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WEYL's LAW 1. Introduction Let Ω ⊂ R N Be a Bounded Domain, Then We

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WEYL's LAW 1. Introduction Let Ω ⊂ R N Be a Bounded Domain, Then We WEYL'S LAW MATT STEVENSON Abstract. These are notes for a talk given in the Student Analysis Seminar at the University of Michigan. The n Laplacian on a bounded domain in R has a discrete set of Dirichlet eigenvalues, accumulating only at 1. Let N(λ) be the number of eigenvalues less than λ, then Weyl's law asserts that the first term of the asymptotic expansion of N(λ) depends only on the dimension and the volume of the domain. Here we sketch the 1912 proof of Weyl, following the expositions of [1], [5], [6]. 1. Introduction Let Ω ⊂ Rn be a bounded domain, then we are interested in solving the eigenvalue problem ∆' = λϕ on Ω (with either Dirichlet or Neumann boundary conditions). The spectral theorem asserts that there exists an 2 orthonormal basis fung of L (Ω) consisting of (Dirichlet or Neumann) eigenfunctions, where the corresponding eigenvalues λ1 ≤ ::: ≤ λn ≤ ::: ! +1 accumulate only at +1 and each have finite multiplicity. Let N(λ) := #fλj ≤ λg be the eigenvalue counting function, then it is natural to consider the asymptotic behaviour of N(λ) as λ ! +1. Consider the following simple examples. Example. Let Ω = [0; a], then the Dirichlet eigenvalue problem becomes the following: ( d2 dx2 '(x) = λϕ(x); '(0) = '(a) = 0: kπ Solving the boundary value problem, the Dirichlet eigenfunctions are 'k(x) = sin a x with corresponding kπ 2 Dirichlet eigenvalues λk = a , for k 2 N. Consider the eigenvalue counting function: ( ) ( p ) kπ 2 a λ a p N(λ) = # fk 2 : λ < λg = # k 2 : < λ = max k 2 : k < ∼ λ. N k N a N π π Example. Let Ω = [0; a] × [0; b], then using separation of variables we can deduce that the Dirichlet eigenfunc- jπ kπ jπ 2 kπ 2 tions of the Laplacian are 'j;k(x; y) = sin a x sin b y with eigenvalues λj;k = a + b , for j; k 2 N. The eigenvalue counting function is then given by ( ) jπ 2 kπ 2 N(λ) = #f(j; k) 2 × : λ < λg = # (j; k) 2 × : + < λ : N N j;k N N a b Let E = f(x; y) 2 2 :( p x )2 + ( p y )2 ≤ 1; x ≥ 0; y ≥ 0g be the area in the first quadrant cut out by the λ R λa/π λb/π ellipse, then N(λ) is the number of integer lattices points in Eλ. To each pair (j; k) 2 N × N such that λj;k < λ, we can associate the unit-area square [j − 1; j] × [k − 1; k]. Then, p p X π λa λb area(Ω) N(λ) = area([j − 1; j] × [k − 1; k]) ≤ area(E ) = = λ. λ 4 π π 4π (j;k)2N×N : λj;k<λ Date: October 31, 2014. 1 2 MATT STEVENSON 0 To get a lower bound on N(λ), translate Eλ down by 1 unit and to the left by 1 unit to get the set Eλ. As before, we compute areas to find that p p p p π λa λb λa λb area(Ω) perimeter(Ω)p N(λ) ≥ − − = λ − λ. 4 π π π π 2π 2π area(Ω) Therefore, we may conclude that the eigenvalue counting function is asymptotically given by N(λ) ∼ 4π λ. These results can be generalized to give the first term of the asymptotic expansion of the eigenvalue counting function in terms of the dimension and the volume of the domain. Lorentz first conjectured this result at a conference at G¨ottingenin 1910, whence Hilbert predicted it would not be solved in his lifetime. Weyl, as a graduate student, solved it 4 months later. The modern formulation of this result is the following. Theorem. (Weyl, 1911) Let N(λ) be the Dirichlet eigenvalue counting function on a bounded domain Ω, then n=2 N(λ) ∼ cnvol(Ω)λ !n n where cn = (2π)n is a constant depending only on the dimension n and !n is the volume of the unit ball in R . Remark. The proof of Weyl's law for a 2-dimensional rectangle generalizes to n-dimensional rectangles. 