Boundary value problems and Heisenberg uniqueness pairs

Abstract

We describe a general method for constructing Heisenberg uniqueness pairs $(\Gamma ,\Lambda )$ in the euclidean space $\mathbb {R}^{n}$ based on the study of boundary value problems for partial differential equations. As a result, we show, for instance, that any pair made of the boundary $\Gamma$ of a bounded convex set $\Omega$ and a sphere $\Lambda$ is an Heisenberg uniqueness pair if and only if the square of the radius of $\Lambda$ is not an eigenvalue of the Laplacian on $\Omega$. The main ingredients for the proofs are the Paley–Wiener theorem, the uniqueness of a solution to a homogeneous Dirichlet or initial boundary value problem, the continuity of single layer potentials, and some complex analysis in $\mathbb {C}^{n}$. Denjoy’s theorem on topological conjugacy of circle diffeomorphisms with irrational rotation numbers is also useful.

Appendix

We give a proof of Theorem 4.1 about the continuity of single layer potential with weakly singular kernels.

Proof of Theorem 4.1

First, when $x_{0}\not \in \Gamma$, $\Phi g$ is continuous at $x_{0}$ since

• the map $x\mapsto \varphi (x-y)g(y)$ is continuous at $x_{0}$, for all $y\in \Gamma$,

• $|\varphi (x-y)g(y)|\leqslant Mg(y)\in L^{p}(\Gamma )\subset L^{1}(\Gamma )$, where M is the sup of $|\varphi (x-y)|$ when x is in a neighborhood of $x_{0}$ that does not intersect $\Gamma$, and $y\in \Gamma$.

Second, assume $x_{0}\in \Gamma$. Let $\textrm{T}\Gamma _{x_{0}}$ and $n_{x_{0}}$ be the $(n-1)$-dimensional tangeant space to $\Gamma$ at $x_{0}$, and the normal unit vector to $\Gamma$ at $x_{0}$. We write $y=x_{0}+\tau _{y}+\eta _{y}n_{x_{0}}\in \Gamma$ with $\tau \in \textrm{T}\Gamma _{x_{0}}$, and we let $\eta =\gamma (\tau )$ be the equation of $\Gamma$ in a neighborhood of $x_{0}$. For some small enough $\delta >0$, let

$$\begin{aligned} \Gamma (x_{0},\delta )=\{y\in \Gamma ,~\tau _{y}\in B(x_{0},\delta )\}, \end{aligned}$$

where $B(x_{0},\delta )$ is the ball of radius $\delta$ in $\textrm{T}\Gamma _{x_{0}}$. Finally, for x sufficiently close to $x_{0}$, we write

$$\begin{aligned} x=x_{0}+\tau _{x}+\eta _{x}n_{x_{0}}. \end{aligned}$$

Then, we have

$$\begin{aligned} \Phi g(x)-\Phi g(x_{0})= & {} \int _{y\in \Gamma (x_{0},\delta )}\varphi (x-y)g(y)d\sigma (y) -\int _{y\in \Gamma (x_{0},\delta )}\varphi (x_{0}-y)g(y)d\sigma (y)\nonumber \\{} & {} +\int _{y\in \Gamma \setminus \Gamma (x_{0},\delta )}(\varphi (x-y)-\varphi (x_{0}-y))g(y)d\sigma (y). \end{aligned}$$

(5.1)

For the first two integrals, for x possibly equal to $x_{0}$, we have, by Hölder inequality, with q the exponent conjugate to p,

$$\begin{aligned}{} & {} \left| \int _{y\in \Gamma (x_{0},\delta )}\varphi (x-y)g(y)d\sigma (y)\right| \\{} & {} \quad \leqslant \left( \int _{y\in \Gamma (x_{0},\delta )}|\varphi (x-y)|^{q}d\sigma (y)\right) ^{1/q} \left( \int _{y\in \Gamma (x_{0},\delta )}|g(y)|^{p}d\sigma (y)\right) ^{1/p}. \end{aligned}$$

The last integral is finite since $g\in L^{p}_{d\sigma }(\Gamma )$. For the second one, note that

$$\begin{aligned} |\varphi (x-y)|\leqslant \frac{C}{|x-y|^{\nu }}\leqslant \frac{C}{|\tau _{x}-\tau _{y}|^{\nu }}. \end{aligned}$$

Moreover,

$$\begin{aligned} d\sigma (y)^{2}=d\tau ^{2}+d\eta ^{2}=(1+\gamma '(t)^{2})d\tau ^{2}\leqslant 4d\tau ^{2} \end{aligned}$$

for $\delta$ small (recall that $\gamma '$ is continuous and $\gamma '(0)=0$). Hence

$$\begin{aligned} \left| \int _{y\in \Gamma (x_{0},\delta )}|\varphi (x-y)|^{q}d\sigma (y)\right| \leqslant 2C\int _{B(x_{0},\delta )} \frac{d\tau }{|\tau _{x}-\tau |^{\nu q}}, \end{aligned}$$

(with $\tau _{x}=0$ if $x=x_{0}$) where the last integral over the $(n-1)$-dimensional ball $B(x_{0},\delta )$ is convergent since

$$\begin{aligned} 1+\nu /(n-1-\nu )<p \iff \nu q<n-1. \end{aligned}$$

Hence, by taking $\delta$ small, it can be made as small as we wish (uniformly in $\tau _{x}$).

It remains to show that the third integral in (5.1) can be made small. For x close to $x_{0}$ so that $x\notin \Gamma \setminus \Gamma (x_{0},\delta )$, the function

$$\begin{aligned} x\mapsto \int _{y\in \Gamma \setminus \Gamma (x_{0},\delta )}\varphi (x-y)g(y)d\sigma (y) \end{aligned}$$

is continuous at $x_{0}$ (same argument as in the first case), and thus, the third integral is small when x is sufficiently close to $x_{0}$. $\square$