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Low energy nodal solutions to the Yamabe equation
Journal of Differential Equations ( IF 2.4 ) Pub Date : 2020-05-01 , DOI: 10.1016/j.jde.2019.11.043 Juan Carlos Fernández , Jimmy Petean
Journal of Differential Equations ( IF 2.4 ) Pub Date : 2020-05-01 , DOI: 10.1016/j.jde.2019.11.043 Juan Carlos Fernández , Jimmy Petean
Given an isoparametric function $f$ on the $n$-dimensional sphere, we consider the space of functions $w\circ f$ to reduce the Yamabe equation on the round sphere into a singular ODE on $w$ in the interval $[0,\pi]$, of the form $w" + (h(r)/\sin r)w'+\lambda(\vert w\vert^{4/n-2}w - w)=0$, where $h$ is a monotone function with exactly one zero on $[0,\pi]$ and $\lambda>0$ is a constant. The natural boundary conditions in order to obtain smooth solutions are $w'(0)=0$ and $w'(\pi )=0$. We show that for any positive integer $k$ there exists a solution with exactly $k$-zeroes yielding solutions to the Yamabe equation with exactly $k$ connected isoparametric hypersurfaces as nodal set. The idea of the proof is to consider the initial value problems on both singularities $0$ and $\pi$, and then to solve the corresponding double shooting problem, matching the values of $w$ and $w'$ at the unique zero of $h$. In particular we obtain solutions with exactly one zero, providing solutions of the Yamabe equation with low energy, which can be computed easily by numerical methods.
中文翻译:
Yamabe 方程的低能节点解
然后解决相应的双重射击问题,匹配$w$和$w'$在$h$唯一零处的值。特别地,我们获得了恰好一个零的解,提供了低能量的 Yamabe 方程的解,可以通过数值方法轻松计算。
更新日期:2020-05-01
中文翻译:
Yamabe 方程的低能节点解
然后解决相应的双重射击问题,匹配$w$和$w'$在$h$唯一零处的值。特别地,我们获得了恰好一个零的解,提供了低能量的 Yamabe 方程的解,可以通过数值方法轻松计算。