2. Proof of Weyl's Law for Bounded Domains When analyzing the solutions to PDEs, it is more natural to work in a Sobolev space rather than an Lp-space. Recall that H1(Ω) is the closed subspace of L2(Ω) consisting of weakly-differentiable functions; it carries an inner product Z hu; viH1 := ru · rv + uv; Ω which makes H1(Ω) into a Hilbert space. Remark that the space D(Ω) of compactly-supported smooth functions 1 1 1 on Ω is contained in H (Ω); their completion with respect to the H -norm is denoted H0 (Ω), and it is the closed subspace of H1(Ω) consisting of trace-zero functions. These two spaces are the appropriate ones in which to frame our eigenvalue problem. Let u 2 C2(Ω)\C(Ω) be a Dirichlet eigenfunction of −∆ with eigenvalue λ, then for any v 2 D(Ω) integration by parts gives that Z Z Z Z ru · rv = u@ν v + (−∆u)v = λuv: Ω @Ω Ω Ω 1 We say that u 2 H0 (Ω) is a weak solution of the Dirichlet eigenvalue problem if it satisfies the above equality 1 1 for any v 2 H0 Ω. Similarly, u 2 H (Ω) is a weak solution of the Neumann eigenvalue problem if it satisfies 1 1 1 the above equality for all v 2 H (Ω). Let V denote either H (Ω) or H0 (Ω) and let fλng denote Neumann or Dirichlet eigenvalues, as appropriate. The spectral theorem applied to the resolvent of the Laplacian asserts that there exists an orthonormal basis 2 fung of L (Ω) consisting of eigenfunctions of −∆, where the eigenvalues fλng each have finite multiplicity and accumulate only at +1. Moreover, the eigenfunctions fung form an orthonormal basis of V with respect to the H1-inner product. There is a variational characterization of these eigenvalues, called the Minimax Principle. Lemma. (Minimax Principle, Version 1) Let V and fλng be as above. Then, λn = inf ρ(u) = sup ρ(u); u2H n−1 u2span(u1;:::;un) kruk2 ? L2 where Hn−1 = V \ span(u1; : : : ; un−1) and ρ(u) := kuk2 is the Rayleigh quotient. L2 WEYL'S LAW 3 Proof. For any u 2 V , write 1 1 2 X X 2 kukL2 = hu; ui = h hu; uniun; ui = jhun; uij ; n=1 n=1 using the fact that hun; uni = 1. Then, the norm of the gradient can be expressed as Z Z 1 1 2 X X 2 krukL2 = ru · ru ≈ −∆u · u = h−∆ hu; uniun; ui = λnjhu; unij ; Ω Ω n=1 n=1 where the second equality is not entirely justified, as we do not know a priori that u is twice weakly-differentiable. If u is now taken in Hn−1, then hu; uki = 0 for k = 0; : : : ; n − 1. Consequently, 1 1 1 2 X 2 X 2 X 2 2 krukL2 = λkjhu; ukij ≥ λn jhu; ukij = λn jhu; ukij = λnkukL2 ; k=n k=n k=1 where the second-to-last equality follows by adding back the zero terms. Rearranging, for any u 2 Hn−1, we have the inequality ρ(u) ≥ λn, and hence the inequality holds for the infimum over u 2 Hn−1. Furthermore, un 2 Hn−1 and ρ(un) = λn, so the infimum is attained by a function of Hn−1, so we have equality. Lemma. (Minimax Principle, Version 2) λn = inf sup ρ(u); X2Φn(V ) u2X where Φn(V ) = fn-dimensional linear subspaces X ⊂ V g. Proof. As span(u1; : : : ; un) 2 Φn(V ), version 1 of the Minimax Principle gives the inequality λn = sup ρ(u) ≥ inf sup ρ(u): X2Φ (V ) u2span(u1;:::;un) n u2X Conversely, for any X 2 Φn(V ), there is nonzero v 2 X \ Hn−1 (by dimension count); hence, ρ(v) ≥ inf ρ(u) = λn =) sup ρ(v) ≥ λn =) inf sup ρ(v) ≥ λn: u2Hn−1 v2X X2Φn(V ) v2X Lemma. Let µn be the Dirichlet eigenvalues on Ω and let νn be the Neumann eigenvalues on Ω, then νn ≤ µn. 1 1 1 1 Proof. As H0 (Ω) ⊂ H (Ω), it follows that Φn(H0 (Ω)) ⊂ Φn(H (Ω)). Then, version 2 of the Minimax Principle says that νn is given by minimizing the same expression as for µn over a larger space, and hence νn ≤ µn. Lemma. (Domain Monotonicity) If Ω1 ⊂ Ω2, then µn(Ω2) ≤ µn(Ω1). 1 1 1 1 Proof. Remark that H0 (Ω1) ⊂ H0 (Ω), so Φn(H0 (Ω1)) ⊂ Φn(H (Ω)). Again applying version 2 of the Minimax Principle, we conclude that µn(Ω2) ≤ µn(Ω1). Proof. (Of Weyl's Law) The proof is in 2 steps: first, we assume that the domain is a union of finitely-many cubes, and we know Weyl's law holds on each cube by our earlier considerations. Secondly, we approximate the domain by from the interior and from the exterior by unions of finitely-many cubes. Step 1: Assume the bounded domain Ω is a finite union of almost-disjoint cubes fQigi. Let fµk(Qi)gk;i and fνk(Qi)gk;i be the collections of all Dirichlet and Neumann eigenvalues on all of the cubes, respectively. Order these sets to get all of the Dirichlet eigenvalues fµ~kgk and all of the Neumann eigenvalues fν~kgk, i.e. so thatµ ~1 ≤ µ~2 ≤ ::: andν ~1 ≤ ν~2 ≤ :::. 4 MATT STEVENSON 1 2 1 1 2 1 Define Hf0 (Ω) = fu 2 L (Ω): ujQi 2 H0 (Qi) for all ig and Hf (Ω) = fu 2 L (Ω): ujQi 2 H (Qi) for all ig, then there is the chain of inclusions 1 1 1 ~1 Hf0 (Ω) ⊂ H0 (Ω) ⊂ H (Ω) ⊂ H (Ω): As in the Minimax Principles, there is a variational characterization of theµ ~k's andν ~k's as µ~k = inf sup ρ(u); ~1 X2Φk(H0 ) u2X and ν~k = inf sup ρ(u): ~1 e X2Φk(H ) u2X P 2 2 Here,ρ ~(u) = kruk 2 =kuk 2 is a modified Rayleigh quotient.
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