Submanifolds that are level sets of solutions to a second-order elliptic PDE
Submanifolds that are level sets of solutions to a second-order elliptic PDE
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作为二阶椭圆偏微分方程解水平集的子流形
DOI：
10.1016/j.aim.2013.08.026
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
D. Peralta
中科院分区：
文献类型：
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作者：
A. Enciso;D. Peralta
研究了rn中的非紧超曲面可以是调和函数模C∞微分同态的正则水平集的刻画问题，以及对其他偏微分方程的一些推广。我们证明了一个通用的充分条件，该条件特别表明，任何连通分量都是非紧的非奇异代数超曲面都可以通过rn的微分同构变换到调和函数的零集的分量并上。我们使用的技术将鲁棒但不显式的局部构造与适当的全局逼近定理相结合。考虑到Berry和Dennis提出的问题的应用，我们还研究了水平集的交集。
We consider the problem of characterizing which noncompact hypersurfaces in R n can be regular level sets of a harmonic function modulo a C∞ diffeomorphism, as well as certain generalizations to other PDEs. We prove a versatile sufficient condition that shows, in particular, that any nonsingular algebraic hypersurface whose connected components are all noncompact can be transformed onto a union of components of the zero set of a harmonic function via a diffeomorphism of R n. The technique we use combines robust but not explicit local constructions with appropriate global approximation theorems. In view of applications to a problem posed by Berry and Dennis, intersections of level sets are also studied.