Conditional energy-level unique continuation hypothesis

PDF p. 5: boxed assumption (UCPw)

Assume the Axiom of Choice: every family of nonempty sets has a choice function. Let $T>0$, $n\ge 2$, and let $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$ be the connected exterior of a nonempty compact obstacle with smooth boundary. Let $\lambda \in \mathbb{R}\setminus \{0\}$ and fix a trajectory $w$ satisfying (W): $w\in C([0,T];H_D^2(\mathrm{\Omega })\cap W^{2,\mathrm{\infty }}(\mathrm{\Omega }))$ and $i{\mathrm{\partial }}_{t}w+\mathrm{\Delta }w+\lambda \mathrm{\mid }w{\mathrm{\mid }}^{2}w=0$ in distributions on $(0,T)\times \mathrm{\Omega }$. Here $H_D^2(\mathrm{\Omega })=H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$, and the path is continuous in the sum norm on the intersection.

Choose $R>0$ with $\mathrm{\Theta }\subset B_R(0)$ and set $\omega =\{x\in \mathrm{\Omega }:\mathrm{\mid }x\mathrm{\mid }>R\}$. Put $V_1=2\lambda \mathrm{\mid }w{\mathrm{\mid }}^2$ and $V_2=\lambda w^2$. A function $v\in C([0,T];H_0^1(\mathrm{\Omega }))$ is an energy-level solution of the adjoint equation if

$i{\mathrm{\partial }}_{t}v+\mathrm{\Delta }v+V_{1}v-V_2\bar{v}=0\text{in }{\mathcal{D}}^{\mathrm{\prime }}((0,T)\times \mathrm{\Omega })\mathrm{.}$

The space $H_0^1(\mathrm{\Omega })$ is the $H^1$-norm closure of smooth compactly supported functions; membership at every time imposes the homogeneous Dirichlet condition in this solution class. Continuity is in the $H_0^1$ norm. The distributional equation means equality on every smooth compactly supported space-time test function, with distributional derivatives.

The hypothesis (UCPw) for this fixed $w$, $T$ and $\omega$ is the following implication: every such energy-level solution $v$ that is zero almost everywhere on $(0,T)\times \omega$ is identically zero as an $H_0^1(\mathrm{\Omega })$-valued path on $[0,T]$.

Reference trajectory hypothesis W

Linearized equation and its real-duality adjoint

Control region and nested real cutoffs

Complex integer-order Sobolev notation

Continuous and mixed time-space function spaces

Distributional equation convention for complex Schrödinger solutions

The Axiom of Choice

The Sobolev space H¹ is a Hilbert space

Hilbert space

Given: AC, the fixed data in the Definition, the trajectory $w$, and a path $v\in C([0,T];H_0^1(\mathrm{\Omega }))$.

Goal: Justify the space-time interpretation and representative independence of the predicate, without asserting its truth for every trajectory.

1.1 F2 verifies that $H_0^1(\mathrm{\Omega })$, with the restricted $H^1$ norm, is a Banach target for the stated path space. Put $Q=(0,T)\times \mathrm{\Omega }$. By the Sobolev norm bound, $v$ is continuous as an $L^2(\mathrm{\Omega })$-valued path. It is bounded there and uniformly continuous by F2, so for each integer $m\ge 1$ choose a finite interval partition with left-endpoint sampled step path $s_m$ satisfying ${\sup }_{0\le t\le T}\mathrm{\parallel }{s}_m(t)-v(t){\mathrm{\parallel }}_2\le 2^{-m}$; assign $s_m(T)=v(T)$. Choose measurable spatial representatives of the countably many sampled classes using F3. They form a jointly measurable finite time-step function $U_m$ on $Q$ with spatial class $s_m(t)$. Tonelli and the norm inequalities give $\mathrm{\parallel }{U}_m{\mathrm{\parallel }}_{L^2(Q)}\le T^{1\mathrm{/}2}({\sup }_t\mathrm{\parallel }v(t){\mathrm{\parallel }}_2+2^{-m})$ and $\mathrm{\parallel }{U}_m-U_l{\mathrm{\parallel }}_{L^2(Q)}\le T^{1\mathrm{/}2}(2^{-m}+2^{-l})$. The factors are sigma-finite and the measure is the Euclidean cylinder measure by F4. [given, F2, F3, F4, F6, F7, choose, construct]

