一类平均曲率型方程的水平集的常秩定理
发布时间：2019-06-20 22:39
【摘要】：解的凸性是偏微分方程和几何分析研究中的一个重要课题,其主要研究方法分为宏观方法和微观方法.对于一般椭圆和抛物方程,我们自然地想研究其解的相关凸性,例如解的凸性和解的水平集的凸性.建立相应的常秩定理通常是研究凸性的重要方法.本文针对一类椭圆偏微分方程解的微观凸性给出一个常秩定理,本文的主要结果如下.定理.令Ω是具有常曲率(∈≥0)空间形式Mn中的一个光滑有界连通区域.令u ∈C4(Ω)∩C2((?))是平均曲率型方程的解,这里Ⅱ(x,u)≥ 0满足结构条件3HαHβ+ 4∈H2 δαβ ≤ 2HHαβ.如果|%絬| ≠ 0,在Ω中,u的所有水平集沿%絬方向是凸的,则u的水平集的第二基本形式在Ω的每一个点处一定有相同的秩.
[Abstract]:The convexity of solutions is an important topic in the study of partial differential equations and geometric analysis. For general elliptical and parabolic equations, we naturally want to study the relevant convexity of their solutions, such as the convexity of solutions and the convexity of level sets. It is usually an important method to study convexity to establish the corresponding constant rank theorem. In this paper, a constant rank theorem is given for the microconvexity of solutions of a class of elliptical partial differential equations. The main results of this paper are as follows. Theorem. Let 惟 be a smooth and bounded connected region in the form of space Mn with constant curvature (鈮，
本文编号：2503566
[Abstract]:The convexity of solutions is an important topic in the study of partial differential equations and geometric analysis. For general elliptical and parabolic equations, we naturally want to study the relevant convexity of their solutions, such as the convexity of solutions and the convexity of level sets. It is usually an important method to study convexity to establish the corresponding constant rank theorem. In this paper, a constant rank theorem is given for the microconvexity of solutions of a class of elliptical partial differential equations. The main results of this paper are as follows. Theorem. Let 惟 be a smooth and bounded connected region in the form of space Mn with constant curvature (鈮，
本文编号：2503566
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