【摘要】：解的凸性是偏微分方程和幾何分析研究中的一個(gè)重要課題,其主要研究方法分為宏觀方法和微觀方法.對于一般橢圓和拋物方程,我們自然地想研究其解的相關(guān)凸性,例如解的凸性和解的水平集的凸性.建立相應(yīng)的常秩定理通常是研究凸性的重要方法.本文針對一類橢圓偏微分方程解的微觀凸性給出一個(gè)常秩定理,本文的主要結(jié)果如下.定理.令Ω是具有常曲率(∈≥0)空間形式Mn中的一個(gè)光滑有界連通區(qū)域.令u ∈C4(Ω)∩C2((?))是平均曲率型方程的解,這里Ⅱ(x,u)≥ 0滿足結(jié)構(gòu)條件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,則u的水平集的第二基本形式在Ω的每一個(gè)點(diǎn)處一定有相同的秩.
[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，

本文編號：2503566