Lower bounds for lowest eigenvalues

Transcription

1 Lower bounds for lowest eigenvalues Peter Dole DRAFT Version 0.3A dated September 994 UNDER CONSTRUCTION GNU FDL Introduction In this abortive paper, we will discuss how to find lower bounds for lowest eigenvalues. The aim will be to get across the basic ideas, so we will discuss onl the simplest cases, and we won t worr about matters like exactl how man derivatives we are demanding of our functions. The work described here is joint work with Bob Brooks and Bob Kohn. Caveat. I seem to recall that I was changing the notation around when I turned to other things, so there ma be some inconsistencies. Also, ou ll note the the stor breaks off abruptl near the end. Continuous version Consider a nice bounded domain Ω in the Euclidean plane. A positive number λ is an eigenvalue of the positive Laplacian ( ) = x + Copright (C) 994 Peter G. Dole. Permission is granted to cop, distribute and/or modif this document under the terms of the GNU Free Documentation License, as published b the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.

2 if there exists a function u (other than the zero function) such that and u = λu u Ω = 0. We label the eigenvalues 0 < λ 0 < λ λ.... The lowest eigenvalue λ 0 is simple, and the corresponding eigenfunction u 0 does not change sign in the interior of Ω. B taking u 0 0 and u 0 =, we get a unique lowest eigenfunction u 0. The lowest eigenvalue λ 0 is the minimum of the Raleigh quotient: λ 0 = u min u = min u = u. These minima are attained when u = u 0, and since u 0 0 we can, if we wish, add the additional constraint u 0 to the classes of functions over which we take these minima. Theorem. λ 0 = sup(λ v, v v λ) = sup inf( v v ). v Proof. Suppose v v λ. Then λ u = = u ( v) u v u u v u v u u + uv u.

3 Thus On the other hand, if we put λ 0 sup(λ v, v v λ). v 0 = u 0 u 0 then and v 0 = u 0 + u 0 u 0 u 0 v 0 v 0 = u 0 u 0 = λ 0. Corollar. Let γ 0 = sup(γ w, w γ, w ). Then Let Proof. Suppose λ 0 γ 0 4. w γ, w. v = γ w. Then ) v v γ γ ( γ = γ 4. 3

4 Then Corollar (Cheeger s inequalit). Let χ 0 = min D Ω length( D). area(d) λ 0 χ 0 4. Proof. B the max-flow min-cut theorem, γ 0 = χ 0. Generalized continuous version Now introduce conductivit tensor σ and capacit function ρ: In this setting the Laplacian becomes and the eigenvalue equation becomes u = ρ (σ u) (σ u) = λρu. Now λ 0 = u (σ u) min ρu = min ρu = u (σ u) and our theorem becomes: Theorem. λ 0 = sup(λ v, v v σ v λρ) = sup inf( v ρ ( v v σ v)). 4

5 In this case, the choice of v that achieves the supremum is Discrete version v 0 = σ u 0 u 0. Let Ω = Ω intω be a graph with adjacenc matrix C(x, ) and valence function D(x) = C(x, ). A positive number λ is an eigenvalue of the discrete Laplacian, where if and u(x) = D(x) C(x, )(u(x) u()) u = λu u Ω = 0. We label the eigenvalues 0 < λ 0 < λ λ.... The lowest eigenvalue λ 0 is simple, and the corresponding eigenfunction u 0 does not change sign in the interior of Ω. B taking u 0 0 and x u 0 (x) =, we get a unique lowest eigenfunction u 0. The lowest eigenvalue λ 0 is the minimum of the Raleigh quotient: λ 0 = x, C(x, )(u(x) u()) min x D(x)u(x) = min x u(x) = x, C(x, )(u(x) u()). These minima are attained when u = u 0, and since u 0 0 we can, if we wish, add the additional constraint u 0 to the classes of functions over which we take these minima. 5

6 Theorem. λ 0 = sup{λ η : intω Ω [0, ), intω η(x,)η(,x) Ω C(x,)( η(x,)) λd(x) } = sup η H inf x intω C(x, )( η(x, )), D(x) Ω where H = {η, intω Ω [0, ) intω η(x, )η(, x) }. Proof. Suppose η H and for ever x intω, For an u with u Ω = 0, λ x u(x) x C(x, )( η(x, )) λ. u(x) C(x, )( η(x, )) = C(x, )u(x) C(x, )η(x, )u(x) x, = C(x, ) [ u(x) + u() η(x, )u(x) η(, x)u() ] x, C(x, ) [ u(x) + u() η(x, )u(x) ] x, η(x, ) u() = C(x, ) u(x) + u() η(x, )u(x) x, η(x, ) u() u(x)u() = C(x, ) (u(x) u()) η(x, )u(x) x, η(x, ) u() C(x, )(u(x) u()). x, (This isn t quite right, of course, since we ve been too cavalier about what happens when Ω, but not to worr... ) 6

7 On the other hand, if we set we have η 0 (x, ) = u 0() u 0 (x) C(x, )( η 0 (x, )) = ( C(x, ) u ) 0() u 0 (x) = u 0(x) u 0 (x) = λ 0. If we think of setting v(x, ) = C(x, )( η(x, )), we can get a form of this principle that is in a certain sense the discrete limit of the continuous variational principle: Corollar. λ 0 = sup{λ v : intω Ω R, v(x, ) C(x, ), Ω v(x,) λd(x) } v(x,) v(,x)+v(x,)v(,x)/c(x,) 0 Proof. Set η(x, ) = v(x, ) C(x, ). More later We ve run out of steam, and we haven t even covered the discrete Cheeger inequalit et.... 7