Is this Stefan-type problem an open problem? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T12:26:31Z http://mathoverflow.net/feeds/question/88655 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88655/is-this-stefan-type-problem-an-open-problem Is this Stefan-type problem an open problem? Kanna Nakamura 2012-02-16T18:18:32Z 2012-02-16T18:18:32Z <p>I am looking for a weak-formulation that would give me an existence and uniqueness of a solution of a Stefan-type problem. It is basically a 2-phase Stefan problem in 2D except that the free-boundary does not have the nice "solution $u=0$ " criteria.</p> <p>Instead, it has a robin boundary condition $$u+\partial_{\nu}u = g,$$ where $g$ is a given function and $\partial_{\nu}$ denotes the derivative in direction of the normal vector to the free boundary, at <em>the free boundary</em> (not fixed boundary).</p> <p>(The above robin condition at the free boundary is assigned in addition to the typical condition on the velocity of the free boundary: $v = \partial_{\nu} u^+ - \partial_{\nu}u^-$.)</p> <p>Does anyone know if this is an open problem? Or does anyone know a work related to this? I will greatly appreciate any reference to a 2-phase Stefan-type problem that is not the classical Stefan problem. Thank you!</p>