Conformally invariant systems of nonlinear PDE of Liouville type

Chanillo, Sagun; Kiessling, Michael K.‐H.

doi:10.1007/bf01902215

Cited by 87 publications

(103 citation statements)

References 21 publications

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“…It is easy to show (see the Appendix) that there is no solution to (23) if λ ≤ 0, so λ > 0. On the other hand, due to the classification of all solutions of (23) (see, for example, [19,21] and [15]), we know that v is the function given in (10). It follows that…”

Section: Compactness and Existence For λ ∈ (−∞ 8π)mentioning

confidence: 99%

Harnack Type Inequality: the Method of Moving Planes

Li¹

1999

Communications in Mathematical Physics

273

271

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Abstract:A Harnack type inequality is established for solutions to some semilinear elliptic equations in dimension two. The result is motivated by our approach to the study of some semilinear elliptic equations on compact Riemannian manifolds, which originated from some Chern-Simons Higgs model and have been studied recently by various authors. IntroductionLet (M, g) be a compact Riemann surface without boundary, V be a positive function on M , W be a function with M W dv g = 1. Throughout the paper dv g denotes the volume element of g, g denotes the Laplace Beltrami operator with respect to g. For λ ∈ R, we seek a solution ofClearly M W dv g = 1 is a necessary condition for (E u ) λ to have a solution. If we set ξ = u − log M V e u dv g for a solution of (E u ) λ , then ξ satisfies

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Section: Compactness and Existence For λ ∈ (−∞ 8π)mentioning

confidence: 99%

Harnack Type Inequality: the Method of Moving Planes

Li¹

1999

Communications in Mathematical Physics

273

271

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“…The Bennett equations also constitute a Liouville system, but are not covered by the theorem of [10] because their coefficient matrix is generally not symmetric, has some negative elements, and is always rank 2. The present paper develops the necessary generalizations of [10] to overcome the first two peculiarities of the Bennett equations, but the rank 2 restricts the proof to a system of two equations. By adapting the treatment of single PDEs with more general nonlinearities developed in [36] (cf.…”

Section: Introductionmentioning

confidence: 99%

“…In this fully conformal case the conformal orbit of the finite current solutions is connected and itself invariant [29]. Invariance under the Euclidean conformal group holds also for the Liouville systems studied in [10], but their conformal orbit of finite mass solutions is generally not connected, and each component not invariant under inversions [15]. Toda systems in R 2 , which are Liouville systems with symmetric coefficient matrix given by the SU(N) Cartan matrix, are studied in [22,23].…”

Section: Introductionmentioning

confidence: 99%

“…The distribution of negative and positive signs in the SU(2) Cartan matrix is opposite to that in our Bennett equations, and sure enough, our radial symmetry proof fails in this case. Interestingly, in this case one can show that radial symmetry is in fact broken by some solutions, see the bifurcation argument with n = 2 in (1.7) of [10], and see [23] for the construction of the complete solution family with finite masses. Another interesting topic not discussed further here is whether the translation invariance along the beam can be broken, as is suggested by various dynamical beam instabilities [44].…”

Section: Introductionmentioning

confidence: 99%

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Symmetry Results for Finite-Temperature,¶Relativistic Thomas-Fermi Equations

Kiessling

2002

Communications in Mathematical Physics

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In the semi-classical limit the relativistic quantum mechanics of a stationary beam of counter-streaming (negatively charged) electrons and one species of positively charged ions is described by a nonlinear system of finite-temperature Thomas-Fermi equations. In the high temperature / low density limit these Thomas-Fermi equations reduce to the (semi-)conformal system of Bennett equations discussed earlier by Lebowitz and the author. With the help of a sharp isoperimetric inequality it is shown that any hypothetical particle density function which is not radially symmetric about and decreasing away from the beam's axis would violate the virial theorem. Hence, all beams have the symmetry of the circular cylinder. * In celebration of the 70 th birthday of Joel L. Lebowitz.

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“…For λ = 1 the fourth-order equation (1.1) is equivalent to the nonlinear system of second-order elliptic PDEs −∆u(x) = K(x)e 2u(x) , (1.4) −∆K(x) = e 2u(x) , (1.5) which describes a conformally flat surface over R 2 with metric g = e 2u g 0 and Gauss curvature function K ≡ K g generated in a self-consistent manner. While a considerably literature has accumulated about the celebrated prescribed Gauss curvature problem where K is given and only u has to be found by solving (1.4), see [2,3,5,6,8,10,11,12,13,14,15,16,17,23,26,27,30,31] and further references therein, the literature on selfconsistent Gauss curvature problems is relatively sparse [7,18,22,24]. In particular, we are not aware of any previous study of the self-consistent Gauss curvature problem (1.4), (1.5), equivalently the conformal plate buckling equation.…”

Section: Introductionmentioning

confidence: 99%

The conformal plate buckling equation

Chanillo¹,

Kiessling²

2001

Comm Pure Appl Math

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The linear equation ∆ 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 ∆ g 2 u = 1, where ∆ g = e −2u ∆ is the Laplace-Beltrami operator for the metric g = e 2u 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 secondorder ∆u(x) + K g (x) exp(2u(x)) = 0 and ∆K g (x) + exp(2u(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(2u) dx = κ, finite area exp(2u) dx = 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 α = 2κ(4π − κ). 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.

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Conformally invariant systems of nonlinear PDE of Liouville type

Abstract: We establish a strict isoperimetric inequality and a Pohozaev-Rellich identity for the system. . , N}, under certain reasonable conditions on the γ i,j and u i . Thus we prove that under these conditions, all solutions u i are radial symmetric and decreasing about some point.

Cited by 87 publications

References 21 publications

Harnack Type Inequality: the Method of Moving Planes

Harnack Type Inequality: the Method of Moving Planes

Symmetry Results for Finite-Temperature,¶Relativistic Thomas-Fermi Equations

The conformal plate buckling equation

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