By M. Bocher

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2 √ In terms of the eccentricity ( 1 − 2 ) = e−2(t) ) we have λ = λ0 + 1 λ0 (λ0 − 2)( 32 4 + 6 ) + O( 8 ) as the eigenvalue near λ0 in the ellipse of area π and eccentricity . D. Joseph [13] computed the series for this eigenvalue but, after reduction to area π , his series has 32 λ0 − 5 in place of λ0 − 2 in the coefﬁcient of 6 (our series agrees to order 4 ). When λ0 is the principal eigenvalue, the coefﬁcent of 6 should be increased about 3%, which has only slight effect on the values in the table [13, p.

3. A Bifurcation Problem with Two-Dimensional Kernel. We seek small solutions u of u + f (x, u) + λu = α(x) + β(x)u in , u = 0 on ∂ where f (x, u) = O(u 2 ) as u → 0, f and λ are ﬁxed ( f is C 3 , λ is constant), and (α,√ β, ) are parameters near (0, 0, unit disc of R2 ). We suppose λ > 0 satisﬁes J1 ( λ) = 0, so the linearization at α = 0, β = 0, = D (= √ unit disc) has two-dimensional kernel with basis {φ1 , φ2 }, φ1 + iφ2 = J1 ( λr )eiθ in polar 2 coordinates. We impose a “non-degeneracy” condition on m(x) ≡ ∂∂u 2f (x, 0), satisﬁed for most choices of m, namely: For some complex constants A, B, C D m(x)(φ1 + iφ2 )(ξ1 φ1 + ξ2 φ2 )2 = Aξ12 + Bξ1 ξ2 + Cξ22 , ξ ∈ R2 and we require Aξ12 + Bξ1 ξ2 + Cξ22 = 0 whenever (ξ1 , ξ2 ) = (0, 0).

Suppose therefore (t) = h(t, ), and λ = λ(t), v = v(t, x) are smooth functions such that v + λv = 0 in v = 0 on ∂ (t), (t), with v(0, ·) ≡ 0, and let v˙ = ∂v/∂t; then ( ˙ = 0 in + λ)v˙ + λv v˙ + V · ∇v = 0 on ∂ (t) (t), where ∂h/∂t = V (t, h), h(0, ·) = i . When t = 0, v = v(0, ·) = some c1 , . . , cm , not all zero and m ˙ λ(0)c k = ˙ λ(0)φ kv = − cj j=1 ∂ V·N m 1 c j φ j for ∂φ j ∂φk · ∂N ∂N ∂φ k Let M 0jk = ∂ V · N ∂ Nj ∂φ at t = 0, c = col(c1 , . . , cm ) and it follows ∂N ˙λ(0)c + M 0 c = 0, c = 0, so λ(0) is an eigenvalue of −M 0 .

### An Introduction to the Study of Integral Equations by M. Bocher

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