论文标题
无限域中的签名本征函数的过度确定问题
An overdetermined problem for sign-changing eigenfunctions in unbounded domains
论文作者
论文摘要
我们研究了$ω\ subset \ mathbb {r}^2 $的非无限无界域的存在，其中方程\ begin {align} - λu_{xx} -u_ {xx} -u_ {tt}＆= u \ qfext {in $，} ω$，} \ nonumber \ end {align}是可解决的，可解决条件\ begin {align} \ frac {\ partial u} {\ partialη} = -1 \ quad \ quad \ quad \ text {on $ \ partialω^+$ $} u} {\partialη} =+1 \ quad \ text {on $ \ partialω^ - $。} \ end end {align}对于每个Integer $ m \ geq 0 $，我们证明存在一个无界域$ω\ subset \ subset \ subset \ subsec \ subset \ mathb {r $ n r $ n r $ n of $ 0 \ leqslant \ ell \ leqslant 2M $，上述问题承认定期签名解决方案。我们构建的域是在$ \ mathbb {r}^2 $中的第一个坐标中周期性的，并且它们会从合适的条上分叉。
We study the existence of non-trivial unbounded domains of $Ω\subset \mathbb{R}^2$ where the equation \begin{align} - λu_{xx} -u_{tt} &= u \qquad \text{in $Ω$,}\nonumber u &=0 \qquad \text{on $\partial Ω$,}\nonumber \end{align} is solvable subject to the conditions \begin{align} \frac{\partial u}{\partial η} =-1\quad \text{on $\partial Ω^+$} \quad \textrm{and}\quad \frac{\partial u}{\partial η} =+1\quad \text{on $\partial Ω^-$.} \end{align} For every integer $m\geq 0$, we prove the existence of a family of unbounded domains $Ω\subset \mathbb{R}^2$ indexed by $0 \leqslant\ell\leqslant 2m$, where the above problem admits periodic sign-changing solutions. The domains we construct are periodic in the first coordinate in $\mathbb{R}^2$, and they bifurcate from suitable strips.
