The conformal plate buckling equation
Sagun Chanillo, Michael K.‐H. Kiessling · Communications on Pure and Applied Mathematics · 2001
Abstract The linear equation Δ2u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ℝ2 has the particularly interesting nonlinear generalization Δg2u = 1, where Δg = e−2u Δ is the Laplace‐Beltrami operator for the metric g = e2ug0, with g0 the standard Euclidean metric on ℝ2. This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+Kg(x) exp(2u(x)) = 0 and Δ Kg(x) + exp(2u(x)) = 0, with x ∈ ℝ2, describing a conformally flat surface with a Gauss curvature function Kg 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 ∫ Kg exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K+ ∈ L1(B1(y)), uniformly w.r.t. y ∈ ℝ2. 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 maxK = K*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc.