On counterexamples to unique continuation for critically singular wave equations

Abstract

We consider wave equations with a critically singular potential $ξ\cdot σ^{-2}$ diverging as an inverse square at a hypersurface $σ= 0$. Our aim is to construct counterexamples to unique continuation from $σ= 0$ for this equation, provided there exists a family of null geodesics trapped near $σ= 0$. This extends the classical geometric optics construction of Alinhac-Baouendi (i) to linear differential operators with singular coefficients, and (ii) over non-small portions of $σ= 0$ - by showing that such counterexamples can be further continued as long as this null geodesic family remains trapped and regular. As an application to relativity and holography, we construct counterexamples to unique continuation from the conformal boundaries of asymptotically Anti-de Sitter spacetimes for some Klein-Gordon equations; this complements the unique continuation results of the second author with Chatzikaleas, Holzegel, and McGill and suggests a potential mechanism for counterexamples to the AdS/CFT correspondence.

Figures (5)

• Figure 1: Illustration of the setting of Theorem \ref{['Theorem']}, namely, of the geometry of $\Omega$ in our chosen $( \sigma, \bar{y}, s )$-coordinates. The eikonal function $\varphi$ defines null hypersurfaces (in $\textcolor{gray}{grey}$) that intersect the boundary $\{ 0 \} \times \mathcal{I}$. The gradient of $\varphi$ generates the null geodesic beams $\mathcal{N}$ (in red), along which our counterexample will propagate.
• Figure 2: These illustrations represent the change of coordinates from $(\rho, t, x)$ in (A) to $(\sigma, s, y)$ in (B). In (B), the gradient of $\varphi$ is aligned with the coordinate vector field $\partial_s$. Null geodesics in both graphics are drawn in red.
• Figure 3: (A) illustrates both the conformal boundary $\sigma = 0$ and the deformed boundary $\tilde{\sigma} = 0$ used to construct the localised counterexamples of Corollary \ref{['corollary_planar_2']}. (B) illustrates a family of null geodesics in pure AdS spacetime \ref{['aads_pure_conf']}, projected to the $\tau$-$\chi$-plane; in particular, these geodesics (in red) start from the conformal boundary, remain near the boundary, and return to the boundary after time $\pi$.
• Figure 4: (A) illustrates, at a fixed time $\tau$, the portion $\rho < \rho_0$ of $\mathcal{M}_{P, \epsilon}$ (shaded in blue), on which the restricted counterexample $u_\ast$ is defined. (B) shows the support of $u_\ast$ (shaded in blue), projected to the $\tau$-$\omega^d$-plane; since the support lies away from $\textcolor{red}{\omega^d = 0}$, then $u_\ast$ can be zero-extended to $\omega^d \geq 0$.
• Figure 5: Construction of the eikonal function $\textcolor{red}{\varphi}$ and the coordinate $\sigma$ in the general aAdS case. The choice of a foliation $x_1$ in $\mathcal{H}$, a spacelike section of $\mathcal{I}$, allows one to construct null hypersurfaces as level set of a smooth function $\varphi$ by extending along a family of null geodesics (black dotted curves). The level sets of $\sigma$ are obtained by extending the sets $\Sigma \cap \lbrace \rho = \rho_0 \rbrace$ along the same geodesics.

Theorems & Definitions (59)

• Definition 1.1
• Definition 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Corollary 1.6
• Corollary 1.7
• Definition 2.1
• Definition 2.2
• Proposition 2.3