DCDS
Topological degree method for the rotationally symmetric $L_p$-Minkowski problem
Jian Lu Huaiyu Jian
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 971-980 doi: 10.3934/dcds.2016.36.971
Consider the existence of rotationally symmetric solutions to the $L_p$-Minkowski problem for $p=-n-1$. Recently a sufficient condition was obtained for the existence via the variational method and a blow-up analysis in [16]. In this paper we use a topological degree method to prove the same existence and show the result holds under a similar complementary sufficient condition. Moreover, by this degree method, we obtain the existence result in a perturbation case.
keywords: topological degree existence of solutions. Monge-Ampère equation Minkowski problem centro-affine Gauss curvature
CPAA
The regularity for a class of singular differential equations
Huaiyu Jian Xiaolin Liu Hongjie Ju
Communications on Pure & Applied Analysis 2013, 12(3): 1307-1319 doi: 10.3934/cpaa.2013.12.1307
We find an iteration technique and thus prove the optimal global regularity for the boundary value problem of a class of singular differential equations with strongly singular lower terms at the boundary. As applications, we obtain the regularity for the radial solutions of Ginzburg-Landau equations and harmonic maps.
keywords: Differential equantion regularity. singular
DCDS
Properties of translating solutions to mean curvature flow
Changfeng Gui Huaiyu Jian Hongjie Ju
Discrete & Continuous Dynamical Systems - A 2010, 28(2): 441-453 doi: 10.3934/dcds.2010.28.441
In this paper, we study the convexity, interior gradient estimate, Liouville type theorem and asymptotic behavior at infinity of translating solutions to mean curvature flow as well as the nonlinear flow by powers of the mean curvature.
keywords: asymptotic behavior Elliptic equation mean curvature flow convex solution gradient estimate.
CPAA
Traveling fronts of curve flow with external force field
Huaiyu Jian Hongjie Ju Wei Sun
Communications on Pure & Applied Analysis 2010, 9(4): 975-986 doi: 10.3934/cpaa.2010.9.975
This paper aims to classify all the traveling fronts of a curvature flow with external force fields in the two-dimensional Euclidean space, i.e., the curve is evolved by the sum of the curvature and an external force field. We show that any traveling front is either a line or Grim Reaper if the external force field is constant. However, we find that the traveling fronts are of completely different geometry for non-constant external force fields.
keywords: nonlinear ODE traveling front Curve flow phase analysis.
CPAA
Translating solutions to mean curvature flow with a forcing term in Minkowski space
Hongjie Ju Jian Lu Huaiyu Jian
Communications on Pure & Applied Analysis 2010, 9(4): 963-973 doi: 10.3934/cpaa.2010.9.963
We study the existence, uniqueness and asymptotic behavior of rotationally symmetric translating solutions to mean curvature flow with a forcing term in Minkowski space. As a result, a part of conjectures in [1] is proved.
keywords: elliptic equation asymptotic behavior. symmetric solution Spacelike hypersurface

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