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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 . 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.
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.
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.
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.
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  is proved.
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