditions for a PDE - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T01:39:14Z http://mathoverflow.net/feeds/question/51282 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51282/non-normal-derivative-boundary-conditions-for-a-pde Non-Normal derivative boundary conditions for a PDE John 2011-01-06T05:06:37Z 2011-01-06T15:35:06Z <p>For a second order PDE (lets say the Laplace equation), is there a problem with specifying neumann boundary conditions, which instead of being specified in the direction normal to the boundary are instead specified in some other direction.</p> <p>For example, could one specify the derivative in the direction of the boundary.</p> <p>Would this lead to a unique solution?</p> <p>This seems like a stupid question, but I couldn't find any information on it.</p> http://mathoverflow.net/questions/51282/non-normal-derivative-boundary-conditions-for-a-pde/51286#51286 Answer by drbobmeister for Non-Normal derivative boundary conditions for a PDE drbobmeister 2011-01-06T07:41:14Z 2011-01-06T07:46:30Z <p>Suppose for definiteness we work with Laplace's equation $\triangledown^{2}u =0$ on the unit disk in $R^{2}$. Assuming things are somewhat smooth, suppose one specified the tangential, instead of the normal, derivative of $u$, i.e. specified $\partial{u} / \partial{\theta}$ on the unit circle. Picking any point $\theta_{0}$ on the circle, one could integrate in $\theta$ to obtain $u(\theta)$ on the circle:</p> <p><code>$u(\theta) = \int_{\theta_{0}}^{\theta} \partial{u}/\partial{\theta}(\alpha) d \alpha + u(\theta_{0})$</code>.</p> <p>So a tangential derivative really specifies a Dirichlet boundary condition. The additive constant in the above integral merely corresponds to a constant solution in the disk, i.e. it shifts the solution $u$ by a constant amount.</p> <p>You can in general freely specify Dirichlet <em>or</em> Neumann conditions, but not both. So take your pick, a tangential <em>or</em> a normal derivative for the boundary condition.</p> http://mathoverflow.net/questions/51282/non-normal-derivative-boundary-conditions-for-a-pde/51319#51319 Answer by Michael Renardy for Non-Normal derivative boundary conditions for a PDE Michael Renardy 2011-01-06T15:35:06Z 2011-01-06T15:35:06Z <p>There is an extensive literature on oblique derivative boundary conditions. A Google search with the keyword "oblique derivative" will get you started.</p>