how does the basis of an inner product space change when the domain is deformed - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T16:11:03Z http://mathoverflow.net/feeds/question/12878 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12878/how-does-the-basis-of-an-inner-product-space-change-when-the-domain-is-deformed how does the basis of an inner product space change when the domain is deformed fredjalves 2010-01-24T22:46:11Z 2010-01-31T10:34:27Z <p>Assume we have a complete orthogonal system on a domain $D$, given by the eigenfunctions of the Laplacian on $D$. For example, the set ${e^{int}}$ on $[-\pi, \pi]$, or the spherical harmonics on the unit ball. Now consider a domain $D'$, which is "close" to D in some sense (the boundary of $D$ is close to the boundary of $D'$ in some suitable norm). </p> <p>Are the eigenfunctions of the Laplacian on $D$ close, in some sense, to the eigenfunctions of the Laplacian on $D'$? Does knowing the basis of $D$ help approximate the basis of $D'$ ? Any known results along these or similar lines appreciated. </p> http://mathoverflow.net/questions/12878/how-does-the-basis-of-an-inner-product-space-change-when-the-domain-is-deformed/12880#12880 Answer by jc for how does the basis of an inner product space change when the domain is deformed jc 2010-01-24T23:42:06Z 2010-01-24T23:42:06Z <p>The study of eigenfunctions and eigenvalues of the Laplacian is a well-developed field called <a href="http://en.wikipedia.org/wiki/Spectral%5Fgeometry" rel="nofollow">spectral geometry</a>. You might start with the first few lectures in the course notes <a href="http://www.math.ucdavis.edu/~saito/courses/LapEig/" rel="nofollow">here</a> or the book by Craioveanu et al. "Old and new aspects in spectral geometry". </p> <p>Perhaps some experts can point you to better introductory references.</p>