2.1 By F5 there is a measurable representative $U$ of the $L^2(Q)$ limit. Letting $l$ tend to infinity in the preceding norm bound gives $\mathrm{\parallel }{U}_m-U{\mathrm{\parallel }}_{L^2(Q)}\le T^{1\mathrm{/}2}{2}^{-m}$. Tonelli first in the time-space factors and then with counting measure gives ${\int }_0^T{\sum }_{m\ge 1}\mathrm{\parallel }{U}_m(t,\cdot )-U(t,\cdot ){\mathrm{\parallel }}_2^{2}dt\le T{\sum }_{m\ge 1}{4}^{-m}<\mathrm{\infty }$. All sections and section norms are taken outside the common null exceptional set; Tonelli for $\mathrm{\mid }U{\mathrm{\mid }}^2$ also makes $U(t,\cdot )$ an $L^2$ function almost everywhere in time. Thus $U_m(t,\cdot )\to U(t,\cdot )$ in $L^2(\mathrm{\Omega })$ for almost every $t$. Their classes $s_m(t)$ converge to $v(t)$ at every time, so definiteness and uniqueness of norm limits give $[U(t,\cdot )]=v(t)$ for almost every $t$. This constructs the required space-time representative with $U\in L^2(Q)$. [F3, F4, F5, F6, F7, step 1.1]

3.1 If $\tilde{U}$ is another jointly measurable function with spatial class $v(t)$ for almost every $t$, the indicator of $\{U\mathrm{\ne }\tilde{U}\}$ has spatial integral zero for almost every $t$. Tonelli makes that disagreement set product-null, and F4 makes it Euclidean null. Hence $U=\tilde{U}$ almost everywhere on $Q$. Its restriction to $(0,T)\times \omega$ has the same property, so the almost-everywhere vanishing condition depends only on the path. The construction for $w$ in F1 has the same independence. [F1, F4, F6, step 2.1]

4.1 Use the representative $W$ from F1 and set $P_1=2\lambda \mathrm{\mid }W{\mathrm{\mid }}^{2}U$ and $P_2=\lambda W^2\bar{U}$. These functions are jointly measurable and satisfy $\mathrm{\parallel }{P}_1{\mathrm{\parallel }}_{L^2(Q)}\le 2\mathrm{\mid }\lambda \mathrm{\mid }{M}^2\mathrm{\parallel }U{\mathrm{\parallel }}_{L^2(Q)}$ and $\mathrm{\parallel }{P}_2{\mathrm{\parallel }}_{L^2(Q)}\le \mathrm{\mid }\lambda \mathrm{\mid }{M}^2\mathrm{\parallel }U{\mathrm{\parallel }}_{L^2(Q)}$. Every compact $K\subset Q$ has finite measure; for $f=U,P_1,P_2$, Hölder gives ${\int }_K\mathrm{\mid }f\mathrm{\mid }\le \mathrm{\mid }K{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(Q)}<\mathrm{\infty }$. F8 therefore makes all three functions regular distributions. Their almost-everywhere classes, and hence these distributions, are independent of the representatives by step 3.1 and F1. F8 now defines $i{\mathrm{\partial }}_t{T}_U+{\mathrm{\Delta }}_x{T}_U+T_{P_1}-T_{P_2}=0$ as equality on every smooth compactly supported test function, exactly the adjoint equation of F1 in the stated solution class. [F1, F4, F7, F8, step 2.1, step 3.1, algebra]

5.1 The space-time equation and vanishing condition are thus well-defined properties of $v$. Its homogeneous Dirichlet convention is membership in the $H^1$ closure $H_0^1$ at each time, as specified in F2, without a pointwise boundary assertion. The conclusion is equality to the zero path on the closed interval, so includes the endpoints. These properties define the conditional implication for the fixed data and observation region in F1. The argument proves neither that (UCPw) holds for all trajectories satisfying (W) nor existence or uniqueness of adjoint solutions. [given, F1, F2, F8, step 3.1, step 4.1] ∎

The original article is cited only for attribution of (UCPw) and equation (1.11) on PDF page 5. The Definition is unchanged. The numbered Justification is a local argument using the named mathematical suppliers, not a proof attributed to the original authors. The current trajectory supplier already supplies the representative for $w$; the additional construction here supplies the energy-level path $v$. No new item, physical premise, publication stamp or independent review is proposed.

The original article is cited only for attribution of (UCPw) and equation (1.11) on PDF page 5. The Definition is unchanged. The numbered Justification is a local argument using the named mathematical suppliers, not a proof attributed to the original authors. The current trajectory supplier already supplies the representative for $w$; the additional construction here supplies the energy-level path $v$. No new item, physical premise, publication stamp or independent review is proposed.