Is this chapter about analysis? Well, yes and no. The topic is resonance modes, which are the eigenfrequencies (eigenvalues, up to a square root) of the Laplace operator for polygonal or other regions. Usually, one goes into a certain amount of analysis to show that this notion makes good theoretical sense, but here we think of the subject as spectral geometry, which means focusing on the interaction between the resonance modes and the geometry of the region. In fact, we really only look at the lowest resonance mode. There is a substantial math (and physics) relation between this and the previous chapter, but I can’t bring that out here. Instead, we only observe one parallel, which is that tiling-by-reflection comes back as an idea. Next, the extremal characterization of the lowest eigenvalue is something that every student of mathematics should know, and also particularly useful in this situation. Finally, we look at Steiner symmetrization, which is exactly the kind of topic this course is aiming for: entirely elementary, requiring a bit of creativity to use, and surprising in its appearance here. I am ashamed to say I had never heard of it prior to preparing this chapter!
A general reference is Chapters 5 and 7 of G. Polya and G. Szegö, Isoperimetric inequalities in mathematical physics, Princeton Univ. Press, 1951. I’ve also benefited from the more recent survey R. Laugesen and B. Siudeja, Triangles and other special domains, in: Shape optimization and spectral theory, De Gruyter Open, 2017, p. 149–200. I read about the Ritz (or Rayleigh-Ritz) method in J. Levandoski, Lecture notes for Math 220B (Applied PDE), Stanford, Summer 2003.
