A new approach to some nonlinear geometric equations in dimension two

Fengbo Hang; Xiaodong Wang
May 2006
Calculus of Variations & Partial Differential Equations;May2006, Vol. 26 Issue 1, p119
Academic Journal
We give new arguments for several Liouville type results related to the equation -?u = Ke2u. The new approach is based on the holomorphic function associated with any solution, which plays a similar role as the Hopf differential for harmonic maps from a surface.


Related Articles

  • A Geometric Problem and the Hopf Lemma. II. YanYan Li; Nirenberg, Louis // Chinese Annals of Mathematics;Apr2006, Vol. 27 Issue 2, p193 

    A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in Rn+1, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane Xn+1=constant in case M satisfies: for...

  • The geometric Neumann problem for the Liouville equation. Gálvez, José; Jiménez, Asun; Mira, Pablo // Calculus of Variations & Partial Differential Equations;Jul2012, Vol. 44 Issue 3/4, p577 

    Let Ω denote the upper half-plane $${\mathbb{R}_+^2}$$ or the upper half-disk $${D_{\varepsilon}^+\subset \mathbb{R}_+^2}$$ of center 0 and radius $${\varepsilon}$$. In this paper we classify the solutions $${v\in\;C^2(\overline{\Omega}\setminus\{0\})}$$ to the Neumann problemwhere $${K, c_1,...

  • Equiconvergence theorems in Sobolev and Hölder spaces of eigenfunction expansions for Sturm-Liouville operators with singular potentials. Sadovnichaya, I. V. // Doklady Mathematics;Apr2011, Vol. 83 Issue 2, p169 

    The article presents a study on the equiconvergence theorems in Sobolev and Hölder spaces of Eigenfunction expansions with fractional exponent for Sturm-Liouville operators with singular potentials. It mentions several studies regarding the equiconvergence of expansions including the rate of...

  • Entire spacelike hypersurfaces of prescribed scalar curvature in Minkowski space. Bayard, Pierre // Calculus of Variations & Partial Differential Equations;Jun2006, Vol. 26 Issue 2, p245 

    We prove existence and uniqueness of entire spacelike hypersurfaces in the Minkowski space with prescribed negative scalar curvature, and with given values at infinity which stay at a bounded distance of a lightcone.

  • A diagnostic procedure based on local influence. Hongtu Zhu; Heping Zhang // Biometrika;Sep2004, Vol. 91 Issue 3, p579 

    Cook's (1986) normal curvature measure is useful for sensitivity analysis of model assumptions in statistical models. However, there is no rigorous approach based on the normal curvature for addressing two fundamental issues: to assess the extent of discrepancy between an assumed model and the...

  • Ruled surfaces of Weingarten type in Minkowski 3-space. Dillen, Franki; Sodsiri, Wijarn // Journal of Geometry;2005, Vol. 83 Issue 1/2, p10 

    In this paper, we study ruled Weingarten surfaces M : x ( s, t) = α( s) + tβ ( s) in Minkowski 3-space on which there is a nontrivial functional relation between a pair of elements of the set { K, K II , H, H II }, where K is the Gaussian curvature, K II is the second Gaussian...

  • Ricci Curvature and Fundamental Group. Xin, Yuanlong // Chinese Annals of Mathematics;Apr2006, Vol. 27 Issue 2, p113 

    By refined volume estimates in terms of Ricci curvature, the two results due to J. Milnor (1968) are generalized.

  • A criterion for correct solvability in Lp(R) of a general Sturm-Liouville equation. N. Chernyavskaya; L. Shuster // Journal of the London Mathematical Society;Aug2009, Vol. 80 Issue 1, p99 

    We consider an equation where f ∈ Lp(ℝ) for p ∈ (1, ∞) with the following conditions: By a solution of the above-mentioned equations, we mean any function y that is...

  • Semiclassical approximation for a nonself-adjoint Sturm-Liouville problem with a parabolic potential. Pokotilo, V. I.; Shkalikov, A. A. // Mathematical Notes;Oct2009, Vol. 86 Issue 3/4, p442 

    The article presents a study which examines the nonself-adjoint Sturm-Liouville problem associated with the Orr-Sommerfeld operator. The researchers in the study provide proof and evidence of the theorem used in defining the problem. Results of the study and explanation of spectral curves and...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics