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In this paper, a generalization of the $L_{p}$-Christoffel-Minkowski problem is studied. We consider an anisotropic curvature flow and derive the long-time existence of the flow. Then under some initial data, we obtain the existence of smooth solutions to this problem for $c=1$., arXiv admin note: substantial text overlap with arXiv:2011.11408

In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean n-space. This flow involves k-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz-Christoffel-Minkowski problem., 19 pages. arXiv admin note: text overlap with arXiv:2005.02376

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 establish the pointwise upper and lower bounds of the gradients of solutions to a class of elliptic systems, including linear systems of elasticity, in a general narrow region and in all dimensions. This problem arises from the study of damage analysis of high-contrast composite materials. Our results show that the damage may initiate from the narrowest place., 21 pages, to appear in Quart. Appl. Math

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.

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.

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.