Lecture 11: 1 October 2010 Using the Matlab numgrid and delsq functions (see also this demo), similar to pset 4, computed in an "L" shaped domain (Dirichlet boundaries). 2׏the first few eigenvalues and eigenvectors of -So far, we know that if A=A* (that is, the operator is real-symmetric/Hermitian/self-adjoint), it has real λ, orthogonal eigenfunctions, and is usually diagonalizable (complete basis of eigenfunctions). Note that A=A* depends not only on A, but also on the boundary conditions and the inner product 〈⋅,·〉. If A is positive definite (〈u,Au〉>0 for u≠0), then we further know that even in cases we can't solve (like the "L" 2׏λ>0. This tells us a lot about operators like -domain). However, there is something more we can say. Presented the min–max theorem (also called the variational theorem). If we define the Rayleigh quotient R(u)=〈u,Au〉/〈u,u〉 for A=A* [noting that R(u) is always real], then for any u≠0 it follows that λmin≤R(u)≤λmax, i.e. R(u) is bounded above and below by the maximum and minimum eigenvalues of A (if any). Gave a simple "proof" of this fact assuming that A is diagonalizable (caveat: this is a nontrivial assumption). If u is an eigenfunction, then R(u) is the corresponding eigenvalue. For a PDE operator A, typically the eigenvalues have unbounded magnitude. Say A is positive-definite; in this case there will be no λmax, but there will be a λmin, so the theorem reduces to λmin≤R(u). Suppose we number the eigenvalues in ascending order λ1≤λ2≤λ3≤···. Then: • λ1 is the minimum value of R(u) over all u≠0, achieved for u=u1. • λ2 is the minimum value of R(u) over all u≠0 orthogonal to u1, achieved for u=u2. • ... • λk is the minimum value of R(u) over all u≠0 orthogonal to uj with j<k, achieved for u=uk. ׏׏For the case of A=-2, R(u)=∫| u|2/∫|u|2, so we conclude that: uk is the function that oscillates as little as possible [to minimize the numerator of R(u)] while satisfying the boundary conditions and being orthogonal to the lower-λ eigenfunctions. Using this fact, we can actually make reasonable guesses for what the first few eigenfunctions look like, qualitatively. Showed this for with the "L" domain, in Matlab. 2׏- We can even do this for PDEs in inhomogeneous media, which are much harder to solve with Dirichlet boundaries, where w(x) is some 2׏analytically. For the case of A=-[1/w(x)] function >0, analogous to pset 2 this is self-adjoint for 〈u,v〉=∫w u v. In this case, [to2, so we conclude that uk is the function that oscillates as little as possiblew|u|∫/2u|׏R(u)=∫|minimize the numerator of R(u)] and is as concentrated as possible in the higher-w regions [to maximize the denominator] while satisfying the boundary conditions and being orthogonal to the lower-λ eigenfunctions. However, minimizing the numerator and maximizing the denominator are contradictory criteria. For example, suppose that w(x)=1 everywhere except for a small region where w(x)=w0>1. In order to concentrate in this small region, u(x) will have to have bigger slope (sacrificing the numerator). As w0 increases, we expect the denominator to "win"and the concentration to increase, while for w0 close to 1 the eigenfunctions should be similar to script that I wrote.mmdemo.m. Showed an example of this, in Matlab, with an 2׏the case of -Running: mmdemo(linspace(1,20,200), 3); makes an animation of the first three eigenfunctions in the "L" domain for w0 going from 1 to 20 in a small circular region at the upper-right of the plot. As w0 increases, we see the eigenfunctions being "sucked" into this circular region, as they try to maximize the denominator of R(u) above all other concerns. (In fact, for w0→∞, they are completely localized in the circular region, and approach the Bessel solutions we saw for the Laplacian in the cylinder.) Further reading: See, for example min-max theorem in Wikipedia, although this presentation is rather formal. Unfortunately, most of the discussion you will find of this principle online and in textbooks is either (a) full of formal functional analysis or (b) specific to quantum mechanics +V for some "potential-energy" function V(x)]. You can find 2׏[where the operator is A=-another Laplacian demo here.MIT OpenCourseWare http://ocw.mit.edu 18.303 Linear Partial Differential Equations: Analysis and Numerics Fall 2010 For information about citing these materials or our Terms of Use, visit: