In this paper, we first investigate a new locally constrained mean curvature flow (1.5) and prove that if the initial hypersurface M0 is of smoothly compact starshaped, then the solution Mt of the flow (1.5) exists for all time and converges to a sphere in C ∞ -topology. Following this flow argument, not only do we achieve a new proof of the celebrated sharp Michael-Simon inequality for mean curvature in Euclidean space R n+1 , but we also get the necessary and sufficient condition for the establishment of the equality.In the second part of this paper, we study a mean curvature type flow (1.7) of static convex hypersurfaces in Euclidean space R n+1 , and prove that the flow (1.7) has a unique smooth solution Mt for all time t ∈ [0, +∞), and the static convexity of the hypersurface is preserved along the flow (1.7). Moreover, Mt converges exponentially to a sphere of radius R in C ∞ -topology as t → +∞. By exploiting the properties of this flow, we develop and present a new sharp Michael-Simon inequality for kth mean curvature.
In this paper, a Hessian type system is studied. After converting the existence of an entire solution to the existence of a fixed point of a continuous mapping, the existence of entire k-convex radial solutions is established by the monotone iterative method. Moreover, a nonexistence result is also obtained.
In this paper, the Dirichlet problem of Hessian type system is studied. After converting the existence of an entire solution to the existence of a fixed point of a continuous mapping, the existence of entire radial solutions is established by Schaefer fixed point theorem.
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