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 $p=2$.

This problem was solved in 2015 by Dipierro snd Karakhanyan. The paper is published in Advances of Mathematics, 2018, vol. 328, 40-81. The authors used a new dichotomy method, boundary Harnack inequalities, and stratification argument. Their proof applies to more general nonlinear problems. ...