OpenAI September 24, 2026

Abstract (from Open-AI):

We prove the strict hot spots conjecture for smooth bounded simply connected planar domains. More precisely, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so all its global maxima and minima lie on the boundary. This holds even when the eigenvalue is multiple. ...

This work constructs a bounded simply connected noncircular domain $\Omega\subset\mathbb{R}^2$ with real-analytic Jordan boundary and a nonconstant function $u$ such that $$ (\Delta+k^2)u=0\quad\hbox{in }\Omega, \qquad u=1,\qquad \partial_\nu u=0 \quad\hbox{on }\partial\Omega $$ for some $k\in(31.967007261,31.967007293)$. Thus $u$ is a Neumann eigenfunction which is constant on the boundary, and $\Omega$ is a counterexample to Schiffer's conjecture. Green's identity gives $$ \widehat{\mathbf 1_\Omega}(k\omega)=0 \qquad(\omega\in\mathbb S^1), $$ so $\Omega$ fails the Pompeiu property and is also a counterexample to the planar Pompeiu conjecture for bounded simply connected Lipschitz domains. ...

We consider the minimization problem

\[ \min\left\{\int_{B_1}|Du|^2:\ u\in W^{1,2}(B_1;\mathbb R^n),\quad u=x\ \text{on }\partial B_1,\quad |u|\ge a\right\}, \quad 0< a< 1. \] ...

Two counterexamples have been published online on arXiv. Probably would need verification. ...

We first prove that the largest subsolution $u$ of the problem exists and is star-shaped with respect to all points in $D$. This in turn implies that the free boundary $\partial \{u >0\}$ is a locally Lipschitz graph. Classical results of free boundary are used to show that the free boundary is $C^{1,\alpha}$. From here we prove uniqueness by a scaling argument , a comparison principle and using the smoothness of the free boundary. We then use quasi-concave envelope $u^*$ of $u$ which is a subsolution of the problem but larger than our solution. This contradicts the fact that our solution is the largest one and we conclude that $u^*=u$. ...

In [1], the authors considered the periodic homogenization of second-order elliptic systems in divergence form with oscillating Dirichlet data or Neumann data of first order. They proved that the homogenized boundary data belongs to $W^{1,p}$ for any $1< p< \infty$. In particular, this implies that the boundary layer tails are Hölder continuous of order $\alpha$ for any $\alpha \in (0, 1)$. ...

Here we will show there exists a map $P_\Omega: W^{1,p}_0(D) \rightarrow W^{1,p}_0(\Omega)$ that satisfies the desired properties of the question and 'behaves' like a projector in the sense that $P^2_\Omega = P_\Omega$ and $P = I$ on $W^{1,p}_0(\Omega)$. Yet $P_\Omega$ is only a linear map if $p=2$.