Stability of the spectrum for perturbations of the boundary - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:15:42Z http://mathoverflow.net/feeds/question/98789 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98789/stability-of-the-spectrum-for-perturbations-of-the-boundary Stability of the spectrum for perturbations of the boundary Piero D'Ancona 2012-06-04T17:24:51Z 2012-07-20T23:46:35Z <p>Consider the Laplace operator on a smooth bounded open set with Dirichlet boundary conditions. I need some result of the following type: if one perturbs the boundary in a suitable sense to be determined, say depending on a small parameter $\epsilon$, then it is possible to order eigenvalues and eigenfunctions so that they are smooth functions of $\epsilon$ and spatial variables. At first this seemed a very natural result, but the total scarcity of references suggests otherwise, and I am starting to think that this kind of result is rather difficult, if one wants to achieve some generality. Has anyone encountered any result in this direction?</p> http://mathoverflow.net/questions/98789/stability-of-the-spectrum-for-perturbations-of-the-boundary/98791#98791 Answer by Denis Serre for Stability of the spectrum for perturbations of the boundary Denis Serre 2012-06-04T17:59:32Z 2012-06-04T17:59:32Z <p>This is true, as long as your domain depends smoothly upon <strong>one</strong> real parameter. Say that you are insterested in the $n$ first eigenvalues. Using a Lyapunov-Schmidt procedure, you may reduce to the situation of an $n\times n$ symmetric matrix $S(\epsilon)$. Then look at Kato's book in the Grundlehren series.</p> <p>If instead your domain depends on two or more parameters, the matrix $S$ will depend on several variables, and the eigenvalues will not be smooth functions, unless they remain simple.</p> http://mathoverflow.net/questions/98789/stability-of-the-spectrum-for-perturbations-of-the-boundary/98792#98792 Answer by Rafe Mazzeo for Stability of the spectrum for perturbations of the boundary Rafe Mazzeo 2012-06-04T18:08:17Z 2012-06-04T18:08:17Z <p>Look at Perturbation of the boundary in boundary-value problems in partial differential equations'' by Dan Henry, London Math Society Lecture Notes #318</p> http://mathoverflow.net/questions/98789/stability-of-the-spectrum-for-perturbations-of-the-boundary/102786#102786 Answer by marrocos for Stability of the spectrum for perturbations of the boundary marrocos 2012-07-20T23:46:35Z 2012-07-20T23:46:35Z <p>Hello, My name is Marcus Morocco. My doctoral thesis was on exactly these issues. I calculated the expressions for the first and second derivatives of eigenvalues ​​and eigenvectors of the laplace opelador ccom Neumann boundary condition. If interest can send the file.</p>