:point_right: On November 21, we discussed Weyl’s theorem on the asymptotics of eigenvalues of Dirichlet’s laplacian, including the necessary ingredients of the proof:

  • an explicit computation of eigenvalues for rectangles;

  • the min-max characterization of eigenvalues (plus a remark, on the best constant in Poincare’s inequality);

  • the inequality between Neuman and Dirichlet eigenvalues;

  • domain monotonicity;

  • Weyl’s theorem for special domains: finite unions of rectangles with pairwise disjoint interiors;

  • the derivation of the general case from the special one, based on the domain monotonicity.

We ended up with an informal discussion of a possible proof of regularity of eigenfunctions (solve Problems 2 and 3, posted on Nov. 20, to fill in all the gaps and details).

You will find this proof of Weyl’s theorem in Courant-Hilbert’s famous monograph Methods of mathematical physics (Volume 1, Chapter VI, Par. 4), or in W.A. Strauss Partial differential equations: an introduction (Chapter 11, Section 6). In both books the proof is written up in a elementary manner, for two and three dimensions - this might be more readable to you than my general presentation. An advanced popular discussion of this theorem and its relations to the heat equation, presented from a different point of view, can be found in Mark Kac’s paper Can one hear the shape of a drum?.