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In this paper, we derive the pointwise upper bounds and lower bounds on the gradients of solutions to the Lame systems with partially infinite coefficients as the surface of discontinuity of the coefficients of the system is located very close to the boundary. When the distance tends to zero, the optimal blow-up rates of the gradients are established for inclusions with arbitrary shapes and in all dimensions.

In this paper, we study a fully nonlinear inverse curvature flow in Euclidean space, and prove a non-collapsing property for this flow using maximum principle. Precisely, we show that upon some conditions on speed function, the curvature of the largest touching interior ball is bounded by a multiple of the speed.

In this paper, we consider the Dirichlet problem for hypersurfaces M = graph u of anisotropic prescribed mean curvature H = H ( x , u , N ) on unbounded domain Ω, where N is the unit normal to M at ( x , u ) . As a corollary of the result, we obtain the existence of translating solutions to the mean curvature flow with a forcing term on unbounded domains. The approach used here is a modified version of classical Perron's method, where the solutions to minimal surface equation are used as supersolutions and a family of auxiliary functions is constructed as local subsolutions.

In this paper, we use the Perron method to prove the existence of viscosity solutions to a class of Monge–Ampere equations on exterior domains in R n ( n ≥ 2 ) with prescribed asymptotic behavior at infinity. This extends the Caffarelli–Li’s result on det ( D 2 u ) = 1 . We also obtain the existence of entire convex viscosity solutions to the Monge–Ampere equations with prescribed asymptotic behavior at infinity.

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 use the Perron method to prove the existence of viscosity solutions to a class of Monge–Ampere equations on exterior domains in R n ( n ≥ 2 ) with prescribed asymptotic behavior at infinity. This problem comes from the study of Gauss curvature flow and its generalization, the flow by powers of Gauss 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 [1] is proved.