The conformal plate buckling equation

✍ Scribed by Sagun Chanillo; Michael K.-H. Kiessling

• Publisher: John Wiley and Sons
• Year: 2002
• Tongue: English
• Weight: 546 KB
• Volume: 55
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3010

✦ Synopsis

Abstract

The linear equation Δ^2^u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ℝ^2^ has the particularly interesting nonlinear generalization Δ~g~^2^u = 1, where Δ~g~ = e^−2__u__^ Δ is the Laplace‐Beltrami operator for the metric g = e^2__u__^g~0~, with g~0~ the standard Euclidean metric on ℝ^2^. This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δ__u__(x)+K~g~(x) exp(2__u__(x)) = 0 and Δ K~g~(x) + exp(2__u__(x)) = 0, with x ∈ ℝ^2^, describing a conformally flat surface with a Gauss curvature function K~g~ that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ K~g~ exp(2__u__)d__x__ = κ, finite area ∫ exp(2__u__)d__x__ = α, and the mild compactness condition K~+~ ∈ L^1^(B~1~(y)), uniformly w.r.t. y ∈ ℝ^2^. We show that asymptotically for |x|→∞ all solutions behave like u(x) = −(κ/2π)ln |x| + C + o(1) and K(x) = −(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and
$\alpha = \sqrt{2\kappa(4\pi - \kappa)}$. We also show that for each κ ∈ (2π, 4π) there exists a K^*^ and a radially symmetric solution pair u, K, satisfying K(u) = κ and max__K__ = K^*^, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc.