Schiffer猜想的反例
Counterexamples to Schiffer's Conjecture
AI总结：
作者在R²中构造无穷多个非球形N重对称平面区域，通过将N拓展为任意实数的松弛问题结合分支理论，否定了Schiffer猜想和Pompeiu问题
AI中文摘要：
Schiffer猜想指出，若欧氏空间Rⁿ中的光滑区域Ω存在拉普拉斯算子的 Neumann 本征函数，且该本征函数在边界上为常数，则该区域必为球。该猜想与Pompeiu问题密切相关，Pompeiu问题的结论是：若非零函数在Ω的任意刚体运动上的积分均为零，则Ω必为球。我们在R²中对这两个猜想均给出了否定答案，构造了无穷多个非球形的平面区域Ω，使其满足上述两个猜想的条件。我们构造的区域具有N重对称性，其中N为足够大的数。我们的方法基于一种新策略，即考虑一个松弛问题，其中N可以是任意实数（仅当N为自然数时，该松弛问题对应Schiffer问题）。随后我们将分支理论应用于该松弛问题，证明局部分支分支的大小可独立于N选取。该结果使我们能够得出结论：当N足够接近某个整数时，从该N出发的分支可达到整数N的取值。
英文摘要：
The Schiffer conjecture states that if a smooth domain $Ω\subset \mathbb{R}^n$ admits a Neumann eigenfunction of the Laplacian which is constant at the boundary, then the domain is a ball. It is intimately related to Pompeiu's problem, stating that if a nonzero function integrates zero over any rigid motion of $Ω$, then $Ω$ is a ball. We disprove both conjectures in $\mathbb{R}^2$, constructing infinitely many planar domains $Ω$ which are not balls and satisfy the conditions above. Our domains are $N$-fold symmetric, with $N$ sufficiently large. Our approach is based on a novel strategy of considering a relaxed problem where $N$ can be any real number (which corresponds to the Schiffer problem only when $N$ is a natural number). We then apply bifurcation theory to this relaxed problem, showing that the size of the local bifurcation branch can be taken independently of $N$. This result allows us to conclude that branches starting with $N$ sufficiently close to an integer reach integer values of $N$.
